On Smooth Surfaces of Degree ≥ 11 in the Projective Fourspace

Dissertation zur Erlangung des Grades

des Doktors der Naturwissenschaften der Mathematisch-Naturwissenschaftlichen Fakultät

der Universität des Saarlandes von

Sorin Popescu

Saarbrücken

Tag des Kolloquiums:  24.09.1993
Dekan:  Prof. Dr. Gerd Schmidt
Berichterstatter:  Prof. Dr. Wolfram Decker

Prof. Dr. Frank-Olaf Schreyer

Contents

0. Notations and basic results  1
1. The Eagon-Northcott complex method  9
2. A K3 surface of degree 10  18
3. Smooth surfaces of degree 11  25
I. Surfaces with d = 11, π = 8  25
II. Surfaces with d = 11, π = 9  28
III. Surfaces with d = 11, π = 10  30
IV. Surfaces with d = 11, π = 11  38
V. Surfaces with d = 11, π = 12  52
VI. Surfaces with d = 11, π = 13  66
VII. Surfaces with d = 11, π = 14  72
4. Examples of smooth surfaces of degree 12  77
5. Examples of Enriques surfaces of degree 13  84
6. A K3 surface of degree 14  87
7. Construction of smooth surfaces with d = 15, π = 21, χ = 0  95
A. Appendix  118

Introduction

Smooth varieties with small invariants have got renewed interest in recent years, primarily due to the finer study of the adjunction mapping by Reider, Sommese and Van de Ven [R], [So], [SV], [VdV]. For the special case of smooth surfaces in IP^{4} the method goes back to the italian geometers, who at the turn of the century used it for the study of the surfaces of degree less than 7, or sectional genus π ≤ 3. Later on, for bigger invariants, there are contributions by Commesatti and especially Roth. For example, in [Ro], Roth tried to establish a classification of smooth surfaces with π ≤ 6, but his lists are incomplete since he misses the non-special rational surfaces of degree 9 and the minimal bielliptic surfaces of degree 10. Nowadays, through the effort of several mathematicians (see references below), a complete classification of smooth surfaces in IP^{4} has been worked out up to degree 10.

But, apart the general framework of classification problems concerning codimension two varieties, there is also another strong motivation for the interest in these surfaces. Namely, in a recent paper Ellingsrud and Peskine [ElP1] proved the conjecture of Hartshorne that there are only finitely many families of special surfaces in IP^{4}. More specifically, given an integer a < 6, they show that the degree of smooth surfaces with K^{2} < aχ is bounded. In particular, there are only finitely many families of smooth surfaces in IP^{4}, not of general type. However, the question of an exact degree bound is still open. A recent work of Braun and Floystad [BF] improves the initial bound (∼ 10000) of Ellingsrud and Peskine to d ≤ 105, but it is believed that the degree of the smooth, non-general type surfaces in IP^{4} should be less or equal to 15. A similar finiteness result for 3-folds in IP^{5} was recently proved in [BOSS], but the real degree bound is supposed to be much higher in this case. Nevertheless, examples of smooth 3-folds in IP^{5} not of general type are known only up to degree 18.

Another reason for the interest in studying surfaces in IP^{4} originates in the small number of known liaison classes of such surfaces. On the other side, Proposition 0.32 below also accounts for such a spareness behavior, so each new specimen of liaison classes is of real interest. In this direction, the work of Decker, Ein and Schreyer [DES] provides a powerful and effective method of construction of surfaces in IP^{4}. Their method, whose basic idea is the application of Beilinson’s spectral sequence to construct the ideal sheaf I_{S} from the cohomology modules H^{1} and H^{2}, will be recalled in detail in chapter 2 below.

The aim of this paper is to provide various examples of smooth surfaces in IP^{4} of degree ≥ 10 and to describe their geometry. On one side we give an account of a rough attempt of classification of smooth surfaces of degree 11 in IP^{4} and provide constructions for 22 different families of surfaces in this degree, being able in some cases to show also uniqueness. Among the described families, 10 are of surfaces not of general type. Three of these examples were previously described in [DES]. On another side we provide a series of examples of smooth surfaces in IP^{4}, not of general type, in degrees varying from 12 up to 15. We’ve tried to work out examples of higher degrees but we failed in this attempt. The methods of construction we used are mainly the Eagon-Northcott approach of [DES] and liaison techniques. The most remarkable families we found are:

- minimal proper elliptic surfaces of degree 12 and sectional genus π = 13, - non-minimal K3 surfaces of degree 14 and sectional genus 19, and

- non-minimal abelian surfaces of degree 15, sectional genus 21, lying on only one quintic hypersurface and thus not coming via a (5, 5) liaison from the Horrocks-Mumford torus of degree 10. We remark here that the quintic elliptic scrolls, the bielliptic surfaces of degree 10 and 15, the minimal abelian surfaces of degree 10 and the non-minimal abelian surfaces of degree 15 arising of these via liaison are essentially the only other known smooth irregular surfaces in IP^{4} (up to taking the pullback through a finite morphism IP^{4} → IP^{4}).

At this point it may be appropriate to recall some references for the list of the smooth surfaces in degrees less or equal to 10 in IP^{4}. The classification and construction of surfaces of degree ≤ 7 is initiated in [Ro] and completed up to degree 8 in [Io1], [Io2], [Ok2], [Ok3], [Ok4], supplemented for the case of rational surfaces of degree 8, sectional genus 5 by [Al1]. In degree 9, the rational surfaces are described in [Al1] and [Al2], the Enriques surfaces with π = 6 in [Cos] and [CV], while the classification and description of the liaison classes is completed in [AR]. In degree 10, the classification in terms of numerical invariants and the description of most of the surfaces is achieved in the beautiful thesis of K. Ranestad [Ra1]. The existence and the uniqueness of bielliptic surfaces of degree 10, π = 6 was taken care by [Ser], the Enriques surfaces of degree 10, π = 8 were first constructed in [DES] and further studied in [Br], while the minimal abelian surfaces were first described about 20 years ago in [HM]. We provide here two constructions of a non-minimal K3 surface of degree 10, π = 9, lying on only one quartic hypersurface, and thus give a positive answer to the last existence case in [Ra1]. The left uniqueness problems, the syzygies and the description of the liaison classes in degree 10 are completed in [PR].

Acknowledgments

The work on this paper was initiated during a stay at the Universities of Kaiserslautern and Bayreuth. It has greatly benefited of the stimulating working atmosphere and the interesting seminars there. I would like also to thank the University of Kaiserslautern for financial support during the academic year 1990-91. Throughout, the work has benefited a lot from the conversations with A. Aure, W. Barth, I. Coanda, G. Ellingsrud, Ch. Peskine and K. Ranestad. Special thanks goes to my advisors W. Decker and F.-O. Schreyer whose support and advice guided me all along this work. I want also to thank W. Decker for his patience and his continuous encouragement which greatly helped me to overcome difficulties and to keep on track during the elaboration of the paper. Last but not least, I want to express my gratitude to C. Banica, who first introduced me to this field.

0. Notations and basic results

We will use standard notations as for instance those in [Ha], [BPV] or [GH]. The invariants of a smooth surface S in IP^{n} are denoted as follows: d = d(S) = H^{2} the degree of S, where H is the hyperplane class π = π(S) the genus of a generic hyperplane section p_{g} = p_{g}(S) = h^{0}(O_{S}(K)) = h^{2}(O_{S}) the geometric genus of S, where K is the canonical class q = q(S) = h^{1}(O_{S}) the irregularity of S χ = χ(S) = χ(O_{S}) the Euler characteristic of the surface s = h^{1}(O_{S}(1)) = h^{1}(O_{S}(H)) the speciality of S, see [Al1] κ(S) the Kodaira dimension of S p_{a}(C) the arithmetic genus of a curve C on S g(C) the geometric genus of a smooth curve C on S. In blowing-up situations we will denote in the same way a divisor downstairs and its total transform on the blow-up. Also a rational curve C with self-intersection C^{2} = −1 will be called a (-1) curve and analogously we’ll speak of a (-2) curve when C^{2} = −2. A (-1) curve will be called a (-1) line if it has degree 1 with respect to the current very ample linear system on the surface. For C a curve on S the adjunction gives a canonical divisor on the curve C:

K_{C} ≡ (C + K)|_{C}

hence the arithmetic genus of C can be computed by the following:

Adjunction formula 0.1. 2p_{a}(C) − 2 = C^{2} + C · K.

P roof. See [Ha Prop. 1.5].⊓ ⊔

For curves C, D and C ∪ D lying on a smooth surface S the formula yields the following addition rule for the arithmetic genus :

(0.2.)  p_{a}(C ∪ D) = p_{a}(C) + p_{a}(D) + C · D − 1.

Theorem (Riemann-Roch) 0.3.

χ(O_{S}(C)) = h^{0}(S, O_{S}(C)) − h^{1}(S, O_{S}(C)) + h^{0}(S, O_{S}(K − C)) = \frac{1}{2}(C^{2} − C · K) + χ(S).

P roof. See [Ha Th.1.6].⊓ ⊔

Hodge index theorem 0.4. If D is an effective divisor on S with D^{2} > 0 and C is a divisor such that D · C = 0, then C^{2} < 0 or C ≡ 0.

P roof. See [Ha Th.1.9].

The index theorem may be used to bound the self intersection number of a curve on S. Namely one has the following:

Corollary 0.5. Let D be an effective divisor with D^{2} > 0 on a smooth surface S. If C is a divisor on S then

C^{2} ≤ \frac{(D · C)^{2}}{D^{2}} .

P roof. Apply the index theorem to C − (\frac{D·C}{D^{2}} )D.⊓ ⊔

For smooth surfaces in IP^{4} with normal bundle N_{S} one obtains the equality:

(0.6.)  d^{2} − c_{2}(N_{S}) = d^{2} − 10d − 5H · K − 2K^{2} + 12χ(S) = 0,

which expresses the fact that S has no double points. This will be in the sequel referred to as the double point formula.

Theorem (Severi) 0.7. All smooth surfaces in IP^{4}, except for the Veronese surface, are linearly normal.

P roof. See [Sev] or [Mo].⊓ ⊔

(0.8.) Multisecants.(see [LB]) Some classical numerical formulas for multisecant lines to a smooth surface in IP^{4} have been recently studied again by Le Barz. Consider the double curve Γ of a general projection of such a surface S to IP^{3} and denote by

δ = \binom{d − 1}{2} − π

the degree of Γ, by

t = \binom{d − 1}{3} − π(d − 3) + 2χ − 2

the number of apparent triple points, i.e. the number of trisecants to S which meet a general point and by

h = \frac{1}{2}(δ(δ − d + 2) − 3t)

the number of apparent double points on Γ. The number of 4-secants to S which meet a general line (if finite) is :

(0.9.)  N_{4}(d, π, χ) = 2\binom{d}{4} + t(d − 3) + h − δ\binom{d − 3}{2}.

The number of 5-secants to S which meet a general plane (if finite) is:

(0.10.) N_{5}(d, π, χ) =\frac{1}{24}d(d − 3)(d − 4)(d^{2} − 15d + 2) − \binom{δ}{2}(d − 4)

− \frac{δ}{6}(d − 2)(d − 4)(d − 21) + h(d − 8) + δt − 3t(d − 3).

Suppose there are no lines on S with positive self-intersection. Then the number of 6secants (if finite) plus the number of exceptional lines on S is:

(0.11.)  N_{6}(d, π, χ) = − \frac{1}{144}d(d − 4)(d − 5)(d^{3} + 30d^{2} − 577d + 786)

+ δ(2\binom{d}{4} + 2\binom{d}{3} − 45\binom{d}{2} + 148d − 317)

− \frac{1}{2}\binom{δ}{2}(d^{2} − 27d + 120) − 2\binom{δ}{3}

+ h(δ − 8d + 56) + t(9d − 3δ − 28) + \binom{t}{2}.

(0.12.) Adjunction theory. In order to determine the possible numerical invariants of the various smooth surfaces in IP^{4} and to see how the constructed surfaces fit into the Enriques-Kodaira classification we will use adjunction theory [So], [VdV], [SV]:

Theorem (Adjunction mapping ) 0.13. Let S be a smooth surface, H be a very ample divisor and K the canonical divisor. Then | H + K | is non-special and has dimension N = π + χ − 2. Furthermore (I) | H +K |= ∅ iff S is either a scroll over a curve of genus q = π, or IP^{2} linearly embedded, or the Veronese surface. (II) If | H + K |≠ ∅, then | H + K | is base point free. In this case

a) (H + K)^{2} = 0 iff S is a Del Pezzo surface or a conic bundle,

b) If (H + K)^{2} > 0, then the map φ_{H+K} : S ⟶ S′ ⊂ IP^{N} is birational onto a smooth surface S′ and is the blowing down of the (-1) lines on S unless

1) S = IP^{2}(p_{1}, …, p_{7}) and H ≡ 6l − ∑_{i=1}^{7} 2E_{i} (the Geiser involution)

2) S = IP^{2}(p_{1}, …, p_{8}) and H ≡ 6l − ∑_{i=1}^{7} 2E_{i} − E_{8}

3) S = IP^{2}(p_{1}, …, p_{8}) and H ≡ 9l − ∑_{i=1}^{8} 3E_{i} (the Bertini involution )

4) S = P(E), where E is a rank 2 indecomposable bundle over an elliptic curve and H ≡ 3B where B is a section with B^{2} = 1 on S.

Related to the above theorem, the following two results impose restrictions on the invariants of a smooth surface in IP^{4}.

Theorem 0.14. [Au1][La]. The only smooth, non-degenerate scrolls in IP^{4} are the rational cubic and the elliptic quintic scrolls.

Theorem 0.15. [ES]. The only smooth conic bundles in IP^{4} are the rational Castelnuovo surfaces of degree 5 and the Del Pezzo surfaces of degree 4.

We derive some easy consequences which will be freely used in the text:

Lemma 0.16. Let S be a smooth surface with κ(S) ≥ 0 and let H be a very ample divisor. Then (H + K)K ≥ 0.

P roof. Clear since K is pseudo-effective.⊓ ⊔

Lemma 0.17. Let S be a smooth surface, let H be a very ample divisor on it and assume that π(S) ≠ q(S). Then | K − H |= ∅, when K^{2} < H^{2}.

P roof. In the above hypothesis | K + H | is nef.⊓ ⊔

Lemma 0.18. Let S be a smooth surface and let H be a very ample divisor on it. Then (HK)^{2} < H^{2} implies that κ(S) ≤ 1.

P roof. We observe first that the above numerical condition implies, via the Hodge index theorem, that K^{2} ≤ 0. Assume now that S would be a surface of general type and consider the adjunction morphism

φ_{H+K} : S ⟶ S_{1}↪^{H_{1}}IP^{N},

which blows down the (−1) lines on S. By using deg S_{1} = H_{1}^{2} = (H + K)^{2} and H_{1}K_{1} = (H + K)K we obtain

H_{1}^{2} = (H^{2} + 2HK + K^{2}) > (HK)^{2} + 2(HK)K^{2} + (K^{2})^{2} = (HK + K^{2})^{2} = (H_{1}K_{1})^{2}

since 2HK + K^{2} = (2H + K)K ≥ 0, K^{2} < 1, and by assumption (HK)^{2} < H^{2}. This means that the numerical hypothesis is preserved through the adjunction process, hence it will hold also on the minimal model S_{min} of S, where the adjunction stops. Therefore we would obtain K_{min}^{2} ≤ 0 which is impossible for a minimal surface of general type. ⊓ ⊔

The same arguments prove also the following:

Remark 0.19. Let S be a smooth surface of general type and let H be a very ample divisor. If (HK)^{2} < pH^{2} for some p ∈ IN^{∗}, then K_{min}^{2} < p.⊓ ⊔

Lemma 0.20. Let S be a smooth non-degenerate surface in IP^{4} with κ(S) = −∞. Then either

a) S is a scroll (d = 3, 5), or

b) S is a conic bundle (d = 4, 5), or

c) S is a Veronese surface (d = 4), or

d) the following inequalities hold

(0.21.)  5χ ≥ 2π − \frac{d^{2} − 7d + 8}{2}

and

(0.22.)  7χ ≥ 2π − \frac{d^{2} − 7d + 4}{2}

Moreover, when the last inequality is strict then also

(0.23.)  26χ ≥ 7π − \frac{3d^{2} + 21d − 14}{2}

P roof. The above inequalities express the fact that the image of S through the adjunction mapping S_{1} = φ_{H+K}(S) ⊂ IP^{π+χ−2} is a non-degenerate surface which has sectional genus π_{1} ≥ q_{1}, and that the adjoint linear system on S_{1} is globally generated when π_{1} ≠ q_{1}.⊓ ⊔

(0.24.) Liaison.(see [PS]) Two surfaces S and S′ in IP^{4} are said to be (geometrically) linked (m, n) if there exist hypersurfaces V and V′ of degree n and respectively m such that V ∩ V′ = S ∪ S′. There are two standard exact sequences of linkage, namely:

0 ⟶ O_{S}(K) ⟶ O_{S∪S′}(m + n − 5) ⟶ O_{S′}(m + n − 5) ⟶ 0
0 ⟶ O_{S}(K) ⟶ O_{S}(m + n − 5) ⟶ O_{S∩S′}(m + n − 5) ⟶ 0.

The first sequence yields the relation between the Euler-Poincaré characteristics:

(0.25.)  χ(S′) = χ(V ∩ V′) − χ(O_{S}(m + n − 5)).

The corresponding sequence for linkage of curves in IP^{3} yields the following relation between the sectional genera:

(0.26.)  π(S) − π(S′) = \frac{1}{2}(m + n − 4)(d(S) − d(S′)).

In terms of syzygies if

0 ⟶ E_{1} ⟶ E_{0} ⟶ I_{S} ⟶ 0

is a resolution with locally free sheaves of the ideal sheaf I_{S} of S, then a mapping cone produces a locally free resolution

0 ⟶ E0^{∨}(−m − n) ⟶ E1^{∨}(−m − n) ⊕ O(−m) ⊕ O(−n) ⟶ I_{S′} ⟶ 0

for the ideal sheaf of the linked surface. In particular we obtain the following isomorphisms between the Hartshorne-Rao modules

(0.27.)  M^{3−i}(S) ≅ M^{i}(S′)^{∗}(5 − m − n)  i = 1, 2

We recall also the relation between the homogeneous ideals of the linked surfaces

(0.28.)  I_{S′} = (I_{V ∩V′} : I_{S})

For a proof of existence via linkage, the following propositions will be used:

Proposition 0.29. If S and T are linked, then S is locally Cohen-Macaulay if and only if T is locally Cohen-Macaulay.

See [PS, Proposition 1.3] for a proof.

Proposition 0.30. If Z is a local complete intersection surface in IP^{4}, scheme theoretically cut out by hypersurfaces of degree d, then Z can be linked to a smooth surface S in the complete intersection of two hypersurfaces of degree d.

For a proof see [PS, Proposition 4.1].

Remark 0.31. (Peskine, cf. [Ra1]) The above proposition remains true if Z is not a local complete intersection in at most a finite set of points but it is locally Cohen-Macaulay and the tangent cone at each of these points is linked to a plane in a complete intersection.

In order to determine the minimal elements of an even liaison class we will need the following version of a lemma from [LR]:

Lemma 0.32. Let Z be a codimension two, locally Cohen-Macaulay subscheme of IP^{n} and define the speciality of Z as e(Z):=max { t | h^{n−2}(O_{Z}(t)) ≠ 0 }.

a) If h^{0}(I_{Z}(e(Z) + n)) = 0, then Z is a minimal element in its even liaison class.

b) If moreover h^{0}(I_{Z}(e(Z) + n + 1)) = 0, then Z is the unique minimal element in its even liaison class.

The proof in [LR] can be easily adapted to the general case.

(0.33.) Reducible curves. One method to work out the classification of the various studied surfaces will be to find special reducible hyperplane sections on them. For the discussion of the components with an arithmetic genus too high in comparison with their degree we use the following

Lemma 0.34. Let C be a curve of degree d and arithmetic genus p on a smooth surface S in IP^{4}. One has the following possibilities:

a) If d ≤ 3, then p ≤ 1 with equality iff C is a plane cubic curve

b) If d = 4, then p ≤ 1 or p = 3 and C is a plane quartic curve

c) If d = 5, then p ≤ 3 or p = 6 and C is a plane quintic curve

d) If d = 6, then p ≤ 6 or p = 10 and C is a plane sextic curve. Moreover, if p = 5 (resp. p = 6) then C decomposes into a plane quintic curve and a line which don’t meet (resp. meet in a point)

e) If d = 7, then p ≤ 6 unless C is a plane septic curve, or C decomposes into a plane sextic and a line which meet in a point (p = 10), or which don’t meet (p = 9), or C decomposes into a plane quintic curve and a plane conic which meet along a subscheme of length two (p = 7)

f) If d = 8, then p ≤ 9 unless C is a plane curve, or C decomposes into a plane septic and a line which meet in a point (p = 15), or which don’t meet (p = 14), or C decomposes into a plane sextic curve and a plane conic which meet along a subscheme of length two (p = 11), or in a point (p = 10), or C decomposes into a plane sextic curve and two skew lines meeting the sextic in points (p = 10).

P roof. See [Ra1]. It is straightforward using Castelnuovo’s bound for the genus of an irreducible curve and formula (0.2). For the last case one can use also the results of [GLP] relating the regularity of the ideal of an irreducible curve with the existence of higher order secant lines.⊓ ⊔

Lemma 0.35. Let S ⊂ IP^{4} be a smooth non-degenerate surface and let π be a plane in IP^{4} cutting S along a curve C. Then C has no multiple components.

P roof. Assume that C would have a multiple component D, i.e. C = 2D + E with E effective and let d = deg D. Then adjunction gives

D^{2} + DK = (d − 1)(d − 2) − 2 = d^{2} − 3d

and

2D(2D + K) = (2d − 1)(2d − 2) − 2.

It follows that D^{2} = d which contradicts Hodge index

D^{2} ≤ DH/deg S < d.  ⊓ ⊔

Lemma 0.36.[PR]. Assume now that S ⊂ IP^{4} contains a plane curve of degree d_{p}. Then a) h^{1}(S, O_{S}(1)) ≥ \frac{1}{2}(d_{p} − 2)(d_{p} − 3) if p_{g} = 0, while

b) h^{1}(S, O_{S}(1)) ≥ \frac{1}{2}(d_{p} − 2)(d_{p} − 3) + 1 − p_{g} if p_{g} ≥ 1.

P roof. Let C be the plane curve on S, let D = H − C and consider the cohomology of the exact sequence

0 ⟶ O_{S}(D) ⟶ O_{S}(H) ⟶ O_{C}(H) ⟶ 0

Then h^{2}(S, O_{S}(D)) = 0 if p_{g} = 0, hence h^{1}(S, O_{S}(H)) ≥ h^{1}(C, O_{C}(H)). In general h^{2}(S, O_{S}(D)) ≤ p_{g} − 1 when p_{g} ≥ 1, so h^{1}(S, O_{S}(H)) ≥ h^{1}(C, O_{C}(H)) − p_{g} + 1.⊓ ⊔

The previous bounds will also be used in connection with the following:

Lemma 0.37.[Bo]. Let C be an effective Cartier divisor on a smooth surface S and let L be a line bundle on C with h^{0}(C, L) ≠ 0. Then there exists a decomposition C = C_{1} + C_{2}, C_{1} ≥ 0, C_{2} > 0 such that

C_{1}C_{2} ≤ deg_{C_{2}}(L_{|C_{2}}).

The following lemmas will be used in the text to produce reducible curves on the studied surfaces.

Lemma 0.38.[Ra2]. Let S be a smooth, (proper) elliptic surface in IP^{4} and let S_{min} be its minimal model. We denote also by m = min{−K^{2}, p_{g} − 1}. a) If p_{g} = 1, then HK + K^{2} ≥ 3.

b) If p_{g} ≥ 2, then

\frac{HK + K^{2} − m}{p_{g} − 1} ≥ 3.

P roof. See [Ra2]. One uses Kodaira’s formula [BPV] for the canonical class of an elliptic fibration and the fact that an elliptic curve has degree at least 3. ⊓ ⊔

Analogously, for general-type surfaces one has

Lemma 0.39.[Ra2]. Let S be a smooth general type surface in IP^{4} with minimal model S_{min}. We denote by K_{min} the total transform on S of the canonical class on S_{min} and by m = min{K_{min}^{2} − K^{2}, p_{g} − 1}. Then there exists a curve D on S of degree

degD ≤ HK + K^{2} − K_{min}^{2} − m

and arithmetic genus

p_{a}(D) = K_{min}^{2} + 1.

(0.40.) Postulation of points. When identifying 6-secant lines to surfaces in IP^{4} we’ll look to their plane sections through such lines and we’ll make use of the following result from [Ra1].

Lemma 0.41. Let π : S → IP^{2} be the morphism obtained by blowing up t points (some possibly infinitely close) in IP^{2}, where 9 ≤ t ≤ 12. Denote the exceptional divisors by E_{1}, ..., E_{t} and consider the linear system

|C| = |4π^{∗}l − ∑_{i=1}^{t} E_{i}|

on S. If dim|C| ≥ 15 − t and |C| has a fixed curve, then there is a curve Γ ≡ π^{∗}l − ∑_{k=1}^{6} E_{i_{k}} or

Γ ≡ 2π^{∗}l − ∑_{k=1}^{10} E_{i_{k}} on S, which is part of the fixed curve of |C|. If dim|C| ≥ 15 − t and |C| has no fixed curve, then t = 12, dim|C| = 3 and |C| has no base

points. Furthermore there is a curve Γ ≡ 3π^{∗}l − ∑_{i=1}^{12} E_{i} on S. If t = 12, dim|C| = 2 and |C| has a fixed curve, then there is a curve Γ ≡ π^{∗}l − ∑_{k=1}^{5} E_{i_{k}}

or Γ ≡ 2π^{∗}l − ∑_{k=1}^{9} E_{i_{k}} or Γ ≡ 3π^{∗}l − ∑_{i=1}^{12} E_{i} on S, which is part of the fixed curve of |C|.

If t = 12, dim|C| = 2 and |C| has no fixed curve, then |C| has at the most one basepoint.

For a proof see [Ra1], (t = 12), and [PR]. The above lemma is, in a certain sense, a special case of the general result in [EP].

1. The Eagon-Northcott complex method

We recall in this chapter a powerful method of construction for surfaces in IP^{4} which was first introduced in [DES]. The rough idea is to realize the surface one wishes to construct as the determinantal locus of a morphism between two vector bundles on IP^{4}. Namely, let E and F be vector bundles on IP^{4} of ranks rkE = r , rkF = r + 1 and, for φ ∈ Hom(E, F) an injective morphism, consider the degeneracy locus D(φ) which is the subscheme given locally by the r-minors of the matrices associated to φ. As a set

D(φ) = { x ∈ IP^{4} | rk φ(x) ≤ r − 1 }.

General facts, see [ACGH], implies that D(φ), when non-empty, has codimension ≥ 2. Moreover, when it has the expected codimension, i.e. equality holds, D(φ) is a locally Cohen-Macaulay surface. Given a morphism φ, one constructs the so called Eagon-Northcott complex:

∧^{r+1}F^{∗} ⊗ ∧^{r}E ⟶^{∧^{r}φ^{∗}(c_{1}(E)−c_{1}(F))} F^{∗}  ⟶^{φ^{∗}} E^{∗} ↓^{id⊗∧^{r}φ}  ↑^{id} ∧^{r+1}F^{∗} ⊗ ∧^{r}F ⟶^{contraction} F^{∗} .

Proposition 1.1. In the above notations, the following statements are equivalent:

a) coker φ is an ideal sheaf.

b) coker φ is either the ideal sheaf of a pure codimension 2 subscheme or O_{IP}^{4}.

c) The Eagon-Northcott complex

0 ⟶ E ⟶^{φ} F ⟶^{∧^{r}φ(c_{1}(F)−c_{1}(E))} O_{IP}^{4}(c_{1}(F) − c_{1}(E))

is exact.

P roof. See [ACGH].⊓ ⊔

For later reference, we state here two well known corollaries of the above proposition:

Corollary 1.2. Let S be a reduced scheme, let E and F be vector bundles on IP^{4} × S of ranks r and respectively r + 1 and let φ : E ⟶ F be an injective morphism. Let also Z = D(φ) and suppose that Z dominates S. Then the set of points where Z_{s} ⊂ IP^{4} × {s} has pure codimension two is open and Z is flat over S above it.

Corollary 1.3. Let E and F be two vector bundles on IP^{4} of ranks r and respectively r + 1 and let φ_{1} and φ_{2} be two morphisms between E and F. Assume also that both determinantal loci S_{i} = D(φ_{i}), i = \overline{1, 2} have the expected codimension two. Then S_{1} and S_{2} lie in the same irreducible component of the Hilbert scheme.

For a proof of both corollaries see for example [BB] or [MDP].

The aim of the method is to realize the desired surface S as a, hopefully smooth, degeneration locus of a morphism φ between suitable chosen vector bundles E and F. In this case, the Eagon-Northcott complex provides a vector bundle resolution for the ideal sheaf of the surface and we can relate the invariants of the surface S with those of E and F. Normalizing, it is is enough to deal with the case of an exact sequence of type:

0 ⟶ E ⟶ F ⟶ I_{S} ⟶ 0

where E, F are as above and S is a surface in IP^{4}. Riemann-Roch without denominators [Fu], gives

1 − i_{∗}(c(N_{S|IP^{4}}^{∗})^{−1}) = c(E − F)

where i : S ↪ IP^{4} is the inclusion and E − F means difference in the Grothendieck group. The homogeneous parts yields the following relations: in degree 2, Porteus´ formula

degS = d = c_{2}(F) − c_{2}(E)

in degree 3

i_{∗}(c_{1}(N_{S|IP^{4}})) = c_{3}(F) − c_{3}(E) − c_{1}(c_{2}(F) − c_{2}(E))  (∗)

in degree 4

i_{∗}(c_{1}(N_{S|IP^{4}})^{2}) = c_{4}(F) − c_{4}(E) − c_{1}(c_{3}(F) − c_{3}(E)) + (c_{1}^{2} − c_{2}(E))(c_{2}(F) − c_{2}(E)) (∗∗)

where c_{1} = c_{1}(E) = c_{1}(F). Assume now that S is smooth. Then adjunction and the self-intersection formula [Ha] give c_{1}(N_{S|IP^{4}}) = 5H + K and c_{2}(N_{S|IP^{4}}) = d^{2}. Substituting in (∗) and (∗∗) one obtains for the sectional genus

2π − 2 = c_{3}(F) − c_{3}(E) − d(4 + c_{1})

and for the self-intersection of the canonical class

K^{2} = c_{4}(F) − c_{4}(E) − (10 + c_{1})(c_{3}(F) − c_{3}(E)) + (c_{1}^{2} − c_{2}(E))(c_{2}(F) − c_{2}(E)) + (25 + 10c_{1})d.

Finally the double point formula, for instance, allows to determine the Euler characteristic χ(O_{S}).

To construct a surface S with the desired invariants we’ll have to find appropriate vector bundles E and F. A general way to construct vector bundles (or more generally coherent sheaves) is to determine first the differentials of the Beilinson’s spectral sequence:

Theorem 1.4.[Bei]. Let G be a coherent sheaf on IP^{n} = P(V). There exists a spectral sequence with E_{1} terms

E_{1}^{pq} = H^{q}(IP^{n}, G(p)) ⊗ Ω_{IP^{n}}^{−p}(−p)

converging to G; i.e. E_{∞}^{pq} = 0 for p + q ≠ 0 and ⊕E_{∞}^{p,−p} is the associated graded sheaf of a suitable filtration of G.

All the E_{1}-terms are in the 2^{nd} quadrant and only finitely many of them are non-zero. Via the canonical isomorphisms induced by contraction, Hom(Ω_{IP^{n}}^{i}(i), Ω_{IP^{n}}^{j}(j)) ≅ Λ^{i−j}V, for i ≥ j, cf. [Bei], the d_{1}-differentials

d_{1}^{pq} ∈ Hom(H^{q}(IP^{n}, G(p)) ⊗ Ω_{IP^{n}}^{−p}(−p), H^{q}(IP^{n}, G(p + 1)) ⊗ Ω_{IP^{n}}^{−p−1}(−p − 1))

can be identified with the natural multiplication maps in

Hom(H^{0}(IP^{n}, O_{IP}^{n}(1)) ⊗ H^{q}(IP^{n}, G(p)), H^{q}(IP^{n}, G(p + 1)).

This also means that to determine the d_{1}-differentials is equivalent to fixing the module structure of ⊕_{p}H^{q}(IP^{n}, G(p)). The higher order differentials d_{r} are induced by maps E_{1}^{pq} ⟶ E_{1}^{p+r,q−r+1}. These maps are not canonically given.

The method is to apply Beilinson’s spectral sequence to the twisted ideal sheaf I_{S}(m) for m = 3 or 4 and to interpret part of the spectral sequence as the spectral sequence for a vector bundle E and the other part as that for a vector bundle F. The differentials will induce a morphism

φ : E ⟶ F

whose degeneracy locus D(φ) will be the desired surface. The E_{1}-terms of the spectral sequence are determined by the dimensions h^{i}(I_{S}(p)) for i = \overline{0, 4}, p = \overline{m − 4, m}. Riemann-Roch gives

Lemma 1.5. Let S be a smooth surface in IP^{4}. Then

χ(I_{S}(p)) = \binom{p + 4}{4} − \binom{p + 1}{2}d + p(π − 1) − χ.

We observe also that some of the cohomology groups vanish :

–  h^{4}(I_{S}(p)) = h^{4}(O_{IP}^{4}(p)) = 0  for all  p ≥ −4;
–  h^{3}(I_{S}(p)) = h^{2}(O_{S}(p)) = h^{0}(O_{S}(K − pH)) = 0  for all  p ≥ 1,

when, for example, either S is not of general type or S is a general type surface with p_{g} ≤ 4, or S is a general type surface with K^{2} < HK;

– h^{2}(I_{S}(p)) = h^{1}(O_{S}(p)) = h^{1}(O_{S}(K − pH)) = 0 for all p ≤ −1

by Kodaira’s vanishing theorem;

– h^{1}(I_{S}(p)) = 0 for all p ≤ 0;
–  h^{1}(I_{S}(1)) = 0 ,

by (0.7), unless if S is the Veronese surface;

–  h^{0}(I_{S}(1)) = 0 ,

unless if S is degenerate, i.e. lies in a hyperplane.

In particular the speciality, cf. [Al1], of a smooth non-degenerate, non-general type surface different from the Veronese surface is given by Riemann-Roch

s = h^{1}(O_{S}(1)) = χ(I_{S}(1)) = π − d + 4 − χ

Related to the postulation of a surface in IP^{4} we mention the following results:

Theorem 1.6.[Ro][Au1]. Let S be a smooth non-degenerate surface in IP^{4} which is contained in a hyperquadric. Then

π = 1 + [\frac{d(d − 4)}{4}]

and S is either the complete intersection of the hyperquadric with another hypersurface, or S is linked to a plane in the complete intersection of the hyperquadric with another hypersurface.

Theorem 1.7.[Au1][Ko]. Let S be smooth surface in IP^{4} which is contained in an irreducible cubic hypersurface V. Then either S is projectively Cohen-Macaulay and linked on V to an irreducible scheme of degree ≤ 3, or S is linked on V to a Veronese surface, or to an elliptic quintic scroll.

Corollary 1.8. A smooth surface S in IP^{4} of degree ≥ 9 which is contained in a cubic hypersurface is necessarily of general type.

Since surfaces on quadric and cubic hypersurfaces are completely classified, for construction purposes we will mainly assume that

h^{0}(I_{S}(2)) = h^{0}(I_{S}(3)) = 0.

To conclude, for a smooth non-degenerate surface S in IP^{4}, not of general type, different from the Veronese surface and not lying on a cubic hypersurface, we can draw the following Beilinson cohomology diagram:

\begin{matrix}↑| & i \\ - - - - - - - ; ∣ 0 ; - - ∣ - - - - ; ∣N + 1 ; - - ∣ - - - - ; ∣ 0 ; - - ∣ - - - - ; ∣ 0 ; - - ∣ - - - - ; ∣ 0 ; - - - - - - - & & \begin{matrix}0 & 0 & 0 & 0 & 0 \\ p_{g} & 0 & 0 & 0 & 0 \\ q & s & h^{1}(O_{S}(2)) \\ 0 & 0 & h^{1}(I_{S}(2)) & h^{1}(I_{S}(3)) & h^{1}(I_{S}(4)) \\ 0 & 0 & 0 & 0 & h^{0}(I_{S}(4)) \end{matrix} & - - - ; - - - ; - - - ; - - - ; - - - ; ⟶_{p} , \end{matrix}

where

N = π + χ − 2.

As previously mentioned, to construct the surface S we will first try to identify the Hartshorne-Rao modules H^{i}(IP^{4}, I_{S}(∗)), i = 1, 2. Sheafification of suitable syzygy modules will provide afterwards the needed vector bundles E and F.

(1.9.) Graded artinian modules. We will briefly recall here some basic facts about graded, finite length modules. Let V = span_{k} < e_{0}, e_{1}, e_{2}, e_{3}, e_{4} >. Let also R = k[x_{0}, x_{1}, x_{2}, x_{3}, x_{4}] be the homogeneous coordinate ring of IP^{4} = IP(V) and let M = ⊕_{n∈ℤ}M_{n} be a graded k-vector space of finite length. A structure of a graded R-module on M is determined by a degree 0, graded morphism of k-algebras f : R → Endgr(M), i.e., by a set of k-linear morphisms

f_{d} : R_{d} → End^{d}(M) = { u ∈ End(M) | u(M_{i}) ⊂ M_{i+d} ∀ i },

for d ∈ IN. More precisely, since R = S^{∗}(V^{∗}), this is equivalent to fix a degree 0, graded k-morphism

u : R_{1}⊗_{k}M(−1) → M  (∗)

satisfying the following commutation relations:

u_{i+1}(h ⊗ u_{i}(g ⊗ α)) = u_{i+1}(g ⊗ u_{i}(h ⊗ α))

for all i, g, h ∈ R_{1} = V^{∗}, α ∈ M_{i}, where we have denoted by u_{i} : R_{1}⊗_{k}M_{i} → M_{i+1} the degree i component of u.

We fix for the sequel on M = ⊕_{n∈ℤ}M_{n} a finite length, graded R-module structure. We’ll denote by

M^{∨} = Hom_{R}(M, R)

the dual module and by

M^{∗} = Hom_{k}(M, k) = ⊕_{n∈ℤ} Hom_{k}(M_{−n}, k)

the k-dual module. We recall also that

M^{∗} ≅ Hom_{R}(M, R^{∗})  (∗)

and that local duality gives

Ext_{R}^{i}(M, R) = 0, for i ≤ 4 and Ext_{R}^{5}(M, R) ≅ M^{∗}(5)

the isomorphisms being homogeneous of degree 0. Let

0⟵ M ⟵^{σ_{0}} L_{0} ⟵^{σ_{1}} L_{1} ⟵^{σ_{2}} L_{2} ⟵ ⋯ ⟵ L_{4} ⟵^{σ_{5}} L_{5} ⟵0  (∗)

be a minimal free resolution of M. We distinguish, for later reference, the following data:

- the support of M

supp M= { n | M_{n} ≠ 0 }

- the Hilbert function

χ_{M}(n)= dim_{k}M_{n}= ∑_{i=0}^{5} (−1)^{i}h^{0}(IP^{4}, L̃_{i}(n))

- the least degree of a generator of M

l_{M}= min{ n ∈ ℤ | M_{n} ≠ 0 }= min{ n ∈ ℤ | h^{0}(IP^{4}, L̃_{0}(n)) ≠ 0 }

- the highest degree of a generator of L_{5} −5

s_{M}= max{ n ∈ ℤ | M_{n} ≠ 0 }= −5 + max{ n ∈ ℤ | h^{0}(IP^{4}, L̃_{5}^{∗}(−n)) ≠ 0 }

- the Mumford-Castelnuovo regularity of the sheafified syzygies

e(Syz_{k}(M))= max_{{i,j \atop j≥k+1}}(e_{ij} − j) if L_{i} = ⊕_{j=1}^{rk L_{i}} R(−e_{ij}) , i = \overline{0, 5},

where we have denoted by

Syz_{k}(M) := (ker σ_{k})^{∼} = (im σ_{k+1})^{∼} , i = \overline{1, 4}

the k-th sheafified syzygy module. A minimal free resolution of M^{∗} is obtained by taking the R-dual of (∗):

0⟵ M^{∗}(5) ⟵^{θ} L5^{∨} ⟵^{σ_{5}^{∨}} L4^{∨} ⟵^{σ_{4}^{∨}} L3^{∨} ⟵ ⋯ ⟵ L1^{∨} ⟵^{σ_{1}^{∨}} L0^{∨} ⟵0

We recall now how to recover a minimal free resolution of M from the knowledge of the multiplication map u : R_{1}⊗_{k}M(−1) → M. In Green’s terms [Gr], this is to compute the Koszul homology groups of M. Tensoring the resolution (∗) by k = R/(x_{0}, x_{1}, x_{2}, x_{3}, x_{4}) over R we obtain a complex of k-vector spaces

0⟵ \overline{M} ⟵^{\overline{σ}_{0}} \overline{L}_{0} ⟵^{\overline{σ}_{1}} \overline{L}_{1} ⟵^{\overline{σ}_{2}} \overline{L}_{2} ⟵ ⋯ ⟵ \overline{L}_{4} ⟵^{\overline{σ}_{5}} \overline{L}_{5} ⟵0

whose homology yields the Tor_{R}^{i}(M, k)’s. Moreover, the resolution (∗) being minimal, we obtain that in fact \overline{L}_{i} ≅ Tor_{R}^{i}(M, k) as graded k-vector spaces, whence

L_{i} = ⊕_{j∈ℤ} (Tor_{R}^{i}(M, k))_{j} ⊗_{k} R(−j − i) for all i = \overline{0, 5}.

To compute the Tor-modules one may use also the resolution of k given by the Koszul complex K_{•}

0⟵ k ⟵^{d_{0}} R ⟵^{d_{1}} Λ^{1}V^{∗} ⊗_{k} R(−1) ⟵^{d_{2}} Λ^{2}V^{∗} ⊗_{k} R(−2) ⟵ ⋯ ⟵^{d_{5}} Λ^{5}V^{∗} ⊗_{k} R(−5) ⟵0.

Tensoring K_{•} over R with M yields the complex K_{•} ⊗_{k} M

0⟵ M ⟵^{δ_{1}} Λ^{1}V^{∗} ⊗_{k} M(−1) ⟵^{δ_{2}} Λ^{2}V^{∗} ⊗_{k} M(−2) ⟵ ⋯ ⟵^{δ_{5}} Λ^{5}V^{∗} ⊗_{k} M(−5) ⟵0

which computes the Tor_{R}^{i}(M, k)’s. We observe also that the differential δ_{1} is nothing else than the multiplicative structure map u : R_{1}⊗_{k}M(−1) → M. To bring everything together and to compute also the morphisms appearing in the minimal resolution of M we use a double complex of graded R-modules, whose terms are

C_{i,j} = M_{t−j} ⊗_{k} Λ^{i+j−t} V^{∗} ⊗_{k} R(−i), i, j ≥ 0,

where, for simplicity, we assumed that M is normalized such as M = ⊕_{i=0}^{t}M_{i} and that, by convention, all exterior powers outside the range 0, …, 5 vanish. As for the differentials the vertical ones are given by δ = (δ_{i+j−t})_{i} ⊗_{k} 1I_{R}(−i) : C_{i,j} → C_{i,j−1} where, as above, δ_{p} are the differentials of K_{•} ⊗_{k} M, and the horizontal ones by d = 1I_{M_{t−j}} ⊗_{k} (d_{i+j−t}(j − t)) : C_{i,j} → C_{i−1,j}. We set as usual D = d + δ.

Lemma 1.10. (d + δ)^{2} = 0.

P roof. Clear.⊓ ⊔

To summarize, we have the following display of the double complex C_{••}:

↓  ↓  ↓
M_{0} ⊗_{k} R ⟵^{1I_{M_{0}} ⊗d_{1}} M_{0} ⊗_{k} Λ^{1} V^{∗} ⊗_{k} R(−1) ⟵^{1I_{M_{0}} ⊗d_{2}} M_{0} ⊗_{k} Λ^{2} V^{∗} ⊗_{k} R(−2) ⟵ ⋯
↓  ↓^{(δ_{1})_{0}⊗1I_{R(−1)}}  ↓^{(δ_{2})_{0}⊗1I_{R(−2)}}
0  ⟵  M_{1} ⊗_{k} R(−1)  ⟵^{1I_{M_{1}} ⊗d_{1}} M_{1} ⊗_{k} Λ^{1} V^{∗} ⊗_{k} R(−2) ⟵ ⋯
↓  ↓  ↓^{(δ_{1})_{1}⊗1I_{R(−2)}}
0  ⟵  0  ⟵  M_{2} ⊗_{k} R(−2)  ⟵ ⋯
↓  ↓  ↓

We remark that, since ⊕_{i} C_{i,j} = M_{t−j} ⊗_{k} K^{•}(j − t), the horizontal lines are exact except

for the most left, non-zero, positions where the cokernels of the differentials are, as graded R-modules, M_{t−j} ⊗_{k} k(j − t). This means that the homology spectral sequence associated to the second filtration of the above complex is degenerated at ′′E^{1} and therefore we get the following

Theorem 1.11. The following assertions hold:

a) The total homology of the double complex C_{••} is

H_{n}^{D}(C_{••}) = \{ \begin{matrix}M & when n = t (as R-modules) \\ 0 & otherwise \end{matrix}

and ⊕_{k=0}^{t}′′E_{k,t−k}^{∞} = ⊕_{k=0}^{t}′′E_{k,t−k}^{1} is the associated graded module to the descending filtration

by degrees of M (i.e. by ⊕_{n≥k}^{t} M_{n}, k ≥ 0).

b) The homology spectral sequence associated to the first filtration computes a minimal free resolution of M, namely

0 ⟵ M = coker σ_{1} ⟵ ⊕_{k=0}^{t}′E_{k,t−k}^{1} ⟵^{σ_{1}} ⊕_{k=0}^{t}′E_{k,(t+1)−k}^{1} ⟵^{σ_{2}} ⋯ ⟵ ⊕_{k=0}^{t}′E_{k,(t+5)−k}^{1} ⟵ 0

where the components of σ_{i} are given by the differentials

d^{k−l} : ′E_{k,(t+i)−k}^{1}⟶′E_{l,(t+i−1)−l}^{1}, k, l = \overline{0, t}, k > l

with an obvious abuse of notations.

P roof. Clear, except may be for the fact that the R-module structure of H_{t}^{D}(C_{••}) = M is the one we started with. But this follows easily, since the vertical differentials are induced by the graded pieces of the multiplicative structure morphism u : R_{1}⊗_{k}M(−1) → M.⊓ ⊔

Remark 1.12. The group GL(5, k) = GL(V^{∗}) acts naturally on R = S^{∗}(V^{∗}) and on the Koszul complex K_{•}, and its action is also compatible with the differentials of the double complex C_{••}. In particular, this fact will allow us to identify later the action of subgroups of GL(5, k) on the minimal free resolution of a module M and to get information on the Betti numbers in case of invariance (e.g., see chapter 7 for an application of this type).

Remark 1.13. For practical construction purposes we’ll need to determine from the double complex C_{••} only the presentation morphism

M = coker σ_{1} ⟵ ⊕_{k=0}^{t}′E_{k,t−k}^{1} ⟵^{σ_{1}} ⊕_{k=0}^{t}′E_{k,(t+1)−k}^{1}.

Since σ_{1} is a direct sum of differentials d^{n}, this can be done in specific cases by the usual ”tic-tac-toe” procedure.

Remark 1.14. In the construction of the complex C_{••} one can in fact take for the vertical differentials any lifts of the multiplication maps

(δ_{1})_{i} : M_{i} ⊗_{k} Λ^{1} V^{∗} ⊗_{k} R(−1)⟶M_{i+1} ⊗_{k} R(−1)

along the Koszul complexes.

One method we shall use to construct vector bundles with prescribed cohomologies is to sheafify syzygy modules. Namely we use the following Proposition 1.15. In the above notation, for all i = \overline{1, 3}, the sheafified syzygy module N_{i} = Syz_{i}(M) is a vector bundle on IP^{4} with cohomology

H^{j}(IP^{4}, N_{i}(∗)) ≅ \{ \begin{matrix}M & for j = i, as graded R-modules \\ 0 & for all j ≠ i, j = \overline{1, 3} \end{matrix} .

Conversely, any vector bundle F with the above intermediate cohomology is stably equivalent with N_{i}, namely

F ≅ N_{i} ⊕ L, with L a direct sum of line bundles

For a proof, see for instance [DES].

We shall make later implicit use of the following well known result concerning extensions of modules

Proposition 1.16. Let E and F be two coherent sheaves on IP^{4} and denote by E = H_{∗}^{0}(E) and by F = H_{∗}^{0}(F). There exists a spectral sequence

E_{2}^{pq} = Ext_{R}^{p}(E, H_{∗}^{q}(F)) ⇒ Ext_{O_{_{IP}4}}^{p+q} (E, F).

In particular the following comparison sequence is exact

0⟶ Ext_{R}^{1}(E, F)^{0} ⟶ Ext_{O_{_{IP}4}}^{1} (E, F) ⟶ Hom_{R}(E, H_{∗}^{1}(F)) ⟶
Ext_{R}^{2}(E, F)^{0} ⟶ Ext_{O_{_{IP}4}}^{2} (E, F).

P roof. See for instance [MDP]. One writes the spectral sequence of a composition of functors [Gro], and uses the adjunction of the two functors and H_{∗}^{0}. The comparison sequence is the associated low-degree sequence.⊓ ⊔

Remark 1.17.

a) The first map in the comparison sequence is associating to an extension class of an extension of graded R-modules the corresponding extension class in Ext_{O_{_{IP}4}}^{1} (E, F) of the short exact sequence obtained via sheafification.

b) The second map is associating to the extension class of an exact sequence

0 ⟶ F ⟶ G ⟶ E ⟶ 0

the cobord map δ : E⟶H_{∗}^{1}(F) in the long exact cohomology sequence.

2. A K3 surface of degree 10

We describe in this chapter two constructions of a smooth, non-minimal K3 surface S ⊂ IP^{4} with d = 10, π = 9. A family of surfaces with these invariants, which all lie on two quartic hypersurfaces, has been constructed by K. Ranestad in [Ra1]. We give an example of a family of surfaces which lie on a single quartic.

In terms of numerical invariants, we recall the following:

Proposition 2.1.[Ra1]. Let S be a smooth surface of degree 10 in IP^{4} with π = 9 and χ = 2. Then either

a) S is a regular, proper elliptic surface with three (−1) lines, or

b) S is a non-minimal K3 surface with three (−1) conics, or

c) S is a non-minimal K3 surface with two (−1) lines and one (−1) quartic.

For a proof, see [Ra1] and also [PR] for the non-existence of a smooth K3 surface with the above invariants and having one (−1) line, one (−1) conic and one (−1) cubic.

K. Ranestad has constructed in [Ra1] examples of surfaces for the cases a) and b) in the above proposition. We give examples for c). First a lemma.

Lemma 2.2.

a) h^{1}(O_{S}(k)) = 0 for k ≥ 2.

b) h^{0}(I_{S}(4)) ≥ 1.

P roof. Severi’s theorem gives h^{1}(O_{S}(1)) = 1. Thus if h^{2}(I_{S}(2)) > 0, then h^{2}(I_{H}(2)) > 0 for at least a web of hyperplane sections H. But the general hyperplane section H in the web is smooth and 2π − 2 = 16 < 20, so O_{H}(2H) is non-special, i.e., h^{1}(O_{H}(2)) = h^{2}(I_{H}(2)) = 0, which is a contradiction; hence by induction a). Part a), Riemann-Roch and (1.6), (1.7) give h^{1}(I_{S}(2)) = 1 and h^{1}(I_{S}(3)) = 3, whence h^{0}(I_{H}(3)) = 1 for at least a pencil of hyperplane sections. But if h^{0}(I_{S}(4)) = 0, then h^{1}(I_{S}(4)) = 0 and thus h^{0}(I_{H}(4)) = h^{1}(I_{S}(3)) = 3 for all H. This is a contradiction since for the hyperplanes in the above pencil h^{0}(I_{H}(4)) ≥ 4.□

Therefore the cohomology diagram of the surface S looks like:

i ↑

1

1  h^{i}(I_{S}(p))
1  3  a

a ⟶

p where a ≥ 1. When a ≥ 2 one shows, see [PR], that in fact a = 2 and S is a smooth K3 surface with three (−1) conics, as constructed in [Ra1]. A construction of this surface using the Eagon-Northcott complex can be found in [DES].

Therefore we assume in the sequel that a = 1 and we look for graded artinian modules M = ⊕_{n≥−2} M_{n} with Hilbert function (1, 3, 1), which we’ll suppose generated in the first non-zero twist, i.e., monogeneous. Any such module has a minimal free presentation of type

0 ← M ⟵ R(2) ⟵^{ψ} 2R(1) ⊕ 5R

where the linear part of ψ is given, without loss of generality, by say x_{0} and x_{1}, and the quadratic by q_{1}, q_{2}, …, q_{5} ∈ k[x_{2}, x_{3}, x_{4}], quadrics in three variables without common zeroes. Therefore the choice of ψ is equivalent to that of a hyperplane section of the Veronese surface in IP^{5}, and there are two types of such curves: the irreducible hyperplane sections leading to the generic module with this Hilbert function and the reducible ones, two conics with a common point, leading to a special module. In both cases let F = Syz_{1}(M) and E = O_{IP}^{4}(−1) ⊕ Ω_{IP^{4}}^{3}(3). By (1.15) H^{1}(F(∗)) = H^{1}(I_{S}(∗ + 4)) and H^{i}(F(∗)) = 0 for i = 2, 3. For a generic module M, F has a minimal free resolution of type

O
0 ← F ⟵ ⊕
15O(−1) 15O(−2)  5O(−3)
↖ ⊕ ⟵ ⊕
O(−3)  2O(−4) ↖ O(−5) ← 0

and for M corresponding to the reducible hyperplane sections of the Veronese surface F has a resolution of type

O
⊕
0 ← F ← 15O(−1) 16O(−2) 7O(−3) O(−4)
⊕ ↖ ⊕ ⟵ ⊕ ⟵ ⊕ ← 0
O(−2)  3O(−3)  3O(−4)  O(−5)

In both cases we have dim_{k} Hom(E, F) = 35 and the degeneracy locus of a generic φ ∈ Hom(E, F) will be a regular surface S of degree 10, with sectional genus 9 and χ = 2. The smoothness can be checked in an example with Macaulay [Mac]. More details on computational aspects can be found in the appendix A of [DES]. To identify the surfaces we use Le Barz’s formula (0.11) which gives N_{6}(10, 9, 2) = 3 for the sum of the number of 6-secant lines plus the number of exceptional lines on S.

We consider first the case of the generic module. A resolution of the ideal sheaf of S is given by the mapping cone between the resolutions of E and F, so it is easily seen that the homogeneous ideal of the surface is generated by quintic hypersurfaces in this case. It follows that S has no 6-secants and exactly three (−1) lines, hence it is a proper elliptic surface as in Proposition 2.1 a). This construction is the one described in [DES], Appendix B, B7.4.

In the case of the special module the mapping cone yields a minimal free resolution for I_{S} of type

O(−4)
⊕
0 ← I_{S} ← 9O(−5) 15O(−6) 7O(−7) O(−8)
⊕ ↖ ⊕ ⟵ ⊕ ⟵ ⊕ ← 0
O(−6)  3O(−7)  3O(−8)  O(−9)

In this case there is precisely one sextic generator for I_{S} and we guess that there is precisely one 6-secant. Indeed the three linear forms

O(−6) ⟵ 3O(−7)

in the bottom row of the resolution are independent (since those in the resolution of F are) and annihilate the module I_{S}/(I_{S})_{≤5}, hence this module has support on a line L. Or one can argue as follows. This line is just the line defined by the forms in the kernel of the multiplication map

H^{0}(IP^{4}, O_{IP}^{4}(1)) ⟶ Hom(H^{1}(I_{S}(3)), H^{1}(I_{S}(4))),

so for all hyperplane sections H containing L we’ll have h^{1}(I_{H}(4)) = 1. On the other hand the cohomology of the exact sequence

0 ⟶ I_{S}(2) ⟶ I_{S}(3) ⟶ I_{H}(3) ⟶ 0

gives 2 ≤ h^{1}(I_{H}(3)) ≤ 3 so each hyperplane H in the net contains at least a plane π with h^{1}(I_{π∩S}(4)) = 1. The generic hyperplane section through L is irreducible, so π cuts S only in points for such a H. Now lemma 0.41 shows that π contains a 6-secant line to the surface since if h^{0}(I_{π∩S}(2)) > 0 the cohomology of the exact sequences

0 ⟶ I_{H}(1) ⟶ I_{H}(2) ⟶ I_{π∩S}(2) ⟶ 0
0 ⟶ I_{S} ⟶ I_{S}(1) ⟶ I_{H}(1) ⟶ 0

yields h^{1}(I_{S}(1)) = h^{1}(I_{H}(1)) > 0, which contradicts Severi’s theorem. Therefore L is a 6-secant to S and, by Proposition 2.1, S is a non-minimal K3 surface with two (−1) lines and one (−1) quartic, as desired.

Remarks 2.3.

a) It is shown in [PR] that any surface of type a) or c) in Proposition 2.1 can be obtained by the above construction. In particular, (1.2) and (1.3) imply that the Hilbert schemes of such surfaces are irreducible and unirational. b) A flat deformation of elliptic surfaces of degree 10, sectional genus π = 9, to a scheme belonging to the irreducible component of the Hilbert scheme containing the K3 surfaces of type c) can be constructed by varying the hyperplane section of the Veronese surface in IP^{5}.

c) In the above setting, all modules M distinguish a plane P, assumed to be given as P = { x_{0} = x_{1} = 0 }. In the case of the elliptic surface, P cuts it along a plane cubic, namely the non-exceptional part of the canonical divisor. For the general K3 surface of type c) P contains the unique 6-secant line L and cuts the surface only in points. ⊓ ⊔

Assume now that S is a surface of type a) or c). S can be linked in the complete intersection of the unique quartic containing it and a quintic hypersurface to an irreducible surface Y of degree 10, with sectional genus π(Y) = 9 and χ(Y) = 0. The cohomology of the liaison exact sequence

0 ⟶ I_{S∪Y} (4) ⟶ I_{Y} (4) ⟶ O_{S}(K) ⟶ 0

gives h^{0}(I_{Y} (3)) = 0 and h^{0}(I_{Y} (4)) = 2. Therefore Y can be further linked in the complete intersection of two quartics to a surface Z of degree 6, sectional genus π(Z) = 1, χ(Z) = 0.

Working out with Macaulay [Mac] the construction of S via the Eagon-Northcott complex one can figure out how the scheme Z should look like. We use in the sequel this description to give a liaison construction of a K3 surface of type c). For the elliptic case we refer to [Ra1] and [PR].

(2.4.) Liaison. Let C be a rational cubic scroll in IP^{4} and let T be a smooth quadric surface cutting C along two lines in its ruling, say L_{1} and L_{2}. Consider next a plane P passing through the directrix L of C, cutting the scroll only along this line and not contained in the hyperplane spanned by the quadric surface. Let now Z = P ∪ T ∪ C. It is easily seen that Z has the right invariants: deg Z = 6 and π(Z) = 1 by formula (0.2). Also Z is locally Cohen-Macaulay and even a local complete intersection except for the points \{p_{i}\} = L ∩ L_{i}, i = \overline{1, 2}. We prove in the sequel that Z can be backwards linked (4, 4) and (4, 5) to a smooth K3 surface of type c). First a lemma.

Lemma 2.5.

a) The scheme X = T ∪ C is a degenerated elliptic quintic scroll in IP^{4} and, in particular, the homogeneous ideal I_{X} is generated by 5 cubics.

b) The homogeneous ideal I_{Z} is generated by 10 quartics and one quintic. Moreover, the quartics cut out the scheme Z outside L and the generic quartic contains L with multiplicity two.

P roof. Let H be the hyperplane spanned by the quadric surface T. For the first part of the lemma we consider the residual exact sequence

0 ⟶ I_{C}(2) ⟶ I_{X}(3) ⟶ I_{H∩X,H}(3) ⟶ 0

of sheaves of ideals on IP^{4}. This sequence remains exact after taking global sections since h^{1}(I_{C}(2)) = 0. Therefore h^{0}(I_{X}(3)) = 5 and it suffices to check whether C is cut out by quadrics and H ∩ X by cubics. The former is clear since C is defined by the 2 × 2 minors of a matrix with linear entries, while H ∩ X = T ∪ L is clearly cut out by the unions of the quadric and the hyperplanes through L. Moreover the cohomology of the above sequence yields

h^{1}(I_{X}(k)) = 0 for all k, h^{2}(I_{X}(k)) = 0 for k ≠ 0, and h^{2}(I_{X}) = h^{2}(I_{T ∪L}) = 1.

Therefore Beilinson’s spectral sequence gives a resolution of the form

0 ⟶ Ω_{IP^{4}}^{3}(3) ⟶ 5O_{IP}^{4} ⟶ I_{X}(3) ⟶ 0

and hence X is a degenerated elliptic quintic scroll. Consider now the exact sequence

0 ⟶ I_{X}(k − 1) ⟶ I_{Z}(k) ⟶ I_{Z∩H′,H′}(k) ⟶ 0

where H′ is a general hyperplane containing P. It remains also exact after taking global sections since h^{1}(I_{X}(k)) = 0 for all k. Now Z ∩ H′ = P ∪ D ∪ f_{1} ∪ f_{2}, where D = T ∩ H′ is a smooth conic and f_{1}, f_{2} are rulings of C, thus h^{0}(I_{Z∩H′,H′}(k)) = h^{0}(I_{D∪f_{1}∪f_{2}}(k − 1)). One easily computes h^{0}(I_{D∪f_{1}∪f_{2}}(2)) = 0, h^{0}(I_{D∪f_{1}∪f_{2}}(3)) = h^{0}(O_{IP}^{3}(3))−2h^{0}(O_{IP}^{1}(3))− h^{0}(O_{IP}^{1}(6)) = 5 and analogously h^{0}(I_{D∪f_{1}∪f_{2}}(4)) = 16. Moreover the homogeneous ideal I_{D∪f_{1}∪f_{2}} is generated by the 5 cubics and one quartic and since, by a), I_{X} is generated by 5 cubics the first assertion of b) follows. For the second part it is enough to observe that all cubics in H^{0}(I_{D∪f_{1}∪f_{2}}(3)) are vanishing on L and that they cut out, scheme-theoretically in fact, D ∪ f_{1} ∪ f_{2} ∪ L.⊓ ⊔

As a consequence of the above lemma and remark 0.31, we obtain that Z can be linked in the complete intersection of two quartic hypersurfaces to an irreducible surface Y, with degY = 10, π(Y) = 9, which contains and is singular along L and which is smooth outside this line. Further Y can be linked (4, 5) to a surface S with the desired invariants. Namely, degS = 10, π(S) = 9 and, from the liaison exact sequences, p_{g} = 1, q = 0. It is easily seen that S is smooth outside L, for a general choice of the linkages. To see the behavior at the intersection with L we’ll work out explicitly this liaison.

We consider the blowing-up

σ : ĨP^{4} = IP(2O_{IP}^{2} ⊕ O_{IP}^{2}(1)) ⟶ IP^{4}

of IP^{4} along the line L, with exceptional divisor E = IP(2O_{IP}^{2}) = IP(3O_{L}) = IP^{2} × L. Let B be a divisor of E corresponding to a section of 3O_{L}(1) and F corresponding to a fibre of the projection σ : IP^{2} × L → L. If a hypersurface V of degree v contains the line L with multiplicity m, then its strict transform Ṽ will meet E along \overline{V}, numerically of type

\overline{V} ≡ mB + (v − 2m)F.

Let V_{1} and V_{2} be two general quartic hypersurfaces containing Z. By lemma 2.5 they have multiplicity two along L, thus \overline{V_{i}} ≡ 2B, i = \overline{1, 2}. On the other hand the strict transforms of P and T cut E along \overline{P} ≡ (B − F)(B − F) ≡ B^{2} − 2BF and \overline{T} ≡ (B − F)2F ≡ 2BF, respectively. Moreover, the strict transform of C cuts E along \overline{C} ≡ B^{2} since the cubic scroll is linked to a plane in the complete intersection of two quadric hypersurfaces having multiplicity one along L.

It follows that, for a general (4, 4) liaison, the strict transform of Y on ĨP^{4} meets E in a curve equivalent to (2B)(2B) − (B^{2} + (B^{2} − 2BF) + 2BF) ≡ 2B^{2}. A local computation shows that the general quintic hypersurface containing Y has multiplicity one along L. Therefore, for a general choice of the (4, 5) liaison, the strict transform of S on ĨP^{4} will meet E in a curve equivalent to 2B(B +3F)−2B^{2} ≡ 6BF. A Bertini argument shows now that for the general choice of the liaison, the surface S residual to Y is smooth. Moreover, since a curve of type BF is blown down on S, it follows that L is a 6-secant line to S.

To show that S is indeed a K3 surface we determine the one dimensional components in the intersections S ∩ T and S ∩ C. The liaison exact sequences (0.24) for Y give P ∩ Y ≡ 3H_{P} − K_{P} − L ≡ 5H_{P}, T ∩ Y ≡ 3H_{T} − K_{T} − [T ∩ C] ≡ 5l_{1} + 3l_{2}, where l_{1} and l_{2} denote the classes of the two rulings of the quadric, and C∩Y ≡ 3H_{C} −K_{C} −[C∩(P ∪T)] ≡ 3(C_{0} +2f)−(−2C_{0} −3f)−C_{0} −2f ≡ 4C_{0} +7f, with C_{0} denoting the numerical class of the directrix L on the scroll C and f the class of a ruling. The one-dimensional components of the intersection scheme S ∩ Z are the residuals (in term of conductor ideals) of the above curves in the complete intersection of Z with the quintic hypersurface used in the liaison of Y with S. Therefore, for a general choice of the liaison, P cuts S only in points, C cuts S along a scheme whose one-dimensional part K_{1} is equivalent to 5H_{C} −[C ∩Y] ≡ C_{0} +3f and T cuts S along a curve K_{2} equivalent to 5H_{T} − [T ∩ C] ≡ 2l_{2}, plus a zero dimensional scheme. On the other hand, a similar computation on S shows that the scheme K_{1} ∪ K_{2} is exactly the canonical divisor of S. Now the liaison exact sequence

0 ⟶ I_{Y ∪Z}(5) ⟶ I_{Y ∪P ∪C}(5) ⟶ O_{T} ⟶ 0

remains also exact after taking global sections. It follows that the quintics in H^{0}(I_{Y} (5)) cut on T a linear system whose fixed part is exactly T ∩ Y. Therefore, for a general choice of the (4, 5) liaison, the curve K_{2} ⊂ Z ∩ S is reduced and hence it is the union of two skew lines, say E_{2} and E_{3}, in the ruling of T which contains L_{1} and L_{2}. Eventually, the adjunction formula on S yields E_{i}^{2} + KE_{i} = 2E_{i}^{2} = 2p_{a}(E_{i}) − 2 = −2, i = \overline{2, 3} and thus E_{2} and E_{3} are exceptional lines on S. By (2.1), it follows that E_{1}:=K_{1} is a (−1) quartic on S and hence S is, as claimed, a K3 surface of type c); i.e., S is embedded by a linear system of type

H = H_{min} − 4E_{1} − E_{2} − E_{3}.

Concerning liaison we want to add the following

Remark 2.6. Z is the unique minimal scheme in the even liaison class of S. Therefore, by the general results in [BBM] and [MDP], S can be viewed as obtained from Z via a basic double link and a flat deformation.

P roof. The arguments used in the proof of Lemma 2.5 give also p_{g}(Z) = 0, q(Z) = 1 and, moreover, that e(Z):=max { t | h^{2}(O_{Z}(t)) ≠ 0 } = −2. Therefore the above claims follow from (0.32) and the general description of a liaison class in [BBM].

3. Smooth surfaces of degree 11

We construct in this chapter several families of smooth surfaces of degree 11 in IP^{4}, providing along also an incomplete and rough attempt of numerical classification. In a certain sense, the classification is to be considered only as a guideline for where one should look to in order to find the desired examples.

We start by recalling the basic relations between the invariants of a smooth surface S of degree 11 in IP^{4}. The double point formula (0.6) takes now the form

K^{2} = 6χ − 5π + 38

and Severi’s theorem (0.7) and Riemann-Roch give

π = χ + 7 + h^{1}(O_{S}(H)) − h^{0}(O_{S}(K − H)).

In addition to these relations Theorem 1.6 says that S is contained in a hyperquadric if and only if π = 1 + [\frac{11(11−4)}{4}] = 20, and in this case S is linked to a plane in the complete intersection of the hyperquadric with a degree 6 hypersurface.

We assume therefore in the sequel that S doesn’t lie on a hyperquadric. A theorem of Gruson and Peskine [GP1], [Au1] provides then π ≤ 15, and furthermore, when equality holds, that S is linked to a plane in the complete intersection of a cubic and a quartic hypersurfaces. Therefore we are left to study only the cases when π ≤ 14. We look now for a lower bound for the sectional genus and we start with the smallest values of π.

Proposition 3.1. If S is a smooth surface of degree 11 in IP^{4}, then π ≥ 8.

P roof. When π ≤ 7 then HK ≤ 1, so the index theorem implies K^{2} ≤ \frac{(HK)^{2}}{11} < 1. On the other hand the double point formula gives

χ = \frac{K^{2} + 5π − 38}{6} < 0

whence S is birationally ruled, which in turn means that K^{2} ≤ 8χ. But this implies 2χ ≥ 38 − 5π ≥ 3 which contradicts the above inequality.⊓ ⊔.

I. Surfaces with d = 11, π = 8

The non-existence of smooth, sectionally non-special surfaces with these invariants was shown in [MR]. We’ll recall their results along the proof of the following proposition which mainly collects the data in [MR]:

Proposition 3.2. A smooth surface S ⊂ IP^{4} of degree 11, with sectional genus π = 8 is necessarily a non-minimal abelian surface with one (−1) line and one (−1) conic.

P roof. In this case HK = 3 so, as above, the double point formula and the index theorem give χ ≤ 0.

Strict inequality means that S is birationally ruled, whence K^{2} ≤ 8χ, which leaves as only possibility χ = −1. Therefore K^{2} = −8 and S is in fact geometrically ruled over a curve of genus 2. But this is impossible by the results in [HR]. It is also possible to see this directly by using adjunction theory since, by Theorem 0.13, S_{1} = φ_{H+K}(S) ⊂ IP^{5} would be a smooth surface with invariants d_{1} = 9, H_{1}K_{1} = −5, π_{1} = 3 > q = 2, K_{1}^{2} = −8, and so adjunction again would give 0 ≤ (H_{1} + K_{1})^{2} = 9 − 10 − 8 = −9, a contradiction.

Thus χ = 0, K^{2} = −2 and, in particular, S is not of general type. The adjunction mapping is birational and the image S_{1} = φ_{H_{1}+K_{1}}(S) ⊂ IP^{6} is a smooth surface of degree d_{1} = 15 and sectional genus π_{1} = 9. Also H_{1}K_{1} = 1 and K_{1}^{2} = −2 + a, where a ≥ 0 is the number of (−1) lines on S. In general we’ll denote by S_{n} the image of the n-th iterated adjunction mapping and, in case it is a surface, by H_{n} the hyperplane section, by K_{n} the canonical divisor, by π_{n} the sectional genus and by a_{n} the number of (−1) lines on S_{n−1}, or equivalently the number of (−1) rational curves of degree n on S. In particular we let S_{0} = S and a_{1} = a.

If p_{g} > 0, then K_{1} is a line and K_{1}^{2} = −1 since (H_{1} +K_{1})K_{1} ≥ 0 and S is not ruled. In this case S is a blown-up abelian surface with one (−1) line and one (−1) conic. The minimal model is embedded by the second adjunction as a surface of degree 16 and sectional genus 9 in IP^{7}.

We assume from now on p_{g} = 0. If a = 0, then (H_{1} + K_{1})K_{1} < 0 so S is birationally ruled. If K_{2}^{2} = −2, then S_{4} = φ_{H_{3}+K_{3}}(S_{3}) ⊂ IP^{3} would be a ruled surface with d_{4} = a_{3} + 3 and π_{4} = a_{3}, for a_{3} ∈ {0, 1, 2}, which is absurd. If K_{2}^{2} = −1, then S_{4} ⊂ IP^{4} would be a surface with d_{4} = 7+a_{3} and π_{4} = 3+a_{3}, with a_{3} ∈ {0, 1}, which is impossible by the classification of such surfaces in [Io1],[Io2],[Ok3],[Ok4]. Finally, if K_{2}^{2} = 0, then S_{2} is minimal and the adjunction process would embed S_{5} ⊂ IP^{4} as a smooth surface with d_{5} = 9 and π_{5} = 5, which is impossible by the classification in [AR].

If a = 1, then the adjoint mapping for S_{1} is again birational on a surface S_{2} ⊂ IP^{7}, having invariants d_{2} = 16, π_{2} = 9, H_{2}K_{2} = 0 and K_{2}^{2} = −1 + a_{2}, where a_{2} ∈ {0, 1} by Hodge index. For a_{2} = 0, S_{5} ⊂ IP^{4+a_{3}} is a smooth surface with d_{5} = 7 + 4a_{3} + a_{4} and π_{5} = 3 + 3a_{3} + a_{4}. Thus if a_{3} = 0, it is a smooth surface of degree 7 or 8 and sectional genus 3 or 4 respectively, in IP^{4}, which is as above a contradiction, while if a_{3} = 1, then S_{6} ⊂ IP^{4} would have invariants d_{6} = 9, π_{6} = 5, K_{6}^{2} = 0, which is absurd by [AR]. Assume now a_{2} = 1. In this case S_{2} is minimal bielliptic because the adjunction process yields surfaces S_{n} ⊂ IP^{7} with the same numerical invariants: d_{n} = 16, H_{n}K_{n} = 0, π_{n} = 9, for all n ≥ 3. In [MR] it is shown that such smooth bielliptic surfaces S_{2} ⊂ IP^{7} exist, while the corresponding surfaces in IP^{4} are necessarily singular.

If a = 2, then K_{1}^{2} = 0 and S_{1} is already minimal. It is not birationally ruled since an argument as in (0.18) shows that the adjunction doesn’t stop. Since K_{1} is not numerically trivial it follows that S would be a blown-up proper elliptic surface with two (−1) lines.

Using Reider’s criterion, in [MR] it is shown that smooth minimal proper elliptic surfaces S_{1} ⊂ IP^{6} with d_{1} = 15, π_{1} = 6, p_{g} = 0 and q = 1 do not exist.⊓ ⊔

Remark 3.3. We have not been able to find either examples nor to prove the non-existence of smooth abelian surfaces as in the above proposition. The author believes they don’t exist.

II. Surfaces with d = 11, π = 9

Adjunction and the double point formula give HK = 5, K^{2} = 6χ−7, thus χ ≤ 1 by Hodge index. On another side, 0 ≤ χ ≤ 1 by (0.20) if S is birationally ruled or rational, while (0.16) yields χ = 1 when κ(S) ≥ 0. Again, as in the previous chapter, we have not been able to construct examples of smooth surfaces with these invariants. In terms of numerical invariants we mention the following remarks:

Proposition 3.4. There are no smooth surfaces S ⊂ IP^{4} with d = 11, π = 9 and χ = 0.

P roof. As argued above, if such a surface exists, then it is birationally ruled over an elliptic curve. In particular h^{1}(O_{S}) = 1 and p_{g} = 0. On another side, Riemann-Roch and Severi’s theorem yield for the speciality h^{1}(O_{S}(1)) = 2. It follows that the variety in ǏP^{4} parametrizing hyperplane sections H for which h^{1}(O_{H}(1)) = 2 is a plane, so there is a line E ⊂ IP^{4} which is the base locus of the net of hyperplanes for which h^{1}(O_{H}(1)) = 2. To prove the proposition we’ll use the following fact [ACGH, p.198, ex. E-1]: Let C be a smooth curve of genus g and let L be a line bundle of degree d. For r = h^{0}(L)−1 the inequality

d ≥ g − d + 2r + h^{1}(L^{⊗2})  (∗)

holds. If E doesn’t lie on S, then the general H in the net is smooth and applying (∗) for C = H and L = O_{H}(K) we obtain

5 ≥ 9 − 5 + 2 + h^{1}(O_{H}(2K)),

which is a contradiction. Therefore E lies on S and is contained in a fibre of the ruling, since otherwise it would dominate the elliptic base of the minimal model of S. If the general section C ∈ |H − E| is not integral, then, by Bertini’s theorem, |H − E| is composed with a pencil of plane curves. It has no base points, since a basepoint would be a singular point on S, thus (H − E)^{2} = 0 and hence E^{2} = −9, p_{a}(C) = E^{2} + 9 = 0, which is impossible for a union of plane quintic curves, or conics. Therefore the general C ∈ |H − E| is a smooth and irreducible curve of degree 10 and genus g(C) = E^{2} + 9. We look now for the degrees of the exceptional curves on S. Let S_{1} denote the image of S under the adjunction map and S_{2} the image of S_{1} under the map defined by |H_{1} + K_{1}|. We compute the following invariants:

S ⊂ IP^{4}  H^{2} = 11  HK = 5  K^{2} = −7  π = 9
S_{1} ⊂ IP^{7} H_{1}^{2} = 14  H_{1}K_{1} = −2  K_{1}^{2} = −7 + a  π_{1} = 7
S_{2} ⊂ IP^{5} H_{2}^{2} = 3 + a  H_{2}K_{2} = −9 + a  K_{2}^{2} = −7 + a + b  π_{2} = −2 + a,

where a and b are non-negative numbers. If a ≥ 4, then π_{2} > q and theorem 0.13 implies that (H_{2} + K_{2})^{2} = −22 + 4a + b ≥ 0, whence in fact a ≥ 5. If a = 5, then b = 2 and S_{2} would be via adjunction a conic bundle over IP^{1}, which is absurd for q = 1; if a = 6, then the adjunction map on S_{2} would have degree (b + 2) over IP^{2}, which is impossible by (0.13), while if a = 7, then b = 0 and the image of S_{2} under the adjunction map would be a surface of degree 6 in IP^{3}, which is again absurd. It follows that a = 3 and π_{2} = 1, hence S_{2} is an elliptic scroll of degree 6 in IP^{5}. In particular, there are only exceptional lines and exceptional conics on S, hence E is either a (−1) line, or (H + K)E = 1 and E is a (−2) line. In the former case, the cohomology of the exact sequence

0 ⟶ O_{E}(E) ⟶ O_{H}(H) ⟶ O_{C}(H) ⟶ 0

yields h^{1}(O_{C}(H)) = h^{1}(O_{H}(H)) = 2. Now again the inequality (∗), applied this time for C and L = O_{C}(K − E), leads to a contradiction. A similar argument rules out the case E^{2} = −2, and thus surfaces with the above invariants do not exist.⊓ ⊔

Proposition 3.5. A smooth surface with invariants d = 11, π = 9 and χ = 1 is either a) rational, or

b) a non-minimal Enriques surface with one (−1) quintic, or

c) a non-minimal regular proper elliptic surface with one (−1) line, or one (−1) conic.

P roof. The first two cases are clear, while regularity follows from the same argument we’ve been using in proposition 3.4. Assume now κ(S) ≥ 1 and let S_{1} denote the image of S through the adjunction mapping. Let H_{1} denote the hyperplane divisor and K_{1} the canonical class on S_{1}. We obtain the following list of invariants:

S ⊂ IP^{4}  H^{2} = 11  HK = 5  K^{2} = −1  π = 9
S_{1} ⊂ IP^{8} H_{1}^{2} = 20  H_{1}K_{1} = 4  K_{1}^{2} = −1 + a  π_{1} = 13
S_{2} ⊂ IP^{12} H_{2}^{2} = 27 + a  H_{2}K_{2} = 3 + a  K_{2}^{2} = −1 + a + b  π_{2} = 16 + a,

where a and b are the number of the (−1)-lines and (−1)-conics on S respectively. We observe that (H_{1}K_{1})^{2} = 16 < 20 = H_{1}^{2}, thus S is not of general type by lemma 0.18. On another side h^{0}(O_{S}(2K)) ≠ 0 by Castelnuovo’s rationality criterion, hence HK_{min} ≥ 2. If HK_{min} = 2, we have h^{0}(O_{S}(2K)) = 1, since an elliptic curve must have degree at least 3, but then an easy argument using Kodaira’s formula for the canonical bundle of an elliptic fibration [BPV] shows in fact that this case cannot occur. Therefore HK_{min} ≥ 3 and S is a non-minimal proper elliptic surface with only one exceptional curve of degree less or equal to 2. ⊓ ⊔ III. Surfaces with d = 11, π = 10

The double point formula gives K^{2} = 6χ − 12 and Hodge index implies K^{2} ≤ 4, thus χ ≤ 2. On the other hand, lemma 0.20 yields χ ≥ 0 when κ(S) = −∞, while the pseudoeffectiveness of K gives χ ≥ 1 for κ(S) ≥ 0.

Proposition 3.6. A smooth surface S ⊂ IP^{4} with d = 11, π = 10 and χ = 1 is either a rational surface or a blown-up Enriques surface, embedded by

H = H_{min} − 2E_{1} − ∑_{i=2}^{6} E_{i}.

P roof. Let S_{1} be the image of S through the adjunction morphism. From (0.13) we obtain the following invariants

S_{1} ⊂ IP^{9} H_{1}^{2} = 19  H_{1}K_{1} = 1  K_{1}^{2} = −6 + a  π_{1} = 11,

where a is the number of (−1) lines on S. Furthermore p_{g} = 0. Otherwise, since S_{1} is not a scroll and (H_{1} + K_{1})K_{1} ≥ 0 by (0.16), a divisor in |K_{1}| would be an exceptional line on S_{1}, and thus the image of S_{1} through adjunction would be a surface with trivial canonical class and χ = 1, absurd. Thus S is regular, and either p_{2} = 0 and S is rational by Castelnuovo’s theorem, or h^{0}(O_{S}(2K_{1})) > 0 and a divisor in |2K_{1}| is twice an exceptional line. In the last case, the image of S_{1} through the adjunction map is a minimal Enriques surface S_{2} ⊂ IP^{10}, with d_{2} = 20 and π_{2} = 11, whereas S is embedded by

H = H_{2} − 2E_{1} − ∑_{i=2}^{6} E_{i}.⊓ ⊔

(3.7.) Determinantal construction. An example of an Enriques surface S with the above invariants and minimal cohomology was constructed in [DES]. Namely, assuming S has the Beilinson cohomology table

i ↑

2  h^{i}(I_{S}(p))
1  5  5
⟶

p

the authors construct the surface as the degeneracy locus of a general morphism φ ∈ Hom(2Ω^{3}(3), G), with G = ker(10O⟶^{ψ}O(2)) and ψ a quite special morphism. Namely, let E ⊂ IP^{4} be an elliptic normal curve, let τ be a non-trivial 2-torsion point on E and denote by \overline{p, p + τ} the line in IP^{4} spanned by p and p + τ. Then (cf. [BHM])

Q = ⋃_{p∈E} \overline{p, p + τ} ⊂ IP^{4}

is an elliptic quintic scroll over the elliptic curve E/τ, which contains E as a 2-section. Corresponding to the other two unramified 2:1 covers of E/τ, Q contains two further elliptic normal curves E′ and E′′ of which it is a 2-translation scroll. The three elliptic curves are mutually disjoint. In this setting, the authors of [DES] take ψ to be defined by the ten hyperquadrics which generate I_{E′} + I_{E′′}. As checked in examples, the rulings of Q are 6-secant lines to the constructed surface S.

(3.8.) Liaison construction. As a completion of the determinantal construction in [DES], we provide in the sequel an equivalent liaison construction for this family of Enriques surfaces.

Let p_{1}, …, p_{5} be five distinct points on the elliptic normal curve E such that p_{1} + p_{2} + … + p_{5} = τ, where τ is a non-trivial 2-torsion point on E, and let π ⊂ IP^{4} be a general plane. There are four independent hyperquadrics containing π and the points p_{i}, and a Bertini argument shows that we can link π in the complete intersection of two general hyperquadrics to a smooth rational cubic scroll T, which meets the elliptic curve E only in the points p_{i}, i = \overline{1, 5}.

Lemma 3.9. The homogeneous ideal I_{T ∪E} is generated by 3 cubic and 5 quartic hypersurfaces.

P roof. Consider the cohomology associated to the residual intersection exact sequences

0 ⟶ I_{E}(k − 2) ⟶^{f} I_{E∪T} (k) ⟶ I_{T ∪Z,V} (k) ⟶ 0,  k ∈ ℤ,

where f is the equation of a general hyperquadric V containing T and Z = V ∩ E \ T ∩ E is a reduced scheme of length 5, disjoint of T. Now h^{1}(I_{E}(m)) = 0, for all m ∈ ℤ (e.g., cf. [Hu]), h^{0}(I_{T,V} (3)) = 8 , h^{0}(I_{T,V} (4)) = 20, while I_{E}(2) and I_{T ∪Z,V} (4) are clearly generated by global sections, so the lemma follows.⊓ ⊔

We remark that a cubic hypersurface containing T ∪ E is obviously irreducible, and by a result of Aure [Au1, Lemma 2.1.6, Lemma 3.1.19], cannot be a cone or have a double plane. Moreover, it has only isolated singularities since its general hyperplane section is a Del Pezzo surface (it cannot be a cone over a smooth plane cubic curve because then the corresponding hyperplane section of T would dominate the base of the cone).

T can be linked in the complete intersection of two cubic hypersurfaces in H^{0}(I_{E∪T} (3)) to a surface B ⊂ IP^{4} of degree 6 and sectional genus 3. For a general choice of the cubics B is smooth, thus it is a Bordiga surface (cf. [Ok2], [Io1]).

Lemma 3.10. The 3 cubic hypersurfaces in H^{0}(I_{E∪T} (3)) intersect along T ∪ E plus the union of five skew lines on B, say E_{1}, E_{2}, …, E_{5}, which are secants to the elliptic curve E.

P roof. B is a rational surface in IP^{4}, embedded by the linear system

|H_{B}| = |4l − ∑_{i=1}^{10} E_{i}|

and, by construction, it contains E and it intersects the rational scroll T along a curve G ≡ H_{B} − K_{B}, of degree 8 and arithmetic genus 5. In terms of the embedding we have

G ≡ 7l − ∑_{i=1}^{10} 2E_{i},

and we may write

E ≡ al − ∑_{i=1}^{10} a_{i}E_{i},

where a ∈ IN^{∗} and a_{i} ∈ IN, for i = \overline{1, 10}. Now G ∩ E = T ∩ E = \{p_{1}, …, p_{5}\}, so

GE = 7a − 2 ∑_{i=1}^{10} a_{i} = 5,

while

H_{B}E = 4a − ∑_{i=1}^{10} a_{i} = 5,

and

E^{2} + K_{B}E = a^{2} − 3a + ∑_{i=1}^{10} a_{i} − ∑_{i=1}^{10} a_{i}^{2} = 0.

We obtain a = 5, ∑_{i=1}^{10} a_{i} = 15 and

∑_{i=1}^{10} \frac{a_{i}(a_{i} − 1)}{2} = 5.

On the other hand, EE_{i} = a_{i} ≤ 2 since E has no trisecants, so the last relation implies that exactly five of the a_{i}’s, say a_{1}, a_{2}, …, a_{5}, are equal to 2, while the rest are equal to 1. Therefore

E ≡ 5l − ∑_{i=1}^{5} 2E_{i} − ∑_{j=6}^{10} E_{j},

and so the residual of G + E in the intersection of B with a general cubic hypersurface of H^{0}(I_{E∪T} (3)) is the unique curve in the linear system

|3H_{B} − E − G| = | ∑_{i=1}^{5} E_{i}|.  ⊓ ⊔

Lemma 3.11. For general choices, B intersects the elliptic quintic scroll Q along E and a zero-dimensional smooth scheme Σ of length 5, outside the elliptic curve.

P roof. T meets Q in 15 points, five of which, by construction, lie on E. Let now ρ : Q → E/τ denote the morphism which defines the ruling of Q and let C_{0} denote the class of a section of Q with minimal self-intersection C_{0}^{2} = 1. Then E ≡ 2C_{0} − αF, where αF is the pullback by ρ of a divisor of degree one on E (cf. [HV]), while H_{Q} ≡ C_{0} + βF, with βF the pullback of a divisor of degree two. Thus, residual to E in the complete intersection of Q with a cubic hypersurface containing E, there is a curve numerically equivalent to C_{0} + γF, with γF the pullback of a divisor of degree 7 on E, and the lemma follows since (C_{0} + γF)^{2} = 14 + 1 = 15.⊓ ⊔

Lemma 3.12. There exists a unique quintic hypersurface V containing both the elliptic scroll Q and the Bordiga surface B.

P roof. We use the residual exact sequences

0 ⟶ I_{Q}(2) ⟶^{f} I_{Q∪B}(5) ⟶ I_{B∪M,V} (5) ⟶ 0

and

0 ⟶ I_{B}(2) ⟶^{g} I_{B∪M}(5) ⟶ I_{E∪M∪L,W} (5) ⟶ 0,

where V = {f = 0} is a general cubic hypersurface containing B ∪ T, W = {g = 0} is a general cubic hypersurface containing the elliptic scroll Q, while M ∼ 3H_{Q} − E and L ≡ 3H_{B} − E are the corresponding residual intersections. Now h^{1}(I_{Q}(2)) = h^{1}(I_{B}(2)) = h^{0}(I_{Q}(2)) = h^{0}(I_{B}(2)) = 0, and it is easily checked that h^{0}(I_{E∪L∪M}(5)) = 2 · 15 + 1 = 31, when the points p_{1}, …, p_{5} are chosen such that the sum p_{1} + p_{2} + … + p_{5} is a non-trivial 2-torsion point on E.⊓ ⊔

We link now B in the complete intersection of V and a general cubic hypersurface in H^{0}(I_{B}(3)) to a surface Z ⊂ IP^{4} of degree 9, sectional genus 9. For general choices, the surface Z is smooth and meets the elliptic quintic scroll along a smooth curve M of degree 10 and genus 1 in the linear system |3H_{Q} − E|, which passes through the five points of the scheme Σ. Therefore the scheme theoretic union Y = Q∪Z is a local complete intersection scheme of degree 14 and sectional arithmetic genus 19.

Lemma 3.13. h^{0}(I_{Y} (4)) = 0 and h^{0}(I_{Y} (5)) = 2.

P roof. One argues as in the proof of lemma 3.12.⊓ ⊔

Therefore, we can link the configuration Y in the complete intersection of the above two quintic hypersurfaces to a surface S ⊂ IP^{4} of degree 11 and sectional genus 10. Moreover, as we’ve checked in an example, the surface S is smooth for a general choice of the initial data.

Lemma 3.14. The surface S has infinitely many 6-secant lines, so Le Barz’s formula doesn’t apply. Namely, the 6-secants to S are precisely the lines in the ruling of Q.

P roof. By (0.24), the scheme S ∪ Z intersects the scroll Q along a curve numerically equivalent to 5H_{Q}−K_{Q}, whence S meets Q along a curve D ≡ 5H_{Q}−K_{Q}−M ≡ 6C_{0}+δF, where δ is the pullback of a divisor of degree 2 on E/τ. Since DF = 6, the rulings of the scroll Q are 6-secant lines to S. There are no other 6-secants to S because Z is a minimal surface of general type, which intersects S along a curve D′ ≡ 5H_{Z} − K_{Z} − M, so for a line L on Z one would have D′L = (5H_{Z} − K_{Z} − M)L ≤ 5H_{Z}L = 5. ⊓ ⊔

Lemma 3.15. E_{1}, E_{2}, …, E_{5} are the exceptional lines of S.

P roof. By lemma 3.10, each line E_{i}, i = \overline{1, 5}, is a secant of E, and thus also of the scroll Q. On another side, B intersects Z along a curve G′ ≡ 3H_{B} −K_{B} ≡ 15l−∑_{i=1}^{10} 4E_{i}, thus a line E_{i} is also a 4-secant to the surface Z in points lying outside the scroll. Altogether, each E_{i}, for i = \overline{1, 5}, is a 6-secant to the configuration Y, and thus it lies on any quintic hypersurface containing Y. In particular, the lines E_{i}, i = \overline{1, 5}, lie on S and it remains to show they are exceptional. The Bordiga surface B is defined by the maximal minors of a 4 × 3 matrix with linear entries, thus Z and also S ∪ Q, being linked with B, are projectively Cohen-Macaulay schemes. In particular the ideal sheaf of S ∪ Q has a minimal free resolution of type

0 ← I_{S∪Q} ⟵ 5O(−5) 3O(−6)
↖ ⊕ ← 0
O(−7)

Dualizing, we see that the minors of the 5 × 3 submatrix with linear entries in the above resolution define the zero-set of a section in H^{0}(ω_{S∪Q}(−2)). It follows that these minors cut out on S an effective divisor in the class |K + D − 2H|, which contains ∑_{i=1}^{5} E_{i} and thus must be equal to, by degree reasons. Now DE_{i} = 2, because the lines E_{i} are only simple secants to the scroll Q, and since

DE_{i} = (2H − K + ∑_{i=1}^{5} E_{i})E_{i} = 2HE_{i} − 2p_{a}(E_{i}) + 2 + 2E_{i}^{2},

we obtain E_{i}^{2} = −1, for i = \overline{1, 5}, and the lemma is proved.⊓ ⊔

Corollary 3.16. The surface S is a non-minimal Enriques surface, embedded by

H = H_{min} − 2E_{0} − ∑_{i=1}^{5} E_{i}.

P roof. In view of proposition 3.6, it is enough to show that h^{0}(O_{S}(2K)) > 0, or equivalently that h^{0}(O_{S}(2K − 2 ∑_{i=1}^{5} E_{i})) = h^{0}(O_{S}(4H − 2D)) ≠ 0. If D′ denotes as in (3.14) the intersection curve of S and Z, then the cohomology of the exact sequence

0 ⟶ O_{S}(K − D − H) ⟶ O_{S}(4H − 2D) ⟶ O_{D′}(4H − 2D) ⟶ 0

together with the vanishing of h^{1}(O_{S}(K − D − H)) = h^{1}(O_{S}(D + H)) = 0, since D +H is ample, imply that it is enough to check that h^{0}(O_{D′}(4H − 2D)) ≠ 0. But, by construction O_{D′}(D) = O_{D′}(M), so we need in fact to show that h^{0}(O_{D′}(4H − 2M)) ≠ 0, and we check this on Z, where since D′ + M ∼ 5H_{Z} − K_{Z} this is equivalent to showing that h^{1}(O_{D′}(H_{Z} + M)) ≠ 0. If N ∼ 3H_{Z} − K_{Z} denotes the intersection of Z and B, then lemma 3.11 and the remarks after lemma 3.12 imply on Z that MN = ME + 5 = 20, where the last intersection number is computed on the scroll Q. Thus M^{2} = −MK_{Z} = −10 and a similar argument shows in fact that O_{M}(H_{Z} − K_{Z}) = O_{M}. In particular O_{M}(H_{Z} + M) = O_{M}(K_{Z} + M + H_{Z} − K_{Z}) = ω_{M} = O_{M} and thus h^{1}(O_{Z}(H_{Z} + M)) = h^{1}(O_{M}(H_{Z} + M)) = h^{1}(O_{M}) = 1 since Z is projectively Cohen-Macaulay. It follows from the cohomology of the exact sequence

0 ⟶ O_{Z}(M + H_{Z} − D′) ⟶ O_{Z}(M + H_{Z}) ⟶ O_{D′}(M + H_{Z}) ⟶ 0

that all we need to check is that the map induced by the multiplication with the equation of the divisor D′

H^{1}(O_{Z}(M + H_{Z} − D′))  ⟶^{D′} H^{1}(O_{Z}(M + H_{Z})) = C I

is trivial. This is a consequence of the commutativity of the diagram

H^{1}(O_{Z}(M + H_{Z} − D′)) ⟶^{D′} H^{1}(O_{Z}(M + H_{Z})) ⟶^{K_{Z}+M} H^{1}(O_{Z}(2M + H_{Z} + K_{Z}))
↓^{K_{Z}+D′}  ↓≅
H^{1}(O_{Z}(M + H_{Z} + K_{Z})) ⟶^{restr.} H^{1}(O_{M}(M + H_{Z} + K_{Z})) ⟶^{0}  H^{1}(O_{M})
⊓ ⊔

As a remark to the above liaison construction, we sketch in the sequel an alternative description of the H^{1}-module of these Enriques surfaces. We need first some extra cohomological information:

Lemma 3.17. h^{0}(O_{S}(2H − ∑_{i=1}^{5} E_{i})) = 1.

P roof. Since, by (1.6), h^{0}(I_{S}(2)) = 0, it is enough to show on the Bordiga surface B that h^{0}(O_{B}(2H_{B} − ∑_{i=1}^{5} E_{i})) = 1. By lemma 3.10 this is equivalent to showing that h^{0}(O_{B}(E − K_{B})) = 1. Now we observe that h^{0}(O_{B}(−K_{B})) = 0, since otherwise E ≡ (2l− ∑_{i=1}^{5} E_{i})+(3l−∑_{i=1}^{10} E_{i}) would be reducible. Hence, by Riemann-Roch, h^{1}(O_{B}(−K_{B})) = 0 and the cohomology of the exact sequence

0 ⟶ O_{B}(−K_{B}) ⟶ O_{B}(E − K_{B}) ⟶ O_{E}(E − K_{B}) ⟶ 0

yields h^{0}(O_{B}(E − K_{B})) = h^{0}(O_{E}(E − K_{B})). Let now as in (3.10) G ≡ H_{B} − K_{B} be the intersection of T and B. Then, by construction G ∩ E = {p_{1}, p_{2}, …, p_{5}}, while O_{E}(K_{B} + E) = ω_{E} = O_{E} by adjunction, and O_{E}(K_{B} +H_{B}) = O_{E}(2H_{B} −G) = O_{E}(p_{1}+p_{2}+…+p_{5}) since 2(p_{1} + p_{2} + … + p_{5}) = 2τ = 0 in the group structure of E. Therefore O_{E}(E − K_{B}) = O_{E}(G + (K_{B} + E) − (K_{B} + H_{B})) = O_{E} and the claim of the lemma follows.⊓ ⊔

Lemma 3.18. h^{1}(O_{S}(1)) = 2 and h^{1}(O_{S}(k)) = 0 for k ≥ 2.

P roof. The first assertion follows from Riemann-Roch and Severi’s theorem. To show that h^{1}(O_{S}(2)) = 0 we consider the cohomology of the exact sequence

0 ⟶ O_{S}(2H − ∑_{i=1}^{5} E_{i}) ⟶ O_{S}(2H) ⟶ O_{∑_{i=1}^{5} E_{i}}(2H) ⟶ 0.

Now h^{1}(O_{∑_{i=1}^{5} E_{i}}(2H)) = ∑_{i=1}^{5} h^{1}(O_{IP}^{1}(2)) = 0, χ(O_{S}(2H − ∑_{i=1}^{5} E_{i})) = 1 by Riemann-

Roch, while h^{2}(O_{S}(2H − ∑_{i=1}^{5} E_{i})) = h^{0}(O_{S}(K + ∑_{i=1}^{5} E_{i} − 2H)) = 0 because H(K + ∑_{i=1}^{5} E_{i} − 2H) = −10 < 0, so lemma 3.17 implies that h^{1}(O_{S}(2H − ∑_{i=1}^{5} E_{i})) = 0 and thus also the desired vanishing. The rest of the claim follows by induction on k since O_{H}(k), for k ≥ 3, is non-special for the general hyperplane section of S.⊓ ⊔

We describe now the multiplicative structure of the dual of the H^{1}-module of I_{S}. Let V ⊂ ǏP^{4} denote the locus of hyperplanes, where the multiplication map H^{1}(I_{S}(3))⟶^{H}H^{1}(I_{S}(4)) has not maximal rank.

Lemma 3.19. V ⊂ ǏP^{4} is a quintic hypersurface; namely the variety of trisecant lines to an elliptic quintic scroll in ǏP^{4}.

P roof. From the liaison exact sequences it follows that h^{0}(I_{S}(4)) = 0, thus Riemann-Roch and lemma 3.18 yield h^{1}(I_{S}(3)) = h^{1}(I_{S}(4)) = 5 and consequently V is a quintic hypersurface in ǏP^{4}. Now for the general plane π containing a ruling f of the elliptic scroll Q one has h^{1}(I_{π∩S}(4)) > 0, and thus also h^{1}(I_{H}(4)) > 0 for the general hyperplane through f because h^{2}(I_{H}(3)) = h^{1}(O_{H}(3)) = h^{1}(O_{S}(3)) = 0. In conclusion, V is the dual variety of the elliptic scroll Q and therefore it is the variety of the trisecant lines to an elliptic quintic scroll in ǏP^{4} (cf. [Seg]).⊓ ⊔

The equation of the variety of trisecant lines to an elliptic quintic scroll was determined in [ADHPR]. Namely, in suitable coordinates y_{0} …, y_{4} of ǏP^{4}, one has

V = {y | det (y_{3i+3j}z_{i−j})_{i,j∈ℤ_{5}} = 0},

for some parameter z ∈ IP^{4}, defined by z_{0} = 2, z_{1} = a, z_{2} = \frac{1}{a} and z_{i} = z_{−i}. (See also chapter 7 for more information on the Moore matrices {y_{3i+3j}z_{i−j}}).

Remark 3.20. We have not been able to give examples, or give proof they do not exist, of rational surfaces as in the proposition 3.6.

Proposition 3.21. If S is a smooth surface of degree 11, with π = 10, χ = 2 in IP^{4}, then S is a regular, minimal proper elliptic surface and its elliptic fibration is given by |2K|.

P roof. Since p_{g} ≥ 1 and HK = 7 > 0, S is either proper elliptic or of general type. Assume that S is a surface of general type and let S_{0} denote the minimal model of S and K_{0} denote its canonical divisor. If S has at least two (−1) curves, then HK_{0} ≤ 5 and p_{a}(K_{0}) = K_{0}^{2} + 1 ≥ 3. Thus, necessarily HK_{0} = 5, p_{a}(K_{0}) = 3, whereas, by (0.34), K_{0} decomposes as the union of a plane quartic curve A and a line L which meet in one point. Now K_{0}^{2} = 2 and A^{2} ≤ 1 by the index theorem, so L^{2} ≥ −1, which means that L is a (−1) line on S_{0}, a contradiction. Therefore S has only one (−1) line or conic E because otherwise HK_{0} ≤ 4, while p_{a}(K_{0}) = K_{0}^{2} + 1 = 2, which is impossible by (0.34). If E is a (−1) conic, then HK_{0} = 5, p_{a}(K_{0}) = 2, so K_{0} spans only a hyperplane in IP^{4}, unless it splits as the union of a plane quartic A and a line L which don’t meet. Now A^{2} + L^{2} = 1 and A^{2} ≤ 1 by Hodge index, so this case leads as above to a contradiction. Thus K_{0} spans only a hyperplane and hence there is a residual curve C ∈ |H − K_{0}|. This curve has degree 6 and arithmetic genus p_{a}(C) = 5, so by (0.34) it is the union of a plane quintic Q and a line L which don’t meet. But C^{2} = 2 and A^{2} ≤ 2 by the index theorem, hence L^{2} ≥ 0, which is again absurd. It follows that K_{0}^{2} = 1 and S has only one exceptional line E. Moreover S is regular, otherwise K_{0}^{2} ≥ 2p_{g} ≥ 4 by [Deb, Th.6.1] and S would have at least four (−1) curves. In particular we obtain that p_{g} = 1. For such a surface, |2K_{0}| has no base points (see [Ca]) and Φ = Φ_{|2K_{0}|} : S_{0} → IP^{2} is a morphism of degree 4. Let Φ̃ : S → IP^{2} denote the composition of Φ with the blowing-down mapping S → S_{0}. The restriction of Φ̃ to a hyperplane section H of S has degree 4 on the image because through four given points of S goes always at least one hyperplane section H. Therefore Φ̃(H) ⊂ IP^{2} is a cubic curve and hence |6K_{0} − H| ≠ ∅. Moreover, since an irreducible cubic has at most one double point, there exists at most one (−2) curve F on S_{0} of degree ≤ 2 in IP^{4}. On the other hand K_{0}(6K_{0} − H) = 0, so a divisor D ∈ |6K_{0} − H| is a sum of E and F with certain multiplicities. But E(6K_{0} −H) = −1 and F (6K_{0} −H) = −F H ≥ −2, so the multiplicities are 1 or 2, while H(6K_{0} − H) = 25, which is again a contradiction. For S an elliptic surface, the assertions of the proposition follow from the double point formula and Kodaira’s formula [BPV] for the canonical class of an elliptic fibration.⊓ ⊔

Remark 3.22. We have not been able to give examples, or give proof they do not exist, of surfaces as in the above proposition.

IV. Surfaces with d = 11, π = 11

The double point formula gives K^{2} = 6χ−17, so Hodge index implies χ ≤ 7, or equivalently χ ≤ 4. For κ(S) ≥ 0, since K is pseudo-effective, (0.16) yields then χ ∈ {2, 3, 4}. In case κ(S) = −∞, lemma 0.20 shows that the only possible value is χ = 1.

A first example of a smooth rational surface with these invariants was constructed in [DES]. We recall here the argument and provide constructions for two further examples.

Proposition 3.23. There exist smooth rational surfaces S ⊂ IP^{4}, with d = 11, π = 11 and embedded by one of the following linear systems

a)  H = 11l − 5E_{0} − ∑_{i=1}^{6} 3E_{i} − ∑_{j=7}^{12} 2E_{j} − ∑_{k=13}^{19} E_{k}

b)  H = 10l − 4E_{0} − ∑_{i=1}^{3} 3E_{i} − ∑_{j=4}^{13} 2E_{j} − ∑_{k=14}^{19} E_{k}

c)  H = 13l − 5E_{0} − ∑_{i=1}^{7} 4E_{i} − ∑_{j=8}^{10} 2E_{j} − ∑_{k=11}^{19} E_{k}.

P roof. We construct the surfaces via the Eagon-Northcott method, so we need first to determine cohomology:

Lemma 3.24. If S ⊂ IP^{4} is a smooth surface with d = 11, π = 11, χ = 1, then h^{1}(O_{S}(1)) = 3, h^{1}(O_{S}(2)) = 1 and h^{1}(O_{S}(k)) = 0, for all k ≥ 3.

P roof. Severi’s theorem gives h^{1}(O_{S}(1)) = 3 and, since h^{1}(O_{H}(2)) = h^{0}(O_{H}(K − H)) = 0 for irreducible hyperplane sections, we deduce that a = h^{1}(O_{S}(2)) ≤ 3. On the other side, Riemann-Roch and (1.6) yield h^{1}(O_{S}(2)) ≥ 1. Consider now a hyperplane section H for which h^{1}(O_{H}(2H)) ≥ 1. By lemma 0.37, we can find a decomposition H = C_{1} + C_{2}, with C_{1} ≥ 0, C_{2} > 0 and C_{1}C_{2} ≤ (K − H)C_{2}. We deduce

2 deg C_{2} ≤ 2p_{a}(C_{2}) − 2  (∗)

and the equivalent inequality

2 deg C_{1} ≥ 2 + (K + H)C_{1} + C_{1}C_{2}.  (∗∗)

By lemma 0.36, any plane curve on S has degree at most five, and the bounds in (0.34) show that the inequality (∗) is impossible, unless C_{2} is a plane quintic, or deg C_{2} ≥ 8. On the other side, since (K + H)C_{1} ≥ 0, it follows from (∗∗) that deg C_{1} ≥ 2. In fact, in case deg C_{1} = 2, the 2-connectedness of the hyperplane sections [VdV] implies that (K +H)C_{1} = 0, and thus, by (0.13), that C_{1} is the union of two (−1) lines (which can also coincide), while C_{1}^{2} = HC_{1} − C_{1}C_{2} = 0, and this is a contradiction. In case deg C_{1} = 3, we obtain again equality in (∗), and namely deg C_{2} = 8, p_{a}(C_{2}) = 9. Furthermore C_{1}^{2} + 1 = (K + H)C_{1} ≥ 0 and thus C_{1}^{2} ≥ −1, whereas KC_{1} = 1 − C_{1}C_{2} ≤ −2 by formula (0.26) and C_{1}^{2} ≤ 0 by Hodge index. It follows that C_{1}^{2} = 0, so h^{0}(O_{S}(C_{1})) ≥ χ(O_{S}(C_{1})) ≥ 2 and thus C_{2} would be a plane curve, which is absurd. Therefore, if a ≥ 2, it follows that S would have a pencil of plane quintics. But this is impossible by Lemma 3.25. S has only finitely many plane curves.

P roof. Since any pencil of plane curves on S is linear, the residual curve in a hyperplane section containing a general element of the family is again a plane curve. But then the hyperplane section is contained in a quadric and this contradicts Severi’s theorem.⊓ ⊔

Assume now that h^{1}(O_{S}(3)) ≥ 1. Then h^{1}(O_{H}(3)) ≥ 1 for at least a web of hyperplane sections H. But the general element in the web is smooth while O_{H}(3H) is non-special since 2π − 2 < 33, and this is a contradiction.⊓ ⊔

Corollary 3.26. Let π ⊂ IP^{4} be the base locus plane of the pencil of hyperplanes for which h^{1}(O_{H}(2)) ≠ 0. Then π ∩ S contains as component a plane quintic curve. Residual to it, there is a pencil of smooth irreducible curves of degree 6, and arithmetic genus ≤ 1. Moreover, when the genus is one, the plane π cuts S along a plane quintic curve with an embedded point, and thus the surface has infinitely many 6-secant lines in this case.

P roof. It follows from the proof of the above lemma that π ∩ S contains as component a plane quintic curve C. Let |D| = |H − C| denote the residual pencil. Then the genus formula (0.2) gives p_{a}(D) = D^{2}, while Hodge index yields the bound D^{2} ≤ 3. In particular, the general member in the pencil |H − C| is smooth. We show in the sequel that the cases D^{2} ∈ {2, 3} cannot occur.

Assume first that D^{2} = 3. Then CD = 3, so the restriction of |D| defines a linear system of degree 3 on C. Since C is not hyperelliptic, it follows that |D|_{|C} is a g_{3}^{1}, and the genus formula tells us that C has, as its sole singularity, a node or an ordinary cusp, the g_{3}^{1} being cut out by the pencil of lines through this double point. In particular h^{0}(O_{C}(H − D)) ≠ 0, while the cohomology of the exact sequence

0 ⟶ I_{H}(1) ⟶ I_{D}(1) ⟶ O_{C}(H − D) ⟶ 0

gives h^{0}(O_{C}(H − D)) = 0 since, obviously, h^{0}(I_{D}(1)) = 0 and h^{1}(I_{H}(1)) = 0 by Severi’s theorem, hence we’ve obtained a contradiction in this case.

If D^{2} = 2, then CD = 4 so either |D| cuts out a g_{4}^{1} on C, or |D|_{|C} has a base point P on C, where D is tangent to the plane π. In the last case, C is again singular and the free part of |D|_{|C} is cut out by the pencil of lines through the double point. One argues now as in the previous case to obtain a contradiction.

Finally, when D^{2} = 1 we obtain CD = 5 and then |D|_{|C} must have a base point, otherwise, since a g_{5}^{1} is cut out on C by the pencil of lines through a point outside the curve, the previous arguments would lead again to a contradiction. The claim of the corollary follows now easily.⊓ ⊔

For construction purposes we assume h^{0}(I_{S}(4)) = 0, thus in other words that the cohomology diagram is minimal

i ↑

3  1  h^{i}(I_{S}(p))
2  1

⟶

p

Then Beilinson’s spectral sequence yields for any such surface S a presentation

0 ⟶ 3Ω^{3}(3) ⟶^{φ} G ⟶ I_{S}(4) ⟶ 0,

where G is the kernel of a morphism

0 ⟶ G ⟶ Ω^{2}(2) ⊕ 2Ω^{1}(1) ⟶^{ψ} O ⟶ 0.

Identifying IP^{4} = IP(V), the morphism ψ is induced by a triple (ψ_{1}, ψ_{21}, ψ_{22}), with ψ_{1} ∈ Λ^{2} V and ψ_{21}, ψ_{22} ∈ V. It is easily seen that there are precisely four different choices which induce surjective ψ’s:

1) (ψ_{1}, ψ_{21}, ψ_{22}) ∈ Λ^{2} V × V × V generic,

2) ψ_{1} = 0 and (ψ_{21}, ψ_{22}) ∈ V × V generic, 3a) ψ_{1} ∈ Λ^{2} V a decomposable tensor, ψ_{21} ∈ V generic, and ψ_{22} = 0, or 3b) ψ_{1} ∈ Λ^{2} V indecomposable, ψ_{21} ∈ V generic, and ψ_{22} = 0. Corresponding, we obtain four families of vector bundles G with minimal free resolutions of type

0 ← G_{1} ← 25O(−1) 15O(−2)  3O(−3)
↖ ⊕ ⟵ ⊕
2O(−3)  3O(−4) ↖ O(−5) ← 0,
25O(−1)  16O(−2)  4O(−3)
0 ← G_{2} ← ⊕ ⟵ ⊕ ⟵ ⊕
O(−2)  3O(−3)  3O(−4) ↖ O(−5) ← 0,
25O(−1)  17O(−2)  6O(−3)  O(−4)
0 ← G_{3a} ← ⊕ ⟵ ⊕ ⟵ ⊕ ⟵ ⊕ ← 0,
2O(−2) 5O(−3) 4O(−4) O(−5)

and respectively

0 ← G_{3b} ← 25O(−1) 15O(−2)  6O(−3)  O(−4)
↖ ⊕ ⟵ ⊕ ⟵ ⊕ ← 0.
5O(−3)  4O(−4)  O(−5)

However, the choice 3b) doesn’t lead to any surface. Namely, in this case, G_{3b} = Ω^{1}(1) ⊕ H, where H = ker(Ω^{2}(2) ⊕ Ω^{1}(1) ⟶^{(ψ_{1},ψ_{21})}O), and there is no sheaf monomorphism φ ∈ Hom(3Ω^{3}(3), G_{3b}) since in the minimal free resolution of H

0 ← H ← 15O(−1) 5O(−2)  O(−3)
↖ ⊕ ⟵ ⊕
5O(−3)  4O(−4) ↖ O(−5) ← 0,

the 5 linear syzygies of the 15 generators involve only 10 of these, and thus the component of φ going to H factorizes through a rank 4 sheaf. In all other three cases, by checking on a computer via [Mac], we find that a general φ ∈ Hom(3Ω^{3}(3), G_{i}), where i ∈ {1, 2, 3a}, defines a smooth surface S_{i} ⊂ IP^{4} with the desired invariants. In case 1) a mapping cone of the resolutions of F = 3Ω^{3}(3) and G_{1} provides the minimal free resolution

0 ← I_{S_{1}} ← 10O(−5) 12O(−6)  3O(−7)
↖ ⊕ ⟵ ⊕
2O(−7)  3O(−8) ↖ O(−9) ← 0

and hence, in particular, we see that S_{1} is cut out by quintic hypersurfaces and thus has no 6-secant lines. Let now Σ_{1} denote the image of S_{1} under the adjunction map, and Σ_{2} denote the image of Σ_{1} under the adjunction map defined by |H_{1} + K_{1}|. Le Barz’s formula gives N_{6} = 7 so there are 7 exceptional lines on S_{1}, and we obtain from (0.13) the following invariants

Σ_{1} ⊂ IP^{10} H_{1}^{2} = 18  H_{1}K_{1} = −2  K_{1}^{2} = −4  π_{1} = 9
Σ_{2} ⊂ IP^{8} H_{2}^{2} = 10  H_{2}K_{2} = −6  K_{2}^{2} = −4 + b  π_{2} = 3,

where b is the number of (−1)-conics on S_{1}. Since (H_{2} + K_{2})^{2} = b − 6 and the adjunction maps Σ_{2} to IP^{2}, theorem 0.13 implies b ∈ {6, 7}. If b = 6, then Σ_{2} is a conic bundle Σ_{2} → IP^{1} with 6 singular fibres. Therefore S_{1} = IF_{e}(p_{1}, p_{2}, …, p_{19}) is a blown-up Hirzebruch surface and we can recover through adjunction its embedding

H = 6C_{0} + (3e + 8)F − ∑_{i=1}^{6} 3E_{i} − ∑_{j=7}^{12} 2E_{j} − ∑_{k=13}^{19} E_{k},

where C_{0} denotes a section with self-intersection −e ≤ 0 and F a fibre of the ruling. Now HC_{0} ≥ 1, so e ∈ {0, 1, 2} and we may choose (via elementary transformations) IP^{2} as minimal model, whence

(1)  H = 11l − 5E_{0} − ∑_{i=1}^{6} 3E_{i} − ∑_{j=7}^{12} 2E_{j} − ∑_{k=13}^{19} E_{k}.

If b = 7, then Σ_{2} is IP^{2} blown-up in 6 points and we obtain

(2)  H = 10l − ∑_{i=1}^{6} 3E_{i} − ∑_{j=7}^{13} 2E_{j} − ∑_{k=14}^{20} E_{k}.

To exclude the linear system (2) we’ll use the information provided by corollary 3.26. Namely, since S_{1} has no 6-secant lines, it follows that the plane distinguished by the H^{2}-cohomology meets the surface along a plane quintic curve C, whose residual pencil |D| = |H − C| is base point free and consists of rational curves of degree 6. Let now D_{1} be the image of a general element in this pencil through the adjunction map on S_{1}. We have H_{1}D_{1} = (H + K)D = 4, and the cohomology of the exact sequence

0 ⟶ O_{S_{1}}(K) ⟶ O_{S_{1}}(K + C) ⟶ O_{C}(K + C) ⟶ 0

gives h^{0}(O_{Σ_{1}}(H_{1} − D_{1})) = h^{0}(O_{S_{1}}(H + K − D)) = h^{0}(O_{S_{1}}(K + C)) = h^{0}(O_{C}(K + C)) = 6. Therefore D_{1} ⊂ IP^{10} has degree 4 and spans a whole IP^{4}, thus it is a rational normal curve in the spanned linear subspace. The image of D_{1} through the map defined by |H_{1} + K_{1}| is then a curve of degree D_{1}(H_{1} + K_{1}) = 2 − D_{1}^{2} ≤ 2, thus Σ_{2} ⊂ IP^{8} is a conic bundle. It follows that in this case S_{1} is a rational surface of type a), i.e., embedded by a linear system of type

H = 11l − 5E_{0} − ∑_{i=1}^{6} 3E_{i} − ∑_{j=7}^{12} 2E_{j} − ∑_{k=13}^{19} E_{k}.

In case 2) the ideal sheaf of S_{2} has syzygies

\begin{aligned}& & & & 10O(−5) & & 13O(−6) & & 4O(−7) \\ 0 & ← & I_{S_{2}} & ← & ⊕ & ⟵ & ⊕ & ⟵ & ⊕ \\ & & & & O(−6) & & 3O(−7) & & 3O(−8) & ↖ & O(−9) & ← 0 \end{aligned} .

This family was first constructed in [DES]. We recall in the sequel from [DES] the description of the embedding in IP^{4}. First observe that the H^{1}-module distinguishes a line L such that I_{S_{2}}/(I_{S_{2}})_{≤5} has support on it. Thus L is the unique 6-secant, and by Le Barz’s formula there are 6 exceptional lines on S_{2}. Denoting as above by Σ_{1} and Σ_{2} the first and the second adjoint surfaces of S_{2} respectively, we obtain via (0.13) the following invariants

Σ_{1} ⊂ IP^{10} H_{1}^{2} = 18  H_{1}K_{1} = −2  K_{1}^{2} = −5  π_{1} = 9
Σ_{2} ⊂ IP^{8} H_{2}^{2} = 9  H_{2}K_{2} = −7  K_{2}^{2} = −5 + b  π_{2} = 2,

where b is the number of (−1)-conics on S_{2}. It follows that (H_{2} + K_{2})^{2} = 0, thus b = 10, and the next adjunction morphism presents Σ_{2} as a conic bundle with 3 singular fibres. Therefore S_{2} = IF_{e}(p_{1}, p_{2}, …, p_{19}) is a blown-up Hirzebruch surface and

H = 6C_{0} + (3e + 7)F − ∑_{i=1}^{3} 3E_{i} − ∑_{j=4}^{13} 2E_{j} − ∑_{k=14}^{19} E_{k}.

HC_{0} ≥ 1, so e ∈ {0, 1, 2} and we may choose IP^{2} as minimal model. In particular

H = 10l − 4E_{0} − ∑_{i=1}^{3} 3E_{i} − ∑_{j=4}^{13} 2E_{j} − ∑_{k=14}^{19} E_{k}

and S_{2} is a surface of type b). We remark that in this case the pencil described in corollary 3.26 is the pullback of the conic fibration on Σ_{2}. Finally, in case 3) we obtain a surface S_{3} with syzygies

10O(−5)  14O(−6)  6O(−7)  O(−8)
0 ← I_{S_{3}} ← ⊕ ⟵ ⊕ ⟵ ⊕ ⟵ ⊕ ⟵ 0
2O(−6)  5O(−7)  4O(−8)  O(−9)

The quintics in the ideal cut out the surface S_{3} plus the plane π distinguished by the H^{2}-cohomology, namely π = span_{k}(e_{0}, e_{1}, e_{2}), if V = span_{k}(e_{0}, …, e_{4}) and ψ_{1} = e_{1} ∧ e_{2}, ψ_{12} = e_{0} in the above construction. π meets S_{3} along a plane quintic curve C with an embedded point P = IP(ke_{0}), so the surface has infinitely many 6-secant lines. The residual pencil |D| = |H − C| has only one base point at P and consists of elliptic curves. One checks in an example that S_{3} has 9 exceptional lines. The adjunction process produces surfaces with the following invariants

Σ_{1} ⊂ IP^{10} H_{1}^{2} = 18  H_{1}K_{1} = −2  K_{1}^{2} = −2  π_{1} = 9
Σ_{2} ⊂ IP^{8} H_{2}^{2} = 12  H_{2}K_{2} = −4  K_{2}^{2} = −2 + b  π_{2} = 5,

where b is the number of (−1)-conics on S_{3}. Now Hodge index gives b ≤ 3, so the next adjoint surface

Σ_{3} ⊂ IP^{4} H_{3}^{2} = 2 + b  H_{3}K_{3} = −6 + b  K_{3}^{2} = −2 + b + c  π_{3} = b − 1

where c is the number of (−1)-twisted cubics on S_{3}, will be a surface of degree at most 5. Let again D_{1} be the image on Σ_{1} of a general element in the pencil |D|. Then H_{1}D_{1} = 5 and as above D_{1} spans a IP^{4}, thus p_{a}(D_{1}) ≤ 1. If p_{a}(D_{1}) = 0, then |D_{1}| has two base points, hence D_{1}^{2} = 2, K_{1}D_{1} = −4 and thus |D_{1}| would map through the second adjunction on a pencil of lines, absurd. Therefore |D_{1}| is a pencil of elliptic curves of degree 5 with only one base point. Similar arguments show in fact that |D| maps via adjunction on a pencil of plane cubic curves on Σ_{3}. Since the rational cubic scroll and the Veronese surface do not possess such pencils, it follows that b = 3 and Σ_{3} is a Castelnuovo surface (see e.g. [Ok2]), i.e., a surface in IP^{4} embedded by

H_{Σ_{3}} = 4l − 2E_{0} − ∑_{i=1}^{7} E_{i}.

Going back through the adjunction process we recuperate the initial embedding in IP^{4}

H = 13l − 5E_{0} − ∑_{i=1}^{7} 4E_{i} − ∑_{j=8}^{10} 2E_{j} − ∑_{k=11}^{19} E_{k},

thus S_{3} is a surface of type c) this time. We remark that in this case |D| = |3l−∑_{i=0}^{7} E_{i}|.⊓ ⊔

Remark 3.27. The above families of rational surfaces lie in different components of the Hilbert scheme. The Eagon-Northcott method used in the previous proposition allows to construct also schemes lying at the intersection of two such components. For example, let ρ : IP^{1} × IP^{4} → IP^{4} be the projection onto the second factor and consider the bundle 𝔊 on IP^{1} × IP^{4} defined as

𝔊 = ker(ρ^{∗}Ω^{2}(2) ⊕ 2ρ^{∗}Ω^{1}(1)⟶^{ψ}O),

where ψ = (te_{0} ∧ e_{2}, se_{0}, te_{1}), and s, t denote the coordinates in IP^{1}. Then 𝔊_{(s:1)} is a bundle on IP^{4} of type G_{2}, for all s ≠ 0, while 𝔊_{(0:1)} is of type G_{3a}. The degeneracy locus of a general morphism φ ∈ Hom(ρ^{∗}Ω^{3}(3), 𝔊) defines an irreducible flat family of surfaces of degree 11, π = 11, χ = 1 in IP^{4}, with general fibre a smooth rational surface of type b), and with the fibre over (0:1) a scheme lying in the intersection of the components of the Hilbert scheme containing rational surfaces of type b) and c), respectively. Explicit computations in one example, via [Mac], show that a general such scheme is the union of the plane span_{k}(e_{0}, e_{1}, e_{2}) and a smooth rational surface of degree 10, sectional genus 8 meeting the plane along a quartic curve.⊓ ⊔

Proposition 3.28. Let S be a smooth surface in IP^{4} with invariants d = 11, π = 11 and χ = 2. Then either

a) S is a blown-up K3 surface, embedded by

H = H_{min} − 5E_{1} − ∑_{i=2}^{5} E_{i},

or

b) S is a blown-up K3 surface, embedded by

H = H_{min} − 4E_{1} − 2E_{2} − ∑_{i=3}^{5} E_{i},

or

c) S is a blown-up K3 surface, embedded by

H = H_{min} − 3E_{1} − ∑_{i=2}^{3} 2E_{i} − ∑_{j=4}^{5} E_{j},

or

d) S is a blown-up K3 surface, embedded by

H = H_{min} − ∑_{i=1}^{4} 2E_{i} − E_{5}.

P roof. We use adjunction to produce a number of possible candidates for the embedding |H| of S in IP^{4}. Let S_{1} denote the image of S under the adjunction map, and S_{2} denote the image of S_{1} under the adjunction map defined by |H_{1} + K_{1}|. We obtain from (0.13) the following invariants

S ⊂ IP^{4}  H^{2} = 11  HK = 9  K^{2} = −5  π = 11
S_{1} ⊂ IP^{11} H_{1}^{2} = 24  H_{1}K_{1} = 4  K_{1}^{2} = −5 + a  π_{1} = 15
S_{2} ⊂ IP^{15} H_{2}^{2} = 27 + a  H_{2}K_{2} = a − 1  K_{2}^{2} = −5 + a + b  π_{2} = a + 14,

where a is the number of (−1)-lines on S and b is the number of (−1)-lines on S_{1}. Since (H_{1}K_{1})^{2} < H_{1}^{2}, lemma 0.18 shows that κ(S) ∈ {0, 1}, and thus S is either a K3 surface or a proper elliptic surface. In particular a ≤ 5 and a + b ≤ 5. On the other hand H_{2}K_{2} = a − 1 ≥ 0, so S has at least one exceptional line. Now, in case S has at most three (−1) lines, then HK_{min} ≤ HK −a−2b−♯(exceptional curves of higher degree) ≤ 2, hence S is necessarily a K3 surface and H, easily reconstructed via the adjunction, is one in the following list of candidates:

1)  H = H_{min} − ∑_{i=1}^{4} 2E_{i} − E_{5}

2)  H = H_{min} − 3E_{1} − ∑_{i=2}^{3} 2E_{i} − ∑_{j=4}^{5} E_{j}

3)  H = H_{min} − ∑_{i=1}^{2} 3E_{i} − ∑_{j=3}^{5} E_{j}

4)  H = H_{min} − 4E_{1} − 2E_{2} − ∑_{i=3}^{5} E_{i}.

When S has four exceptional lines, then K_{1}^{2} = −1 and H_{2}K_{2} = 3. Therefore, either S_{2} is minimal and S is a regular, proper elliptic surface embedded by

5)  H = H_{min} − 2E_{1} − ∑_{i=2}^{5} E_{i}, (elliptic)

or S is a blown-up K3 surface embedded by

6)  H = H_{min} − 5E_{1} − ∑_{i=2}^{5} E_{i}.  (K3)

Finally, if S has five exceptional lines, then S_{1} is minimal, whence S is a proper elliptic surface embedded by

7)  H = H_{min} − ∑_{i=1}^{5} E_{i}. (elliptic)

We go on to study these candidates, excluding all but 1), 2), 4) and 6). First a lemma Lemma 3.29. If S ⊂ IP^{4} is a smooth surface with d = 11, π = 11 and κ(S) ≥ 0, then h^{1}(O_{S}(k)) = 0, for all k ≥ 2.

P roof. We observe that h^{2}(O_{S}(n)) = h^{0}(O_{S}(K − nH)) = 0 for all n ≥ 1 because (K − nH)(K + H) = 6χ − 8 − 20n < 0. Riemann-Roch and Severi’s theorem then give h^{1}(O_{S}(1)) ≤ 2. On the other side h^{1}(O_{H}(2)) = h^{0}(O_{H}(K − H)) = 0 since H(K − H) < 0, thus h^{1}(O_{S}(k)) ≥ h^{1}(O_{S}(k + 1)) for all k ≥ 1, and therefore to prove the lemma it is enough to check that h^{1}(O_{S}(2)) = 0. Assume that h^{1}(O_{S}(2)) > 0. Then the variety V ⊂ ǏP^{4} parametrizing hyperplane sections for which h^{1}(O_{H}(2)) > 0 contains a plane, so there is a line L contained in a net of hyperplanes for which h^{1}(O_{H}(2)) doesn’t vanish. The line L is contained in S and gives rise to a residual exact sequence

0 ⟶ O_{C}(2H − L) ⟶ O_{H}(2H) ⟶ O_{L}(2H) ⟶ 0.

If the general section C ∈ |H − L| is irreducible, then the cohomology of the above exact sequence implies that 2p_{a}(C) − 2 ≥ (H − L)(2H − L), which combined with the genus formula says that L^{2} ≥ −1, whereas L^{2} = −1 because of the assumption on the Kodaira dimension. We obtain p_{a}(C) = 10, and then the cohomology of the exact sequence

0 ⟶ O_{IP}^{1} ⟶ O_{H}(2H) ⟶ O_{C}(2H) ⟶ 0

yields h^{1}(O_{C}(2H)) > 0, which is absurd since C is irreducible and 2HC > 2p_{a}(C)−2 = 18. The case when |H − L| is composed with a pencil can be ruled out as in (3.4). Therefore h^{1}(O_{S}(2)) = 0 and the claim of the lemma follows.⊓ ⊔

From the above lemma, combined with Riemann-Roch and theorems 1.6 and 1.7, we deduce h^{1}(I_{S}(2)) = h^{1}(O_{S}(2)) = 0, and thus h^{0}(I_{H}(3)) = 0 for all hyperplane sections H of S.

In case 3), there exist residual curves D_{i} ∈ |H −E_{i}|, for i = 1, 2, of degree 8 and arithmetic genus 8. Moreover, the cohomology of the exact sequence

0 ⟶ O_{S}(H + E_{i}) ⟶ O_{S}(2H) ⟶ O_{D_{i}}(2H) ⟶ 0

yields h^{1}(O_{D_{i}}(2)) = 0 since h^{2}(O_{S}(H + E_{i})) = h^{0}(O_{S}(K − H − E_{i})) ≤ h^{0}(O_{S}(K − H)) = 0. Thus Riemann-Roch implies that h^{0}(O_{D_{i}}(2)) = 9, and therefore either D_{i} lies on a smooth quadric surface Q_{i}, as a curve of type (3, 5), or D_{i} splits as the union of two plane quartic curves which meet along a scheme of length 3. We sketch in the sequel only the former case, but similar arguments apply to rule out also the second one. Let H_{1} and H_{2} be the hyperplanes spanned by E_{1} and E_{2} respectively, and let π = H_{1} ∩ H_{2} be their intersection plane. Since D_{1}D_{2} = 5, it follows from Bezout’s theorem that the two quadric surfaces share a curve on the plane π. Assume first that Q_{1} and Q_{2} meet both π along the same conic C. Then, since E_{i}D_{j} = 3, for i ≠ j, we deduce that each twisted cubic E_{i} intersects the conic C in a scheme of length 3, and thus either π cuts S along eleven points on a conic, or C is contained in the surface S. In the former case the cohomology of the exact sequences

0 ⟶ I_{H}(1) ⟶ I_{H}(2) ⟶ I_{π∩S}(2) ⟶ 0

0 ⟶ I_{S} ⟶ I_{S}(1) ⟶ I_{H}(1) ⟶ 0

yields h^{1}(I_{S}(1)) = h^{1}(I_{H}(1)) > 0, which contradicts Severi’s theorem. In the last case we may write D_{i} = C + G_{i}, i = 1, 2, with G_{i} curves of type (2, 4) on the quadrics Q_{i}. In this setting we obtain CG_{i} = 6, E_{i}C = 3, E_{i}G_{j} = 1 and thus also G_{1}G_{2} = (H − E_{1} − C)G_{2} = −1, which is absurd for two effective divisors without common components. At last, the case when the two quadrics Q_{i}, i = 1, 2, share only a line in the plane π, can be excluded in a similar way.

In case 5), the canonical divisor K_{min} of the minimal model of S would be a plane cubic curve in IP^{4}. Thus residual to it there is a pencil |D| = |H − K_{min}| of curves of degree 8 and arithmetic genus 8. As above one sees that h^{0}(I_{D}(2)) ≠ 0. But then a hyperplane section of S by a hyperplane containing the plane of K_{min} will be contained in a cubic surface, thus contradicting the fact that h^{0}(I_{S}(3)) = h^{1}(I_{S}(2)) = 0.

In case 7), we obtain HK_{min} = HK_{1} = 4 this time, so the elliptic curve K_{min} spans only a IP^{3}. The residual curve D has degree 7 and arithmetic genus 7, thus splits by (0.34) as D = A + B, where A is a plane quintic curve and B is a conic meeting A along a scheme of length 2. This is a contradiction since, by (0.36), the degree of a plane curve on S is at most 4. We have proved therefore the claim of the proposition.⊓ ⊔

(3.30.) Constructions. We’ll show in the sequel that all the types of surfaces in the above proposition exist. For construction purposes we’ll assume that S lies on no quartic hypersurface, i.e., that the cohomology table of I_{S}() looks like

i ↑

1

2  h^{i}(I_{S}(p))
3  2

⟶ p

Everything is determined by the module structure of the H^{1}-module of the ideal sheaf. Beilinson’s theorem suggests in all cases to take

E = O(−1) ⊕ 2Ω^{3}(3)  and  F = Syz_{1}(H^{1}(I_{S}(∗ + 4)))

and to construct the surface S as the degeneracy locus of a morphism ψ ∈ Hom(E, F). Assuming that the k-dual of H^{1}(I_{S}(∗)) is generated by elements in the first non-zero twist, a minimal free presentation of H^{1}(I_{S}(∗))^{∗} is of type

0 ⟵ H^{1}(I_{S}(∗))^{∗} ⟵ 2R(5) ⟵^{φ} 7R(4) ⊕ aR(3),

where 0 ≤ a ≤ 3 depends on the number of linear syzygies of the linear part of φ. The morphism φ = (φ_{1}, φ_{2}) is given by a 2 × (7 + a) matrix with linear entries in φ_{1} and quadratic in φ_{2}. For a generic choice, φ_{1} has artinian cokernel and we may think of it as containing a 2 × 5 block whose cokernel is supported on 5 points. Assuming that the 5 points are the vertices {p_{i}} of the standard simplex in k^{5} and performing column operations on φ_{1}, we obtain that

φ_{1} = ( \begin{matrix}x_{0} & x_{1} & x_{2} & x_{3} & x_{4} & 0 & 0 \\ a_{0}x_{0} & a_{1}x_{1} & a_{2}x_{2} & a_{3}x_{3} & a_{4}x_{4} & l_{1} & l_{2} \end{matrix} ) ,

with l_{i} linear forms l_{i} = ∑_{j=0}^{4} b_{ij}x_{j}, i = 1, 2. We distinguish four cases:

a) l_{1} and l_{2} are two general linear forms, whereas a = 0, thus the presentation matrix is given by φ_{1} alone in this case. For this choice, the bundle F_{a} has a free resolution of type

10O(−2)
0 ← F_{a} ⟵ 20O(−1) ⟵ ⊕
5O(−3) ↖ 7O(−4) ⟵ 2O(−5) ← 0

and one checks via [Mac] that the general ψ ∈ Hom(E, F_{a}) leads to a smooth surface S_{a} ⊂ IP^{4} with desired invariants and whose ideal sheaf has a minimal free resolution

\begin{aligned}0 & ← & I_{S_{a}} & ⟵ & 9O(−5) & & 8O(−6) \\ & & & & & ↖ & ⊕ \\ & & & & & & 5O(−7) & ↖ & 7O(−8) & ⟵ & 2O(−9) & ← 0 \end{aligned} .

In particular, since the ideal I_{S_{a}} is generated by quintics, S_{a} has no 6-secants and hence N_{6}(11, 11, 2) = 4 is the number of exceptional lines on S_{a}. It follows from proposition 3.28 that S_{a} is non-minimal K3 surface embedded by

H = H_{min} − 5E_{1} − ∑_{i=2}^{5} E_{i}.

This family of K3 surfaces was first constructed in [DES].

b) l_{1} and l_{2} have a common zero at exactly one of the points p_{i}. Then φ_{1} has 6 linear syzygies, thus a = 1 and we take as φ_{2} a general column matrix of quadrics. We get this time a bundle F_{b} with resolution of type

20O(−1)  11O(−2)  O(−3)
0 ← F_{b} ⟵ ⊕ ⟵ ⊕ ⟵ ⊕
O(−2)  6O(−3)  7O(−4) ↖ 2O(−5) ← 0

and the general morphism ψ ∈ Hom(E, F_{b}) gives a smooth surface S_{b} ⊂ IP^{4} with resolution of type

\begin{aligned}& & & & 9O(−5) & & 9O(−6) & & O(−7) \\ 0 & ← & I_{S_{b}} & ⟵ & ⊕ & ⟵ & ⊕ & ⟵ & ⊕ \\ & & & & O(−6) & & 6O(−7) & & 7O(−8) & ↖ & 2O(−9) & ← 0 \end{aligned} .

This time there is precisely one sextic generator of I_{S_{b}}, and the 6 linear forms in the first step of the resolution define a line L in the hyperplane dual to the point p_{i}, which is the support of I_{S_{b}}/(I_{S_{b}})_{≤5} and thus the unique 6-secant of S_{b}. Consequently there are only N_{6} − 1 = 3 exceptional lines, and hence S_{b} is a non-minimal K3 surface embedded by

H = H_{min} − 4E_{1} − 2E_{2} − ∑_{i=3}^{5} E_{i}.

c) l_{1} and l_{2} have a common zero at two of the points p_{i}. In this case φ_{1} has 7 linear syzygies and we take φ_{2} to be a general 2 × 2 matrix of quadrics (a = 2). The bundle F_{c} has a minimal free resolution

20O(−1)  12O(−2)  2O(−3)
0 ← F ⟵ ⊕ ⟵ ⊕ ⟵ ⊕
2O(−2)  7O(−3)  7O(−4) ↖ 2O(−5) ← 0

and the general morphism ψ ∈ Hom(E, F_{c}) gives rise to smooth surface S_{c} ⊂ IP^{4} with syzygies

\begin{aligned}& & & & 9O(−5) & & 10O(−6) & & 2O(−7) \\ 0 & ← & I_{S_{c}} & ⟵ & ⊕ & ⟵ & ⊕ & ⟵ & ⊕ \\ & & & & 2O(−6) & & 7O(−7) & & 7O(−8) & ↖ & 2O(−9) & ← 0 \end{aligned} .

In this case the module I_{S_{c}}/(I_{S_{c}})_{≤5} has support on two skew lines, L_{1} and L_{2}, which are the 6-secants of S_{c}. It follows that there are only N_{6} − 2 = 2 exceptional lines, and hence S_{c} is a non-minimal K3 surface embedded by

H = H_{min} − 3E_{1} − ∑_{i=2}^{3} 2E_{i} − ∑_{j=4}^{5} E_{j}.

d) Finally l_{1} and l_{2} have a common zero at three of the points p_{i}, say p_{0}, p_{1} and p_{2}. Now φ_{1} has 8 linear syzygies and we take φ_{2} to be given by a general 2 × 3 matrix of quadric forms. This leads to a vector bundle with resolution

20O(−1)  13O(−2)  3O(−3)
0 ← F_{d} ⟵ ⊕ ⟵ ⊕ ⟵ ⊕
3O(−2)  8O(−3)  7O(−4) ↖ 2O(−5) ← 0

and the general morphism ψ ∈ Hom(E, F_{d}) degenerates along a smooth surface S_{d} ⊂ IP^{4} with resolution

\begin{aligned}& & & & 9O(−5) & & 11O(−6) & & 3O(−7) \\ 0 & ← & I_{S_{d}} & ⟵ & ⊕ & ⟵ & ⊕ & ⟵ & ⊕ \\ & & & & 3O(−6) & & 8O(−7) & & 7O(−8) & ↖ & 2O(−9) & ← 0 \end{aligned} .

The choice we’ve made distinguishes a plane, namely Π = span_{k}(p_{0}, p_{1}, p_{2}), which meets the surface S_{d} along a smooth plane quartic curve C and the 3 points p_{0}, p_{1} and p_{2} outside it. It follows that the 3 lines L_{ij} = span_{k}(p_{i}, p_{j}), for 0 ≤ i < j ≤ 2, are 6-secant lines to the surface S_{d}, and they are the only ones since Π is the support of I_{S_{d}}/(I_{S_{d}})_{≤5}. Consequently there is only one exceptional line, and thus S_{d} is a non-minimal K3 surface embedded by

H = H_{min} − 2 ∑_{i=1}^{4} E_{i} − E_{5}.  □

Proposition 3.31. A smooth surface S ⊂ IP^{4} with invariants d = 11, π = 11, χ = 3 is either a minimal, general type surface on the Noether’s line, or a regular, non-minimal surface of general type with only one exceptional line.

P roof. The double point formula gives K^{2} = 1, so S is of general type. S is a regular surface, hence p_{g} = 2, otherwise [Deb. Th.6.1] yields K_{min}^{2} ≥ 6, thus HK_{min} ≤ 4 while p_{a}(K_{min}) ≥ 7, which is absurd by (0.34). Assume now that S is not minimal and let E_{1}, …, E_{k} denote the exceptional curves on S. If k ≥ 2, then HK_{min} ≤ 7 and p_{a}(K_{min}) = K_{min}^{2} + 1 ≥ 4 thus a curve D ∈ |K_{min} − E_{1}| would have degree at most 6 and arithmetic genus at least 4. The only possibility is that k = 2, HE_{1} = HE_{2} = 1, whereas D spans only a IP^{3}, HD = 6 and p_{a}(D) = 4. But then a curve in |H − D| would have degree 5 and arithmetic genus 4, which is absurd. Therefore there exists only one exceptional curve E on S. Assume that HE ≥ 2. Then a curve D ∈ |K_{min} − E| has degree ≤ 5 and arithmetic genus 3, thus it spans only a IP^{3} and equality in fact holds. By (0.34) the residual curve H − D splits as H − D = A + B, where A is a plane quintic curve and B is a line disjoint of A. It follows that A^{2} + B^{2} = 2, thus also A^{2} ≥ 3 since B^{2} ≤ −1 and this contradicts Hodge index. In conclusion, if S is not minimal, then it has only one exceptional line.⊓ ⊔

Proposition 3.32. There exist smooth regular non-minimal surfaces of general type S ⊂ IP^{4} with d = 11, π = 11, χ = 3 and with only one exceptional line.

P roof. For construction purposes we assume that S doesn’t lie on a quartic hypersurface, and thus, by lemma 3.29 above, that the cohomology table of I_{S}(∗) is minimal

i ↑

2

1  h^{i}(I_{S}(p))
1  4  3

⟶ p Beilinson’s theorem suggests to take

E = 2O(−1) ⊕ Ω^{3}(3)  and  F = Syz_{1}(H^{1}(I_{S}(∗ + 4)))

and to define S as the degeneracy locus of a morphism φ ∈ Hom(E, F). Thus we have again to determine the module structure of the H^{1}-module of the ideal sheaf. We assume that M = H^{1}(I_{S}(∗)) is monogenous, i.e., that M is the tensor product of a monogenous module M′ with Hilbert function (1, 4, 3) in 4 variables, say over R′ = k[x_{0}, …, x_{3}], with the Koszul complex of one hyperplane, say x_{4}. However, it is easily seen that for the generic choice of such a module no morphism in Hom(E, F) can be injective. The trick is to choose M′ special, and for that we start with four general lines L_{1}, L_{2}, L_{3} and L_{4} in IP^{3} = Proj(k[x_{0}, …, x_{3}]) and define M^{′∗} as the cokernel of

0 ⟵ M′^{∗} ⟵ 3R′(4) ⟵^{(θ_{1},θ_{2})} 8R′(3) ⊕ R′(2),

where θ_{1} is the product γα of a random matrix γ ∈ M_{3,4}(k) with the direct sum α of the four Koszul complexes built on the lines L_{i}, i = \overline{1, 4} in IP^{3}, while θ_{2} is a column matrix of general quadrics. Tensoring M′ with the Koszul complex of x_{4} and sheafifing syzygies we obtain a bundle F with minimal resolution:

15O(−1) 9O(−2) O(−3)
0 ← F ⟵ ⊕ ⟵ ⊕ ⟵ ⊕
4O(−2)  12O(−3)  11O(−4) ↖ 3O(−5) ← 0

and this time a general morphism φ ∈ Hom(E, F) provides a smooth surface S ⊂ IP^{4}, with desired invariants and syzygies

8O(−5) 8O(−6) O(−7)
0 ← I_{S} ⟵ ⊕ ⟵ ⊕ ⟵ ⊕
4O(−6)  12O(−7)  11O(−8) ↖ 3O(−9) ← 0

We check now what kind of surface we did obtain. By construction, it follows from the cohomology of the exact sequence

0 ⟶ I_{S}(2) ⟶ I_{S}(3) ⟶ I_{H}(3) ⟶ 0

that the hyperplane section H_{0}, cut out on S by the hyperplane spanned by the four lines, is the unique one such that h^{0}(I_{H_{0}}(3)) = 1. One checks that for a general choice of the construction data, the unique cubic containing H_{0} is a smooth Del Pezzo surface X ⊂ IP^{3}. Now cohomology in a another twist of the above exact sequence yields h^{0}(I_{H_{0}}(5)) = 11, thus we may link H_{0} in the complete intersection of X with a general quintic to a curve G of degree 4, arithmetic genus −3. On another side, by construction, I_{S}/I_{S}_{≤5} is supported on the four lines, and so we might guess that they are the 6-secants of S. Indeed, G is the union of the four lines we’ve started with, and since we can always choose the basis of Pic(X) such that X ⊂ IP^{3} is embedded by H_{X} ∼ 3l − ∑_{i=1}^{6} E_{i}, whereas the lines L_{i} are of class E_{i}, i = \overline{1, 4} (see [Ha] or [GP2]), we deduce that H_{0} ∼ 15l − ∑_{i=1}^{4} 6E_{i} − 5E_{5} − 5E_{6}. Intersection theory on X shows now that each line L_{i} is a 6-secant to H_{0}, and thus also to S, and these are the only 6-secant lines to S. Since Le Barz’s formula gives N_{6}(11, 11, 3) = 5 we deduce that there is one exceptional line on the surface S. This proves the claim of the proposition.⊓ ⊔ V. Surfaces with d = 11, π = 12

Adjunction and the double point formula give HK = 11 and K^{2} = 6χ − 22, thus Hodge index yields χ ≤ 5. Now lemma 0.20 implies that χ ≥ 0, and equality holds only when S is a (blown-up) ruled surface over an elliptic curve. Moreover, when κ(S) ≥ 0, the pseudo-effectiveness (0.16) of K gives χ ≥ 2.

Lemma 3.33. There are no smooth surfaces S ⊂ IP^{4} with d = 11, π = 12 which are birationally ruled over an elliptic curve.

P roof. We consider, once again, the adjunction mapping whose image

S_{1} = φ_{H+K}(S) ⊂ IP^{10}  has invariants  d_{1} = 11, H_{1}K_{1} = −11, π_{1} = 1,

whence K_{1}^{2} = 0, S_{1} is geometrically ruled and φ_{H+K} is the contraction of 22 exceptional lines on S. Therefore, the formula for the canonical class of a ruled surface shows that the fibers of the ruling of S are embedded as (degenerated) twisted cubics in IP^{4}. Let F be such a ruling. The residual curve D = H − F has degD = 8 and p_{a}(D) = 10, whence, according to lemma 0.34, it contains a plane sextic curve as component. But this contradicts the bound on the degree of a plane curve in (0.36).⊓ ⊔

Proposition 3.34. There are no smooth rational surfaces S ⊂ IP^{4}, with d = 11 and π = 12.

P roof. Assume such surfaces exist. Severi’s theorem gives h^{1}(O_{S}(1)) = 4. On the other hand, if h^{1}(O_{H}(2)) = h^{0}(O_{H}(K − H)) ≠ 0 for the general hyperplane section H of S, then, since H(K −H) = 0, it would follow from Weil’s lemma (see [W] or [So]) that K −H is trivial, which is absurd. Therefore h^{1}(O_{H}(2)) = 0 for the general H, and thus the long cohomology sequence of

0 ⟶ O_{S}(1) ⟶ O_{S}(2) ⟶ O_{H}(2) ⟶ 0

gives h^{1}(O_{S}(2)) ≤ 4. Now h^{0}(I_{S}(3)) = 0 by (1.7) and h^{2}(O_{S}(3)) = h^{0}(O_{S}(K − 2H)) = 0 because p_{g} = 0, hence h^{1}(O_{S}(3)) ≥ 1 since by Riemann-Roch χ(I_{S}(3)) = 1. It follows that the variety V ⊂ ǏP^{4} parametrizing hyperplane sections for which h^{1}(O_{H}(3)) ≠ 0 is non-empty. Let H be a hyperplane section of S corresponding to a point of V. Then lemma 0.37 yields a decomposition H = C_{1} + C_{2}, C_{1} ≥ 0, C_{2} > 0 such that

C_{1}C_{2} ≤ (K − 2H)C_{2}  (∗)

whence

3 deg C_{2} ≤ 2p_{a}(C_{2}) − 2.  (∗∗)

This is readily seen to be impossible for deg C_{2} ≤ 5 by using the bounds for the arithmetic genus in (0.34). Now lemma 0.36 shows that plane curves on S have degree at most 5. Combined with the bounds in (0.34) this implies deg C_{2} ≥ 9, whence deg C_{1} ≤ 2. But the inequality (∗) is equivalent to

3 deg C_{1} ≥ (K + H)C_{1} + C_{1}C_{2} + 11.

Therefore, since (K + H)C_{1} ≥ 0 by (0.16) and C_{1}C_{2} ≥ 1 by the 1-connectedness of H, we obtain deg C_{1} ≥ 4, which is a contradiction. ⊓ ⊔

We discuss in the sequel the possible invariants for a surface S with κ(S) ≥ 0.

Proposition 3.35. Let S ⊂ IP^{4} be a smooth surface with d = 11, π = 12 and χ = 2. Then S is a non-minimal K3 surface embedded by

H = H_{min} − 2E_{1} − ∑_{i=2}^{10} E_{i}.

P roof. Let S_{1} be the image of S through the adjunction map. We compute the following invariants for S_{1} ⊂ IP^{12}: d_{1} = 23, π_{1} = 13, H_{1}K_{1} = 1. Now p_{g} ≥ 1, so |K_{1}| consists of a unique exceptional line and S is a blown-up K3 surface. In particular K_{1}^{2} = −1 and the adjunction morphism φ_{H+K} is blowing down 9 exceptional lines on S.⊓ ⊔

An example of a smooth surface with the above invariants has been first constructed via liaison by K. Ranestad (private communication). We’ll recall here his approach and prove that the general surface can be obtained by this construction. First a remark:

Lemma 3.36. Let S ⊂ IP^{4} be a surface with d = 11, π = 12 and κ(S) ≥ 0. Then

a) h^{1}(O_{S}(1)) > h^{1}(O_{S}(2)) or h^{1}(O_{S}(2)) = 0

b) h^{1}(O_{S}(k)) = 0 for k ≥ 3.

P roof. First observe that h^{2}(O_{S}(n)) = h^{0}(O_{S}(K − nH)) = 0 for all n ≥ 1 because (K − nH)(K +H) = 6χ−11(2n+1) < 0. Thus Severi’s theorem and Riemann-Roch give for the speciality h^{1}(O_{S}(1)) = 5−χ ≤ 3. The argument used in the proof of proposition 3.34 gives here h^{1}(O_{H}(2)) = 0 for a general hyperplane section H, hence h^{1}(O_{S}(2)) ≤ h^{1}(O_{S}(1)) = 5 − χ. Assume first that h^{1}(O_{S}(2)) = h^{1}(O_{S}(1)). Then the variety V ⊂ ǏP^{4} parametrizing hyperplane sections for which h^{1}(O_{H}(2)) ≠ 0 is a determinantal hypersurface of degree ≤ 3. It is not contained in the dual variety of S since otherwise S would be degenerated when deg V = 2 or would have too many plane curves when deg V = 3. Therefore one can find a Lefschetz pencil δ (see [AF], [Z]) such that for H a general member in it h^{1}(O_{H}(2)) ≠ 0 and hence O_{H}(K −H) is trivial. But then Weil’s lemma applies again to show that K −H is trivial, which is absurd. It follows that h^{1}(O_{S}(2)) ≤ 4 − χ ≤ 2 or h^{1}(O_{S}(2)) = 0. We show next that h^{1}(O_{S}(k)) = 0 for k ≥ 3. First of all h^{1}(O_{H}(k)) = 0 when k ≥ 3 and H is a smooth hyperplane section, thus the claim follows by induction for χ = 4, 5. Assume now χ ≤ 3 and h^{1}(O_{S}(3)) ≠ 0. Then the variety W ⊂ ǏP^{4} parametrizing hyperplane sections for which h^{1}(O_{H}(3)) ≠ 0 contains a plane, so there is a line L ⊂ S which is the base locus of a net of hyperplanes for which h^{1}(O_{S}(3)) ≠ 0. If the general divisor C ∈| H − L | is irreducible, then it follows from the exact sequence

0 ⟶ O_{C}(3H − L) ⟶ O_{H}(3H) ⟶ O_{L}(3H) ⟶ 0

that h^{1}(O_{C}(3H − L)) ≠ 0, and thus 2p_{a}(C) − 2 ≥ (H − L)(3H − L). Combined with formula (0.2), which reads

22 = 2p_{a}(C) − 2 − 2L^{2},

this gives L^{2} ≥ 7, which is impossible. If the general divisor C ∈| H − L | is not integral, then | H −L | is composed with a basepoint free pencil of plane curves. We have O_{C}(3H − L) = O_{C}(2H) and thus h^{1}(O_{C}(2H)) > 0. It follows that | H − L | is a pencil of plane quintic curves. This is a contradiction since

2p_{a}(C) − 2 = CK = HK − LK ≤ 11 + 2 + L^{2} = 4.
⊓ ⊔

Lemma 3.37. Let S ⊂ IP^{4} be a smooth surface as in proposition 3.35. Then h^{1}(I_{S}(k)) = 0 for all k ≥ 3.

P roof. Riemann-Roch gives χ(I_{S}(3)) = 0, whence h^{1}(I_{S}(3)) = 0 because h^{0}(I_{S}(3)) = 0 by (1.7) and h^{1}(O_{S}(3)) = h^{2}(O_{S}(3)) = 0 by lemma 3.36. It follows then from the cohomology of the exact sequences

0 ⟶ I_{S}(m − 1) ⟶ I_{S}(m) ⟶ I_{H}(m) ⟶ 0,  m = \overline{3, 4}

that h^{1}(I_{H}(4)) = h^{1}(I_{S}(4)) and h^{1}(I_{H}(3)) = 2 for all hyperplane sections H. Assume now that h^{1}(I_{S}(4)) > 0. Then each hyperplane contains at least one plane π for which h^{1}(I_{π∩S}(4)) > 0. For the general hyperplane this plane section is a finite scheme of length 11. If it contains a subscheme of length 10 which is contained in a conic, then h^{1}(I_{π∩S}(3)) ≥ 3 and thus also h^{2}(I_{H}(2)) = h^{1}(O_{H}(2)) = h^{0}(O_{H}(K − H)) ≥ 1, which in turn implies, as before, that K−H is trivial, whence a contradiction. It follows from lemma 0.41 that π ∩ S contains a subscheme of length 6 which is contained in a line. But this means that the general hyperplane section to S contains 6-secant lines, which is absurd. Therefore h^{1}(I_{S}(4)) = 0 and the lemma follows. ⊓ ⊔

Corollary 3.38. A smooth K3 surface S ⊂ IP^{4} with d = 11, π = 12 is the degeneracy locus of a morphism φ

0 ⟶ O(−1) ⊕ 3Ω^{3}(3) ⟶^{φ} 2Ω^{2}(2) ⊕ 2O ⟶ I_{S}(4) ⟶ 0,

and thus its ideal sheaf has a minimal free resolution of type:

2O(−4)
0 ← I_{S} ⟵ ⊕
4O(−5) ↖ 7O(−6) ⟵ 2O(−7) ← 0.

P roof. The above lemmas give the cohomology table

i ↑

1

3  2  h^{i}(I_{S}(p))

2 ⟶

p

hence the corollary follows from Beilinson’s spectral sequence. ⊓ ⊔

Proposition 3.39. The general smooth K3 surface S ⊂ IP^{4} with d = 11, π = 12 is linked (4, 4) to a reducible surface X_{5} = P ∪ P_{1} ∪ P_{2} ∪ P_{3} ∪ P_{4}, where P is a plane and P_{i} are planes cutting P along lines l_{i}, i = \overline{1, 4}, such that no three lines have common intersection points.

P roof. Let S ⊂ IP^{4} be the degeneracy locus of a general morphism

φ = ( \begin{matrix}φ_{11} & φ_{12} \\ φ_{21} & φ_{22} \end{matrix} ) ∈ Hom(O(−1) ⊕ 3Ω^{3}(3), 2Ω^{2}(2) ⊕ 2O).

It follows from Severi’s theorem and lemma 3.36 that the variety W ⊂ ǏP^{4} parametrizing hyperplane sections for which h^{1}(O_{H}(2)) ≠ 0 is a smooth rational cubic scroll in ǏP^{4}, namely the degeneration locus of the 2 × 3 matrix with linear entries which defines φ_{22} ∈ Hom(3Ω^{3}(3), 2Ω^{2}(2)). The rulings of W, respectively its directrix, correspond to pencils of hyperplanes for which h^{1}(O_{H}(2)) ≠ 0; the pencil parametrized by the directrix l being distinguished by the fact that

dim( ⋂_{H∈H^{0}(O_{l}(1))^{∗}} ker(H^{1}(O_{S}(1))⟶^{·H}H^{1}(O_{S}(2)))) = 2,

where the above intersection is considered as a vector subspace of H^{1}(O_{S}(1)). In the case of the pencils parametrized by rulings of W this dimension is only one. Recall now from (3.35) that S is embedded by a linear system

H = H_{min} − 2E_{1} − ∑_{i=2}^{10} E_{i}.

Consider the residual pencil |D| = |H − E_{1}|. First of all, since DE_{1} = 3, |D| has no fixed component since this would lie in the plane of the conic E_{1}. Therefore the general D is irreducible, of degree 9 and arithmetic genus 10. Moreover, it is in fact a complete intersection of two cubic surfaces since, by Riemann-Roch, h^{0}(I_{D,H}(3)) ≥ 2 and since if D would lie on a quadric, then it would be a curve of type (3, 6) on a smooth quadric surface and thus S would have too many 6-secants. Consider the exact sequence

0 ⟶ O_{S}(H + E_{1}) ⟶ O_{S}(2H) ⟶ O_{D}(2H) ⟶ 0.

From the above discussion O_{D}(2H) = ω_{D} and the natural restriction map H^{0}(O_{IP}^{4}(2)) = H^{0}(O_{S}(2H)) ⟶ H^{0}(O_{D}(2H)) is surjective, so taking cohomology of the exact sequence we see that h^{1}(O_{S}(H + E_{1})) = 1. Now the natural multiplication maps

H^{1}(O_{S}(1))⟶^{·H}H^{1}(O_{S}(2)),

where H is a hyperplane section containing E_{1}, factorize obviously

H^{1}(O_{S}(1))⟶^{·E_{1}}H^{1}(O_{S}(H + E_{1})) ⟶^{·(H−E_{1})}H^{1}(O_{S}(2)),

thus the plane P of the exceptional conic E_{1} must coincide with the base locus of the pencil of hyperplanes parametrized by the directrix l of W. The planes which are base loci of the pencils parametrized by rulings of W build a non-normal cubic hypersurface V ⊂ IP^{4} having P as double plane. V is the dual variety of W ⊂ ǏP^{4}. The general curve D is a complete intersection (3, 3), hence it cuts the plane P along a scheme of length 9. Since DE_{1} = 3 and D^{2} = 6 it follows that P cuts S along E_{1} and a scheme T of length 6 which is the base locus of |D|. Now a self-duality argument implies that T is the set of mutual intersection points of 4 lines l_{i}, i = \overline{1, 4} in the plane P, such that no three of them have common intersection points. It is easy to see now that for each line l_{i}, i = \overline{1, 4}, there is a plane P_{i} in the ruling of V such that P ∩P_{i} = l_{i}. The hyperplane H_{i}, spanned by P and P_{i}, is special with respect to the pencil |D|, in the sense that the two cubics containing the member D_{i} ∈ |D|, lying in H_{i}, have P_{i} as component. Thus D_{i} splits as the union of a plane quintic curve C_{i} in the plane P_{i} and an elliptic normal curve of degree 4. In particular, the union P ∪ P_{1} ∪ P_{2} ∪ P_{3} ∪ P_{4} is part of the variety of 5-secants to S, and thus it is contained in the intersection of all quartic hypersurfaces containing S. The claim of the proposition follows now by degree reasons.⊓ ⊔

Remark 3.40. The cubic hypersurface V ⊂ IP^{4}, in the proof of proposition 3.39, intersects the surface S ⊂ IP^{4} along the nine exceptional lines, twice the exceptional conic and the union of the four plane quintic curves P_{i} ∩ S, i = \overline{1, 4}.⊓ ⊔

Proposition (Ranestad) 3.41. Let P ⊂ IP^{4} be a plane and let P_{i} ⊂ IP^{4} be general planes cutting P along four general lines l_{1}, l_{2}, l_{3}, l_{4}. Then X_{5} = P ∪ P_{1} ∪ P_{2} ∪ P_{3} ∪ P_{4} can be linked (4, 4) to a smooth K3 surface S ⊂ IP^{4}, with d = 11, π = 12.

P roof. We denote by {p_{ij}} = l_{i} ∩ l_{j}, 1 ≤ i < j ≤ 4, the mutual intersection points. One sees easily that X_{5} is a local complete intersection scheme except for the points p_{ij}, where it is only Cohen-Macaulay. Therefore the proposition will follow via the liaison argument in (0.31) once we show that X_{5} is cut out by quartic hypersurfaces. To see this, one proceeds by induction using the residual intersection sequences

0 ⟶ I_{X_{k−1}}(m − 1) ⟶ I_{X_{k}}(m) ⟶ I_{X_{k}∩H_{k},H_{k}}(m) ⟶ 0,  m ∈ ℤ,

where X_{k} = P ∪ ⋃_{i=1}^{k} P_{i} and H_{k} is a general hyperplane through the plane P_{k}. We obtain namely, that I_{X_{k}}(k−1) is globally generated and h^{1}(I_{X_{k}}(k − 1)) = 0 for all k = \overline{1, 5}. Since degX_{5} = 5, π(X_{5}) = 0 and χ(X_{5}) = 1 the linked surface has invariants d = 11, π = 12 and χ = 2 and is therefore, by (3.35), a non-minimal K3 surface. ⊓ ⊔

Proposition 3.42. Let S be a smooth surface in IP^{4} with d = 11, π = 12, χ = 3. Then S is either

a) a regular, proper elliptic surface embedded by

H = H_{min} − 2E_{1} − ∑_{i=2}^{4} E_{i},

or

b) a regular, proper elliptic surface embedded by

H = H_{min} − ∑_{i=1}^{4} E_{i},

or

c) a blown-up, general type Horikawa surface embedded by

H = H_{min} − ∑_{i=1}^{5} E_{i}.

P roof. In this case K^{2} = −4 and the image of S through the adjunction morphism is a surface S_{1} ⊂ IP^{13} with invariants

d_{1} = 29, H_{1}K_{1} = 7, π_{1} = 19, K_{1}^{2} = −4 + a

where a ≥ 0 is the number of (−1) lines on S. In particular, Hodge index gives a ≤ 5. Assume first that S is a surface of general type. Then K_{min}^{2} ≥ 1, so there are at least 5 exceptional curves on S, say E_{1}, …, E_{k}, for some k ≥ 5. Now a curve D_{i} ∈ |K_{min} − E_{i}| has degree ≤ 11 − 5 − 2(k − 5) − 1 = 15 − 2k and arithmetic genus K_{min}^{2} + 1 ≥ k − 3, so it is easily seen that the only possibility is that k = 5 and all of the E_{i}’s are exceptional lines. Lemma 0.39 shows further that S is in this case regular, and thus a surface of type c). If S is not of general type, then it is a regular proper elliptic surface by lemma 0.38, and in particular p_{g} = 2. A similar argument as above shows now that there are at least 3 exceptional lines on S. Therefore, either S is a surface of type b), or there exist only 3 exceptional lines on the surface and necessarily one more exceptional conic, and thus S is of type b).⊓ ⊔

(3.43.) Constructions. We show in the sequel that types a) and c) in the above list of surfaces with d = 11, π = 12, χ = 3 exist. Lemma 3.36, together with the general results (1.6) and (1.7), provide the following cohomology table for the ideal sheaf of S ⊂ IP^{4}

i ↑

2

2  1  h^{i}(I_{S}(p))
1  a

a+1 ⟶

p

Assume first that the cohomology table is minimal, i.e., a = 0. Then Beilinson’s spectral sequence produces a vector bundle resolution of the ideal sheaf of a surface S_{1}

0 ⟶ 2O(−1) ⊕ 2Ω^{3}(3) ⟶^{φ} Ω^{2}(2) ⊕ Ω^{1}(1) ⊕ O ⟶ I_{S_{1}}(4) ⟶ 0,

and one checks, via [Mac], that the degeneracy locus of a general morphism φ is a smooth surface in IP^{4} with the desired invariants and a minimal free resolution

O(−4)
0 ← I_{S_{1}} ⟵ ⊕
8O(−5) ↖ 13O(−6) ← 6O(−7) ← O(−8) ← 0

In particular, I_{S_{1}} is generated by quintic hypersurfaces and thus S_{1} has no 6-secant lines. From Le Barz’s formula, which gives N_{6}(11, 12, 3) = 3, we deduce that there are 3 exceptional lines on the constructed surface, hence S_{1} is a proper elliptic surface of type a). The determinantal construction suggests also the equivalent liaison construction in proposition 3.44 below.

We construct now a family of surfaces with non-minimal cohomology

i ↑

2

2  1  h^{i}(I_{S}(p))
1  1

2 ⟶

p Beilinson’s theorem suggests this time to take

E = 2O(−1) ⊕ 2Ω^{3}(3)  and  F = 2O ⊕ ker(Ω^{2}(2) ⊕ Ω^{1}(1)⟶^{ψ}O),

where ψ is a suitable epimorphism. We take ψ such that the component on Ω^{2}(2) is given by a decomposable element in the exterior algebra, i.e., without loss of generality, we set ψ = (e_{1} ∧ e_{2}, e_{0}), where IP^{4} = IP(span_{k}(e_{0}, …, e_{4})). One checks, via [Mac], that the degeneracy locus of a general morphism φ ∈ Hom(E, F) is a smooth surface S_{2} ⊂ IP^{4}, with syzygies

2O(−4)
⊕
0 ← I_{S_{2}} ⟵ 3O(−5) 5O(−6)  O(−7)
⊕ ↖ ⊕ ⟵ ⊕
2O(−6) 5O(−7)  4O(−8) ↖ O(−9) ← 0

One checks further that the distinguished plane Π = span_{k}(e_{0}, e_{1}, e_{2}) meets the constructed surface along a plane quintic curve with an embedded point at p = IP(ke_{0}), thus S_{2} has an infinity of 6-secant lines, namely all lines in Π going through p. By computing the base locus of the canonical pencil, we find out that there are this time 5 exceptional lines on the surface, thus S_{2} is a blown-up Horikawa surface of type c). We remark that S_{2} can be linked in the complete intersection of two quartic hypersurfaces to a scheme Z of degree 5 and sectional genus 0, which decomposes as Z = 2Π ∪ Q, where 2Π is a double structure on the plane Π and Q is a rational cubic scroll.

Proposition 3.44. A smooth rational surface X ⊂ IP^{4} of degree 9, sectional genus 7 can be linked in the complete intersection of a quartic and a quintic hypersurfaces to a smooth, non-minimal proper elliptic surface S ⊂ IP^{4}, with d = 11, π = 12, χ = 3, which is embedded by

H = H_{min} − 2E_{0} − ∑_{i=1}^{3} E_{i}.

P roof. Smooth rational surfaces X ⊂ IP^{4}, with d = 9 and π = 7 have been studied in [Al2] and [AR]. They lie on a net of quartic hypersurfaces and on six extra independent quintic hypersurfaces, so they can be linked (4, 5) to locally Cohen-Macaulay schemes S ⊂ IP^{4} with invariants d = 11, π = 12, χ = 3. We use the intrinsic description of the linear system of the embedding X ⊂ IP^{4}, which is due to J. Alexander [Al2], to describe this liaison. We’ll also need and recall along the proof the results in [AR]. To begin with, it follows from [Al2] that X is IP^{2} blown-up in 15 points and

H_{X} ≡ 9l − ∑_{i=1}^{6} 3F_{i} − ∑_{j=7}^{9} 2F_{j} − ∑_{k=10}^{15} F_{k},

where F_{i}, i = \overline{1, 15} are the exceptional curves of the blow-up map, and where the 15 points are chosen such that there exists a pencil

|D| = |6l − ∑_{i=1}^{6} 2F_{i} − ∑_{j=7}^{9} F_{j} − ∑_{k=10}^{15} F_{k}|

with base points on F_{7}, F_{8} and F_{9}. Residual to the pencil there is a plane cubic curve H_{X} − D ≡ 3l − ∑_{i=1}^{9} F_{i}, and we’ll denote in the sequel with Π its plane. Since the base points of the pencil |D| are on the lines F_{i}, i = \overline{7, 9}, there exist on X three plane quartic curves

D_{m} ≡ 6l − ∑_{i=1}^{6} 2F_{i} − ∑_{j=7}^{9} F_{j} − ∑_{k=10}^{15} F_{k} − F_{m+6},

for m = \overline{1, 3}. Let Π_{1}, Π_{2}, Π_{3} be the planes of these curves. Each pencil |H_{X} − D_{m}|, m = \overline{1, 3}, has a base point p_{m} in the corresponding plane Π_{m}, thus any line in Π_{m} through p_{m} is a 5-secant to X, and it follows that any quartic containing X must also contain the scheme Π_{1} ∪ Π_{2} ∪ Π_{3}. Now D_{m}(H_{X} − D) = 2 for all m, so each Π_{m} meets Π along a line, and thus Π is also contained in all quartics containing X. The liaison result in [AR] says that the complete intersection of two general quartics containing X decomposes as Π ∪ Π_{1} ∪ Π_{2} ∪ Π_{3} ∪ T, where T is a cubic Del Pezzo surface intersecting Π_{i}, i = \overline{1, 3} along lines and X along the hyperplane section cut on X by the hyperplane spanned by the Del Pezzo surface.

We fix now a general quartic hypersurface containing X, and thus also the union Π ∪ Π_{1} ∪ Π_{2} ∪ Π_{3}. A Bertini argument shows that we can link X on V, via a quintic hypersurface W, to a smooth surface S. Now W cuts each of the planes Π_{i}, i = \overline{1, 3}, along the quartic D_{i} and an extra line, call it E_{i}, which then necessarily lies on S. The same argument shows that Π cuts the surface S along a conic, which we’ll denote in the sequel by E_{0}. On another side, the above discussion shows that there exists on V a pencil of Del Pezzo surfaces T_{(λ:μ)}, all meeting X along hyperplane sections H_{(λ:μ)} of it. Take now the embedding H_{T_{(λ:μ)}} = 3l − ∑_{i=1}^{6} G_{i} of T_{(λ:μ)} such that each Π_{i} intersects T_{(λ:μ)} along the line in the class G_{i}, for all i = \overline{1, 3}. Then the curve H_{(λ:μ)} has numerical class H_{(λ:μ)} ≡ 4H_{T_{(λ:μ)}} − ∑_{i=1}^{3} G_{i} ≡ 12l − ∑_{i=1}^{3} 5G_{i} − ∑_{j=4}^{6} 4G_{j} and thus the pencil T_{(λ:μ)} will cut on S a pencil of elliptic curves M_{(λ:μ)} of degree 6, in the class 3l − ∑_{j=4}^{6} G_{j}. In particular, it follows that S is a proper elliptic surface. More precisely, the residual curve C ≡ H − (5H_{T_{(λ:μ)}} − H_{(λ:μ)}) of a member in the pencil has degree 5 and arithmetic genus p_{a}(C) = M_{(λ:μ)}^{2} + 6, and since M_{(λ:μ)}^{2} ≥ 0 we deduce that M_{(λ:μ)}^{2} = 0 and C is a plane curve. It follows that |M_{(λ:μ)}| coincides with the residual pencil of the plane quintic curve C, and moreover that |M_{(λ:μ)}| = |K_{min}|. In particular the exceptional part of the canonical divisor on S has degree 5, and thus S is an elliptic surface of type a) in the notation of proposition 3.42. Furthermore, it is easily checked that E_{1}, E_{2} and E_{3} are the exceptional lines of S, while E_{0} is the exceptional conic.⊓ ⊔

Proposition 3.45. Let S be a smooth surface in IP^{4} with d = 11, π = 12, χ = 4. Then S is either

a) a blown-up, general type Horikawa surface embedded by

H = H_{min} − E_{1},

and having no 6-secant lines, or b) a minimal, regular, general type surface with only one 6-secant line, or

c) a regular, non-minimal, general type surface embedded by

H = H_{min} − E_{1} − E_{2},

and having infinitely many 6-secant lines, namely all lines in one of the rulings of a smooth quadric surface.

P roof. The double point formula gives K^{2} = 2, thus S is of general type. Furthermore S is regular. Otherwise, the inequality in [Deb, Th.6.1] reads

K_{min}^{2} ≥ 2p_{g} ≥ 8

so S would have at least 6 exceptional curves, whence HK_{min} ≤ 5 while p_{a}(K_{min}) = K_{min}^{2} + 1 ≥ 9, which is absurd. Riemann-Roch and lemma 3.36 yield then the following cohomology table for the ideal sheaf of S

\begin{aligned}i & ↑ \\ & 3 \\ & 1 & h^{i}(I_{S}(p)) \\ 2 & a \\ & a \\ & ⟶ \\ & p \end{aligned} .

We’ll discuss in the sequel the possible values for a and their corresponding linear systems. Let Δ(4) be the locus in ǏP^{4} where (h^{0}(I_{H}(4))) · (h^{1}(I_{H}(4))) ≠ 0.

Lemma 3.46. If S has a 6-secant, then Δ(4) contains a plane.

P roof. Let π be a general plane through the 6-secant. Then h^{1}(I_{π∩S}(4)) > 0 and since h^{1}(O_{H}(3)) = h^{0}(O_{H}(K − 2H)) = 0 for the general hyperplane containing π ∩ S, it follows that h^{1}(I_{H}(4)) > 0 and thus the claim of the lemma. ⊓ ⊔

Lemma 3.47. Any hyperplane section H ∈ Δ(4) has a proper 6-secant or contains a plane curve as component.

P roof. Assume that H has no plane curve as component. Then the sequence

0 ⟶ I_{H}(3) ⟶^{H′} I_{H}(4) ⟶ I_{H∩H′}(4) ⟶ 0

is exact for all hyperplane sections H′. Now h^{1}(I_{H}(3)) = h^{1}(I_{S}(3)) = 2 so taking cohomology of the exact sequence we see that there is at least one plane section π ∩ S in H for which h^{1}(I_{π∩S}(4)) > 0. This plane section is a scheme of length 11. If it contains a subscheme of length 10 which is contained in a conic then h^{0}(I_{π∩S}(3)) > 1. On the other hand, from the above cohomology table h^{0}(I_{H}(3)) = 0, hence taking cohomology of the exact sequence

0 ⟶ I_{H}(2) ⟶ I_{H}(3) ⟶ I_{π∩S}(3) ⟶ 0

we see that h^{1}(I_{H}(2)) ≥ 2. But, since h^{2}(I_{S}(1)) = 1 this implies that h^{1}(I_{S}(2)) > 0, which is a contradiction. Therefore, it follows from lemma 0.41 that π ∩ S contains a subscheme of length 6 which is contained in a line, and the lemma follows.⊓ ⊔

Lemma 3.48. S has only finitely many plane curves.

P roof. Since any pencil of plane curves on S is linear the residual curve in a hyperplane section containing a general element of the family is again a plane curve. But then the hyperplane section is contained in a quadric and this contradicts Severi’s theorem.⊓ ⊔

If a ≥ 3 then Δ(4) = ǏP^{4} and the above lemmas show that the general hyperplane section of S has 6-secants, while S is contained in at least a net of irreducible quartics, a contradiction. Thus we are left to consider the cases where a ∈ {0, 1, 2}. When a = 0 we easily see that Δ(4) = ∅, hence S has no 6-secant lines by lemma 3.46. On the other hand, Le Barz’s formula (0.11) gives N_{6} = 1, so S has exactly one exceptional line E_{1}. If there exists also another exceptional curve E_{2} on S then K_{min}^{2} ≥ 4, HK_{min} ≤ 8, thus a curve D ∈ |K_{min} − E_{1} − E_{2}| would have degree at most 5 and arithmetic genus at least 5. Since in any case p_{a}(D) ≠ 6, this is a contradiction by lemma 0.34 and therefore S is embedded by H = H_{min} − E_{1}. In case a = 1 the above lemmas imply that Δ(4) = IP^{2} and that S has only one 6secant line. Therefore there are no exceptional lines on S this time. In fact S is minimal. The adjunction mapping process and a reasoning similar to the previous case show that the only other alternative is that there exists an exceptional conic E on S. But then a general curve D ∈ |H − E| is irreducible, of degree 9 and arithmetic genus 10 and, since h^{2}(O_{S}(2H − D)) = h^{0}(O_{S}(K − H − E)) = 0, the cohomology of the sequence

0 ⟶ O_{S}(2H − D) ⟶ O_{S}(2H) ⟶ O_{D}(2H) ⟶ 0

shows that h^{1}(O_{D}(2)) = 0, whence finally by Riemann-Roch that D lies on a quadric. It is necessarily a curve of type (3, 6) on a smooth quadric, so S would have too many 6-secants, which is absurd. In the last case a = 2 and Δ(4) is a determinantal hyperquadric in ǏP^{4}. The above lemmas imply that all lines in one of the rulings of the dual quadric surface Q are 6-secants to S. Moreover, if π is a general plane spanned by two concurrent rulings of the quadric Q, then h^{0}(I_{π∩S}(3)) = 1 and h^{0}(I_{π∩S}(4)) = 5. It follows that π meets S in at most two points outside the two chosen rulings, and therefore the quadric Q cuts S along a curve D of type (6, a), with a ≥ 3. On another side, if a = 4, then the residual curve H − D is a line and thus p_{a}(H − D) = 0 yields D^{2} = 24, which would contradict Hodge index. We conclude that Q meets S along a curve D of type (6, 3). Consider now again the residual curve H − D. It has degree 2 and arithmetic genus D^{2} − 6, and Bezout’s theorem gives D(H − D) ≤ 4, i.e., D^{2} ≥ 5, so either equality holds and H − D is the union of two skew lines, or D^{2} = 6 and H − D is a conic. The last case cannot occur, since otherwise the 6-secants of S would fill up a 3-fold, which is impossible for a surface contained in a pencil of irreducible quartics. Therefore H − D = E_{1} + E_{2}, with HE_{1} = HE_{2} = 1 and E_{1}E_{2} = 0, and since E_{1}^{2} + E_{2}^{2} = (H − D)^{2} = −2, while E_{i}^{2} ≤ −1, i = 1, 2, we deduce that E_{1} and E_{2} are exceptional lines on S. An argument similar to that used in case a) shows that there are no further exceptional curves on S, whence S is embedded by H = H_{min} − E_{1} − E_{2}.⊓ ⊔

(3.49.) Constructions. We’ll show in the sequel that all types of surfaces in proposition 3.45 exist. For convenience, we denote by S_{0}, S_{1} and S_{2} surfaces corresponding to types a), b) and respectively c). Then Beilinson’s spectral sequence and the above discussion imply that the ideal sheaf of any of the surfaces has a resolution of type

0 ⟶ 3O(−1) ⊕ Ω^{3}(3) ⟶^{φ_{a}} G_{a} ⊕ aO ⟶ I_{S_{a}}(4) ⟶ 0  (∗)

where a ∈ {0, 1, 2}, and G_{a} is a vector bundle, kernel of a morphism

0 ⟶ G_{a} ⟶ 2Ω^{1}(1) ⟶^{ψ_{a}} aO ⟶ 0.

Also Δ(4) coincides with the locus in ǏP^{4} where the 2 × a matrix defining ψ_{a} is dropping rank. Vice-versa, we use the information on Δ(4) to construct such surfaces, and take G_{0} = 2Ω^{1}(1) and as G_{a}, a = 1, 2, the kernel of a suitable epimorphism ψ_{a} : 2Ω^{1}(1) → aO such that the 2 × a matrix M_{a} defining it degenerates on a plane if a = 1, or on a point quadric cone if a = 2. Namely, without loss of generality, one can take M_{1} = (e_{1}, e_{2})

and M_{2} = ( \begin{matrix}e_{3} & e_{4} \\ e_{0} & e_{1} \end{matrix} ), and define S_{a} as the degeneracy locus of a general morphism φ_{a} as

above in (∗). Smoothness can be checked in examples with [Mac]. From (∗) we obtain also the following minimal free resolutions of the ideal sheaf:

0 ← I_{S_{0}} ← 12O(−5) ⟵ 19O(−6) ⟵ 10O(−7) ⟵ 2O(−8) ← 0

thus S_{0} is cut out by quintics and has no 6-secants,

O(−4)
⊕
0 ← I_{S_{1}} ← 7O(−5) 10O(−6) 3O(−7)
⊕ ↖ ⊕ ← ⊕
O(−6)  3O(−7) 3O(−8) ↖ O(−9) ← 0

where the linear part of the last syzygy are the equations of the unique 6-secant to S_{1}, and

\begin{aligned}& & & & 2O(−4) \\ & & & & ⊕ \\ 0 & ← & I_{S_{2}} & ← & 2O(−5) & & 3O(−6) \\ & & & & ⊕ & ↖ & ⊕ \\ & & & & 4O(−6) & & 10O(−7) & ↖ & 8O(−8) & ⟵ & 2O(−9) & ← 0 \end{aligned} .

The quartics and the quintics in the ideal I_{S_{2}} cut out the union of S_{2} with the quadric of 6-secant lines.

Remark 3.50. A surface S ⊂ IP^{4}, with d = 11, π = 12, χ = 4, of type c) in the above proposition, can be linked (4, 4) to a scheme Z of degree 5, sectional genus 0, which is the union of a double structure on a quadric and a plane which meets the quadric along one of its rulings.

Proposition 3.51. A smooth surface S ⊂ IP^{4}, with d = 11, π = 12, χ = 5 is a regular, minimal surface of general type.

P roof. The same argument we used in proposition 3.45 shows that S is regular. Moreover, Severi’s theorem and lemma 3.36. yield h^{1}(O_{S}(k)) = 0, for all k ≥ 1, thus combining with (1.6) and (1.7) we obtain also h^{1}(I_{S}(2)) = 1 and h^{1}(I_{S}(3)) = 3. It follows that h^{1}(I_{H}(3)) = 3 for some hyperplane sections H, or equivalently h^{0}(I_{H}(3)) = 1, whence h^{0}(I_{H}(4)) ≥ 4, and furthermore h^{0}(I_{S}(4)) ≥ 1. Assume that S is not minimal and let E be an exceptional curve on it. Then a curve D ∈ |K_{min} − 2E| will have degree ≤ HK − 3HE ≤ 8 and arithmetic genus p_{a}(D) = K_{min}^{2} + 1 ≥ 10. Then by lemma 0.34 such a curve would have as component a plane curve of degree at least 6 which is impossible by lemma 0.36. Therefore S is a minimal surface as claimed.⊓ ⊔

(3.52.) Construction. As a consequence of the above discussion the ideal sheaf of the surface S has the following cohomology table

i ↑

5

h^{i}(I_{S}(p))
1  3  2

1 ⟶

p

since if h^{1}(I_{S}(4)) = 3, then h^{0}(I_{S}(4)) = 2 and Beilinson’s spectral sequence implies that the two quartics in I_{S} would have non-trivial linear syzygies, which is impossible if the surface is smooth. Similar arguments, together with the information in the proof of (3.51) allows to exclude the cases when h^{1}(I_{S}(4)) ≥ 4. Beilinson’s theorem suggests to take

E = 5O(−1)  and  F = Syz_{1}(H^{1}(I_{S}(∗ + 4))),

and since a monogenous artinian module with Hilbert function (1, 3, 2) is the tensor product of a similar module on 3 variables with the Koszul complex of a plane, we have without loss of generality

F = ker(4O(−1) ⊕ O⟶^{ψ}O(1)),

with ψ = (x_{0}, x_{1}, q_{1}, …, q_{4}), where q_{i} ∈ k[x_{2}, x_{3}, x_{4}] are general quadrics. The degeneracy locus of a general morphism φ ∈ Hom(E, F) is a smooth surface S ⊂ IP^{4} with the desired invariants and syzygies

O(−4)
⊕
0 ← I_{S} ⟵ 6O(−5) 8O(−6)  2O(−7)
⊕ ↖ ⊕ ⟵ ⊕
3O(−6) 8O(−7)  7O(−8) ↖ 2O(−9) ← 0

The quintics in the ideal I_{S} cut out the surface and an extra plane Π. The plane Π meets the surface along a plane quartic curve and in three other points, thus S has three 6-secant lines, namely the three lines joining pairwise these points. Since S is minimal, this matches Le Barz’s formula which gives N_{6}(11, 12, 6) = 3.

VI. Surfaces with d = 11, π = 13

The double point formula gives K^{2} = 6χ−27, so Hodge index implies χ ≤ 7. On the other side, Lemma 0.20 yields χ ≥ 1. Also, when κ(S) ≥ 0, the pseudo-effectiveness of K gives via (0.15) χ ≥ 3, whence in particular κ(S) ≥ 1.

Proposition 3.53. There are no smooth, rational surfaces S ⊂ IP^{4} with invariants d = 11, π = 13.

P roof. We use the adjunction mapping to exclude this case. Theorem 0.13 gives for the surface S_{1} = φ_{H+K}(S) ⊂ IP^{12}

d_{1} = 16, H_{1}K_{1} = −8, π_{1} = 5, K_{1}^{2} = −21 + a

where a ≥ 0 is the number of (−1) lines on S. Now (0.13) applied for S_{1} implies that (H_{1} + K_{1})^{2} = −21 + a ≥ 0. If a > 21 then S_{2} = φ_{H_{1}+K_{1}}(S_{1}) is a surface in IP^{4} with invariants

d_{2} = −21 + a, H_{2}K_{2} = −29 + a, π_{2} = −26 + a

whence necessarily a ≥ 26. But this is impossible since by Hodge index K_{1}^{2} = −21 + a ≤ \frac{64}{16} = 4. We are left with the case a = 21 which means that S_{1} is a conic bundle having, since K_{1}^{2} = 0, eight singular fibres in the ruling. Let IF_{e} = IP(O_{IP}^{1} ⊕ O_{IP}^{1}(−e)), e ≥ 0 be the relative minimal model of S_{1}, C_{0} a section with minimal negative self-intersection −e and f a ruling. Thus we may write, pulling back C_{0} and f

H ≡ 4C_{0} + (2e + 8)f − ∑_{i=1}^{8} 2E_{i} − ∑_{j=9}^{29} E_{j}

Let now B ≡ f − E_{j}, for some j ≥ 9. Then HB = 3, so B is contained in a hyperplane section of S. Let C be the residual curve. It has degree 8 and arithmetic genus 10, hence by Lemma 0.34 contains a plane sextic curve as component. But this is a contradiction since h^{1}(O_{S}(1)) = 5 and hence, by Lemma 0.36, the maximal degree of a plane curve on S is 5.⊓ ⊔

Proposition 3.54. There are no smooth surfaces S ⊂ IP^{4} with d = 11, π = 13, χ = 3.

P roof. In this case K^{2} = −9 and the image of S through the adjunction morphism is a surface S_{1} = φ_{H+K}(S) ⊂ IP^{14} with invariants

d_{1} = 28, H_{1}K_{1} = 4, π_{1} = 17, and K_{1}^{2} = −9 + a

where a ≥ 0 is the number of (−1) lines on S. Now H_{1}^{2} > (H_{1}K_{1})^{2}, so by lemma 0.18 S is a proper elliptic surface. Moreover, since H_{1}K_{1} = 4, Kodaira’s formula for the canonical class of an elliptic fibration [BPV] implies that p_{g} = 2 and that S is either minimal, or has exactly one (−1) line. In the first case the moving part of |K| is a pencil of elliptic quartic curves and hence the residual of a general member in the hyperplane section it spans will be a curve D of degree 7 and arithmetic genus 9. By Lemma 0.34 such a curve has a plane sextic curve as component and since h^{1}(O_{S}(1)) = 3 this contradicts the bound in (0.36). In the second case the elliptic fibration of S is by plane cubic curves and the residual of a fibre in a hyperplane section containing it will have degree 8 and arithmetic genus 10. Again, by Lemma 0.34, such a curve has a plane sextic as component contradicting the results of (0.36) ⊓ ⊔

Proposition 3.55. There are no smooth surfaces S ⊂ IP^{4} with d = 11, π = 13, χ = 4.

P roof. In the above hypothesis K^{2} = −3. Let S_{min} be the minimal model of S. If S is of general type, then by Noether’s inequality [BPV]

K_{min}^{2} ≥ 2p_{g} − 4 ≥ 4

Thus S has at least 5 exceptional curves E_{i}, i = \overline{1, 5} and HK_{min} ≤ 13 − 5 = 8 and p_{a}(K_{min}) ≥ 3. Since p_{g} ≥ 3 we can find a curve C ∈ |K_{min} − E_{1} − E_{2}|. It has degree HC ≤ 6 and arithmetic genus p_{a}(C) ≥ 3. If one of E_{1} or E_{2} has degree at least 2 then HC ≤ 4 and p_{a}(C) ≥ 3 which is absurd in view of Lemma 0.34. Therefore S has only (−1) lines and it is easily seen that there are two possibilities, either 5 or 6 exceptional lines. Nevertheless, in both cases, arguments as above or as in (0.38) give p_{g} = 3 and q = 0. In the first case we compute K_{min}^{2} = 2, whence HC = 6, p_{a}(C) = 3. If C is irreducible then it spans only a hyperplane and the residual curve will have degree 5 and arithmetic genus 5 which is impossible by Lemma 0.34. It remains that C decomposes as the union of a plane quintic curve A and two skew lines L_{1} and L_{2} or as the union of a plane quartic curve A and a conic Q such that each line (resp. the conic) meets A in one point. Neither of the lines L_{1} or L_{2} is exceptional since, e.g., from L_{1} = E_{1} and CE_{1} = (K_{min} −E_{1} −E_{2})E_{1} = 1 would follow AL_{1} = AE_{1} = 2. Therefore L_{i}^{2} ≤ −2, i = 1, 2 (resp. Q^{2} ≤ −2) whence A^{2} ≥ 0 because C^{2} = 0. On the other side Hodge index gives A^{2} ≤ 1. The residual pencil |H − A| has degree 7 and arithmetic genus 7 + A^{2} ≥ 7 so, by Lemma 0.34, A^{2} = 0 and each of its members decomposes into a plane quintic curve and a conic which meet along a subscheme of length two. This means that the plane spanned by A cuts the surface S along a curve of degree at least 6, thus contradicting the bound in (0.36). In the case of 6 exceptional lines we compute K_{min}^{2} = 3, HC = 5, p_{a}(C) = 4 which is again impossible by Lemma 0.34. The last case we have to discuss is when S is a proper elliptic surface. Then K_{min} is numerically equivalent to nF + ∑_{i=1}^{k} F_{i}, n ≥ p_{g} − 1 ≥ 2, k ≥ 0, where F is a fibre of the elliptic fibration and the F_{i}’s are reduced parts of not necessarily distinct fibers. Since HK_{min} ≤ 13 − 3 = 10 we get HF ≤ 5. If HF = 5 then p_{g} = 2, k = 0 and S has exactly 3 exceptional lines. A curve D ≡ F − E_{1} has degree 4 and arithmetic genus 1, hence it spans only a IP^{3}. But then the residual curve H − D will have degree 7 and arithmetic genus 8 which is impossible by lemma 0.34. If HF ≤ 4, since a curve D ≡ F − E_{1} with E_{1} exceptional curve has degree at least 3, it follows that HF = 4, n = 2, k ≥ 1 and S has 3 exceptional lines. But then HF_{1} ≤ 2 which is absurd for a curve of arithmetic genus 1.⊓ ⊔ Proposition 3.56. A smooth surface S ⊂ IP^{4} with d = 11, π = 13 and χ = 5 is the degeneracy locus of a morphism φ inducing

0 ⟶ 4O(−1) ⊕ Ω^{3}(3) ⟶^{φ} Ω^{2}(2) ⊕ 3O ⟶ I_{S}(4) ⟶ 0.

P roof. To show regularity we proceed as in (3.45). Namely, when q ≥ 1, the inequality in [Deb, Th.6.1] gives K_{min}^{2} ≥ 10, hence HK_{min} ≤ 6 while p_{a}(K_{min}) ≥ 11, which is absurd. We determine next the cohomology table of S. First of all h^{0}(O_{S}(K − H)) = 0 by lemma 0.17, so Riemann-Roch gives h^{1}(O_{S}(1)) = 1. Assume now that h^{1}(O_{S}(2)) ≥ 2. Then there exists a line L ⊂ IP^{4} which is the base locus of a net of hyperplanes for which h^{1}(O_{H}(2)) ≥ 2. The line L lies on S; otherwise the general H in the net is smooth, whence hyperelliptic by Clifford’s theorem and this is impossible by theorem 0.13 and [E]. Therefore the general H in the net decomposes H = L + C, where C is a smooth curve with HC = 10 and g(C) = 13 + L^{2}. Moreover C is irreducible since otherwise |H − L| would be composed with a pencil of plane curves and this leads to a contradiction as in the proof of lemma 3.36. Consider the cohomology of the exact sequence

0 ⟶ O_{C}(2H − L) ⟶ O_{H}(2H) ⟶ O_{L}(2H) ⟶ 0.

h^{1}(O_{H}(2H)) = 2, whereas h^{1}(O_{L}(2H)) = 0, so h^{0}(O_{C}(K − H)) = h^{1}(O_{C}(2H − L)) ≥ 2. Therefore, either (K − H)C = 5 + L^{2} ≥ 3, or L^{2} = −3, h^{1}(O_{C}(2H − L)) = 2 and C is hyperelliptic. When L^{2} ≥ −2, it follows from the exact sequence

0 ⟶ O_{IP}^{1}(L^{2} + 1) ⟶ O_{H}(2H) ⟶ O_{C}(2H) ⟶ 0

that h^{0}(O_{C}(K − H − L)) = h^{1}(O_{C}(2H)) = 2. This implies again that C is hyperelliptic and L^{2} = −1, because C(K − H − L) = 2L^{2} + 4 and, in any case, L^{2} ≤ −1. In both cases we obtained a contradiction, since a hyperelliptic curve C can never be embedded in a projective space with degree at most g(C) + 2 (see for instance [ACGH, p.221]). Therefore Riemann-Roch and (1.6) give h^{1}(O_{S}(2)) = 1 and thus also h^{1}(O_{S}(k)) = 0, for k ≥ 3, since otherwise h^{1}(O_{H}(3)) > 0 for at least a web of hyperplane sections, which is absurd because the general H in the web is smooth and O_{H}(3H) is non special. From (1.7) it follows h^{1}(I_{S}(3)) = 0 and a similar argument to that used in the proof of (3.37) shows that h^{1}(I_{S}(4)) = 0. We obtain the following cohomology table

i ↑

4

1  1  h^{i}(I_{S}(p))

3 ⟶

p and thus the claim of the proposition is a consequence of Beilinson’s spectral sequence.⊓ ⊔

Corollary 3.57. A smooth surface S ⊂ IP^{4} with d = 11, π = 13, χ = 5 is a regular, non-minimal, general type surface with only one exceptional line.

P roof. S has one exceptional line E_{1} since N_{6} = 1 by Le Barz’s formula, and because it follows from the above proposition that there are no 6-secants. If E_{2} is another exceptional curve, then for a curve D ∈ |K_{min} − 2E_{2}| we would have HD ≤ 6 and p_{a}(D) ≥ 5, whence necessary E_{2} is a (−1) conic, K_{min}^{2} = 5 and D decomposes, by (0.34), as D = A + L, where A is a plane quintic and L a line which doesn’t meet A. But L^{2} ≤ −1 and A^{2} ≤ 2 by Hodge index, so D^{2} = A^{2} + L^{2} = 1 implies that L^{2} = −1 and thus that L coincides with E_{1}, which is absurd because DE_{1} = 0.⊓ ⊔

(3.58.) Construction. A surface S ⊂ IP^{4} with the above invariants can be constructed as the degeneracy locus of a general morphism φ ∈ Hom(4O(−1) ⊕ Ω^{3}(3), Ω^{2}(2) ⊕ 3O). In particular the ideal sheaf has a minimal free resolution of type

3O(−4)  O(−5)
0 ← I_{S} ⟵ ⊕ ⟵ ⊕
2O(−5)  4O(−6) ↖ O(−7) ← 0

where the three linear syzygies in the top row are the equations defining the unique exceptional line of S. An equivalent construction, which provides also a smoothness argument, is contained in the following

Proposition 3.59. A smooth, general type surface Y ⊂ IP^{4}, with d = 9 and π = 8 can be linked in the complete intersection of a quartic and a quintic hypersurface to a smooth surface S ⊂ IP^{4} with invariants d = 11, π = 13, p_{g} = 4 and q = 0.

P roof. It follows from [AR] that Y is cut out by quartic hypersurfaces, so S is smooth for a general choice of the liaison. Formula (0.26) and the cohomology of the liaison exact sequence

0 ⟶ O_{S}(K) ⟶ O_{Σ}(4) ⟶ O_{Y} (4) ⟶ 0,

where Σ is the complete intersection (4, 5), give further π(S) = 13, p_{g}(S) = 4, q(S) = 0.⊓ ⊔

Proposition 3.60. A smooth surface S ⊂ IP^{4} with d = 11, π = 13, χ = 6 is a regular, minimal, general type surface, linked in the complete intersection of two quartic hypersurfaces to an elliptic quintic scroll.

P roof. We have K^{2} = 9, so S is of general type. A smooth curve of degree 11 and genus 13 in a projective space spans only a IP^{3}, thus h^{0}(O_{H}(H)) = 4. By Lemma 0.17 h^{0}(O_{S}(K − H)) = 0 so h^{1}(O_{S}(1)) = 0 and the cohomology of the exact sequence

0 ⟶ O_{S} ⟶ O_{S}(H) ⟶ O_{H}(H) ⟶ 0

give q = h^{1}(O_{S}) = 0, i.e., S is a regular surface.

We determine next the cohomology table of S. Since h^{1}(O_{H}(2)) = h^{0}(O_{H}(K − H)) ≤ 1 ( if non-zero, h^{0}(O_{H}(K − H)) ≤ H(K − H) = 2 and in case of equality S would have hyperelliptic hyperplane sections, which is impossible by [So], [E] ) the cohomology of the above exact sequence twisted by H yields h^{1}(O_{S}(2)) ≤ 1. In any case h^{1}(O_{H}(3)) = 0 for all irreducible hyperplane sections H and thus h^{1}(O_{S}(n)) ≥ h^{1}(O_{S}(n + 1)) for all n ≥ 2 and, in particular h^{1}(O_{S}(3)) ≤ 1. But if h^{1}(O_{S}(3)) > 0, then h^{1}(O_{H}(3)) > 0 for at least a web of hyperplane sections, which is absurd, since the general H in the web is smooth, while O_{H}(3) is non-special. It follows that h^{1}(O_{S}(k)) = 0, for all k ≥ 3.

S is not contained in a cubic hypersurface, because otherwise h^{1}(I_{S}(3)) = h^{0}(I_{S}(3)) − χ(I_{S}(3)) ≥ 2, and this is impossible for a surface which, by theorem 1.7, is in the liaison class of the Veronese surface. Therefore h^{0}(I_{S}(3)) = 0 and h^{1}(I_{S}(3)) = 1.

Claim. h^{1}(O_{S}(2)) = h^{1}(I_{S}(2)) = 0.

Proof of the claim. Assume that this is not the case; i.e., h^{1}(O_{S}(2)) = h^{1}(I_{S}(2)) = 1. Then the cohomology table of S has the shape:

i ↑

5

1  h^{i}(I_{S}(p))
1  1  a

a+2 ⟶

p

where a ∈ IN. Since the E^{∞}-terms outside the diagonal in Beilinson’s spectral sequence vanish, there should exist a monomorphism

Ω^{2}(2)⟶^{φ}G ⊕(a + 2)O,

where G is the kernel of a morphism Ω^{1}(1)⟶^{ψ}aO. Thus, by rank considerations, a ≥ 1. On the other hand, if a = 1, then for such a monomorphism to exist, ψ must vanish identically. But then Beilinson’s spectral sequence implies that Z, the codimension two subscheme defined as the degeneracy locus of φ, would be a component of S, which is a contradiction to the smoothness assumption. Therefore, necessary a ≥ 2. We’ll show that this leads to a contradiction. First, taking cohomology of the exact sequence

0 ⟶ I_{S}(2) ⟶ I_{S}(3) ⟶ I_{H}(3) ⟶ 0

we see that h^{1}(I_{H}(3)) = 2, for all hyperplane sections H in a web. Since the general hyperplane section H in the web is smooth and O_{H}(3) non-special, we get h^{0}(O_{H}(3)) = 21, whence h^{0}(I_{H}(3)) = 1. Now, from the cohomology of the above exact sequence twisted by H, we deduce that there are hyperplane sections H_{0} in the web, for which h^{1}(I_{H_{0}}(4)) = 2. But h^{1}(O_{H_{0}}(4)) = 0, since h^{1}(O_{S}(4)) = h^{2}(O_{S}(3)) = 0, thus h^{0}(I_{H_{0}}(4)) = 5. This means that H_{0} is contained in a cubic and an independent quartic surface, so it could be linked in their complete intersection to a curve of degree 1 and arithmetic genus −2, which is a contradiction.⊓ ⊔

Therefore h^{1}(O_{S}(2)) = h^{1}(I_{S}(2)) = 0, and since a morphism Ω^{1}(1) ⟶ O is never surjective (its cokernel has support on one point), we deduce that h^{1}(I_{S}(4)) = 0. Hence we obtain the cohomology table

i ↑

5

h^{i}(I_{S}(p))

1

2 ⟶

p

and thus S is the degeneracy locus of a morphism

0 ⟶ 5O(−1) ⟶ Ω^{1}(1) ⊕ 2O ⟶ I_{S}(4) ⟶ 0.

In particular, S can be linked (4, 4) to an elliptic quintic scroll. Dualizing the last exact sequence we obtain

0 ⟶ O(−5) ⟶ 2O(−1) ⊕ Ω^{3}(3) ⟶ 5O ⟶ ω_{S} ⟶ 0,

so |K| has no base points and S is minimal.⊓ ⊔

Remark 3.61. a) The homogenous ideal of an elliptic quintic scroll is generated by cubic hypersurfaces, so the scroll can be linked (4, 4) to a smooth surface as in the above proposition.

b) The surfaces described in the previous proposition are hyperplane sections of smooth, log-general type, unirational threefolds X ⊂ IP^{5}, with d = 11, π = 13, χ = 1 and κ(X) = −∞ (cf. [BSS] and [Ch]). An easy argument shows also the uniqueness of the examples given in [BSS] for threefolds with these invariants.

Proposition 3.62. There are no smooth surfaces S ⊂ IP^{4} with invariants d = 11, π = 13, χ = 7.

P roof. We compute χ(O_{S}(1)) = 6, so by Riemann-Roch and Severi’s theorem it follows that h^{0}(O_{S}(K − H)) = h^{2}(O_{S}(H)) ≥ 1. But a curve C ∈ |K − H| will have degree 2 and arithmetic genus 2 which is absurd.⊓ ⊔ VII. Surfaces with d = 11, π = 14

In this last case we compute K^{2} = 6χ − 32 and HK = 15, so Hodge index implies χ ≤ 8. Lemma 0.20 yields χ ≥ 1 and, in case κ(S) ≥ 0, the pseudo-effectiveness of K implies via (0.16) that χ ≥ 3 and in particular that κ(S) ≥ 1.

Proposition 3.63. There are no smooth rational surfaces with d = 11, π = 14.

P roof. We use again the adjunction mapping to rule out the various candidates. If S_{1} denotes the image of S under the adjunction mapping, we obtain the following invariants:

S ⊂ IP^{4}  H^{2} = 11  HK = 15  K^{2} = −26  π = 14
S_{1} ⊂ IP^{13} H_{1}^{2} = 15  H_{1}K_{1} = −11  K_{1}^{2} = −26 + a  π_{1} = 3,

where a is the number of the (−1)-lines on S. The adjoint linear system of S_{1} has projective dimension two, so theorem 0.31 implies that either (H_{1} + K_{1})^{2} = a − 33 = 1 and S_{1} is IP^{2} blown-up in one point, or (H_{1} + K_{1})^{2} = a − 33 = 0 and S_{1} is a conic bundle with one singular fibre in the ruling. In the first case we may write

H = 7l − 2E_{0} − ∑_{i=1}^{34} E_{i}.  (∗)

In the second case, if IF_{e} = IP(O_{IP}^{1} ⊕ O_{IP}^{1}(−e)), e ≥ 0, is the relative minimal model of S_{1}, C_{0} is a section of IF_{e} with minimal self-intersection −e and f is the class of a ruling, then working back through the adjunction process we may write

H ≡ 4C_{0} + (2e + 6)f − 2E_{1} − ∑_{i=2}^{34} E_{i}.

Now HC_{0} ≥ 1 so e ∈ {0, 1, 2}, and we may choose IP^{2} as minimal model

H = 8l − 4E_{0} − 2E_{1} − ∑_{i=2}^{34} E_{i}.  (∗∗)

We use cohomological information to exclude both cases. First, Riemann-Roch and Severi’s theorem yield h^{1}(O_{S}(1)) = 6, χ(I_{S}(2)) = 7 and χ(I_{S}(4)) = 11. Now the generic hyperplane section H of S is not hyperelliptic, so h^{1}(O_{H}(2)) = h^{0}(O_{H}(K − H)) ≤ 2 by Clifford’s theorem, and from the cohomology of the exact sequence

0 ⟶ O_{S}(1) ⟶ O_{S}(2) ⟶ O_{H}(2) ⟶ 0

we obtain h^{1}(O_{S}(2)) ≤ h^{1}(O_{S}(1)) + 2 = 8. Since (K − 2H)H < 0, we get also that h^{1}(O_{H}(k)) = 0, for k ≥ 3 and H an irreducible hyperplane section, and thus it follows from the cohomology of the above sequence that h^{1}(O_{S}(k)) ≥ h^{1}(O_{S}(k + 1)) for all k ≥ 2. Therefore h^{1}(O_{S}(4)) ≤ 8, and since χ(I_{S}(4)) = 11 we deduce that S is contained in at least a net of irreducible quartic hypersurfaces. In particular the 5-secants of S can fill at most a surface of degree 5 in IP^{4}.

Assume now that S is embedded by the linear system (∗). Then residual to the curve E_{0} there is a pencil |D| = |H − E_{0}| of curves of degree 9 and sectional genus 12. If the general D is irreducible, then either it is a curve of type (3, 5) on a smooth quadric, or it is linked (2, 5) on a quadric cone with one of its rulings. In both cases the quadric would consist of 5-secants to S, and thus we obtain a contradiction since S would have too many 5-secant lines. If the general element of D is not irreducible, then the only possibility is that D has as base component a line L in this plane. Similar arguments rule also this case out. Finally, assume that S would be embedded by the linear system (∗∗). Then a curve C_{i} = l − E_{0} − E_{i}, for i = \overline{2, 34}, has degree 3 and thus spans only a hyperplane in IP^{4}. Residual to it there is a curve D_{i} of degree 8 and arithmetic genus 11. By (0.34), such a curve decomposes as D_{i} = A_{i} + B_{i}, where A_{i} is a plane sextic curve and B_{i} is a conic meeting A_{i} along a scheme of length 2. Therefore, by Bezout, the plane spanned by A_{i} lies in the intersection of all the quartic hypersurfaces containing S, and since the planes spanned by the various plane sextics are obviously distinct, the surface S would have again too many 5-secants.⊓ ⊔

Proposition 3.64. There are no smooth surfaces S ⊂ IP^{4} with invariants d = 11, π = 14 and χ = 3.

P roof. The claim is a consequence of the following refinement of the Castelnuovo inequality for space curves (cf. [Deb, Prop. 3.1])

Let X be a smooth non-ruled surface, and let L be a line bundle on X such that the induced mapping φ_{L} is birational on the image. Then

4h^{0}(L) − 6 ≤ h^{0}(L^{⊗2})

which we apply for the adjoint linear system |H + K| on S. Namely, in case such a surface would exist, we get h^{0}(O_{S}(H + K)) = 16, while h^{0}(O_{S}(2H + 2K)) = χ(O_{S}(2H + 2K)) = 56 since h^{1}(O_{S}(2H + 2K)) = 0 by Kodaira’s vanishing theorem.⊓ ⊔

Proposition 3.65. There are no smooth surfaces S ⊂ IP^{4}, with d = 11, π = 14 and χ = 4.

P roof. The double point formula yields K^{2} = −8, while p_{g} ≥ 3, thus S has at least 8 exceptional curves which are part of any canonical divisor on S. Let S_{0}, with canonical divisor K_{0}, denote the minimal model of S. Now S_{0} is either elliptic or of general type.

If S_{0} is elliptic, then K_{0} is numerically equivalent to nF + ∑_{i=1}^{k} a_{i}F_{i}, n > 1 and k, a_{i} ≥ 0, where F is a fibre of the elliptic fibration and the F_{i}’s are the reduced parts of the multiple fibers. Since HK_{0} ≤ 7, we deduce that n = 2 and HF = 3. But then there is a curve C ∈ |F − E|, where E is one of the exceptional curves on S, with HC ≤ 2 and p_{a}(C) = 1, which is a contradiction.

If S_{0} is a surface of general type, then Noether’s inequality gives

K_{0}^{2} ≥ 2p_{g} − 4 ≥ 2,

thus S has at least ten (−1) curves E_{i}, i = \overline{1, 10}, while HK_{0} ≤ 5 and p_{a}(K_{0}) = K_{0}^{2} +1 ≥ 3. Since p_{g} ≥ 3, we can find a curve C ≡ K_{0} − E_{1} − E_{2} on S. It has degree HC ≤ 3 and arithmetic genus p_{a}(C) ≥ 3, and thus we have reached again a contradiction to the smoothness of S.⊓ ⊔

Proposition 3.66. There are no smooth surfaces S ⊂ IP^{4}, with d = 11, π = 14 and χ = 5.

P roof. We compute K^{2} = −2, so S has at least two (−1) curves E_{1} and E_{2}. As above, let S_{0}, with canonical divisor K_{0}, denote the minimal model of S.

If S_{0} is elliptic, then we argue as in proposition 3.65 to obtain that K_{0} ≡ nF, where F is a fiber of the elliptic fibration and n ∈ {2, 3}. If n = 3, then HF = 4, HE_{1} = 1 and HE_{2} = 2, hence there exists a residual curve D ∈ |F − E_{2}| of degree 2 and arithmetic genus p_{a}(D) = 1, which is impossible. If n = 2, then HF = 4, HE_{1} = 1 and HE_{2} = 2. The residual curve C ∈ |F − E_{2}| has degree 4 and arithmetic genus 1, so it spans only a hyperplane in IP^{4}. Therefore there is a curve D ∈ |H − C| of degree 7 and arithmetic genus 9. By lemma 0.33, D decomposes into a plane sextic curve A and a conic B, which doesn’t meet A. The curve B is not exceptional, because otherwise it would be equal to E_{2}, while B^{2} = B^{2} + AB = DB = DE_{2} = (H − F + E_{2})E_{2} = 1, which is a contradiction. Therefore B^{2} ≤ −2, and then D^{2} = A^{2} + B^{2} = 2 implies A^{2} ≥ 4, which is impossible by the index theorem (HA = 6 implies that A^{2} ≤ 3).

We are left to consider the case when S_{0} is a surface of general type. Then Noether’s inequality yields K_{0}^{2} ≥ 4, so there are at least 6 exceptional curves E_{i}, i = \overline{1, 6}, while HK_{0} ≤ 9 and p_{a}(K_{0}) = K_{0}^{2}+1 ≥ 5. Since p_{g} ≥ 4 we can find a curve D ≡ K_{0}−E_{1}−E_{2}−E_{3} on S. It has degree HD ≤ 6 and arithmetic genus p_{a}(D) = p_{a}(K_{0}) ≥ 5. Hence, either D is a plane quintic curve, K_{0}^{2} = 5 and S has seven (−1) lines, or HD = 6, p_{a}(D) = 5 and S has six (−1) lines. In the first case, residual to the plane quintic we get a pencil of curves of degree 6 and arithmetic genus 6. By (0.33), such a curve splits as the union of a plane quintic curve A and a line B, which meet in one point. Since the line doesn’t move, it lies in the plane spanned by the quintic, so B(A + B) = B(H − D) = 1 − BD = −4, whence B^{2} = −5. But A^{2} + B^{2} = 1, thus A^{2} ≥ 6, which contradicts the index theorem. In the second case, D decomposes as the union of a plane quintic curve A and a line B, which does not meet A. As above B^{2} ≤ −2, A^{2} + B^{2} = 1, thus A^{2} ≥ 3 and we get again a contradiction by the index theorem.⊓ ⊔

Proposition 3.67. There are no smooth surfaces S of degree 11 in IP^{4}, with π = 14 and χ = 6.

P roof. The double point formula gives K^{2} = 4, so S is of general type. On the other side h^{2}(O_{S}(1)) = h^{0}(O_{S}(K − H)) = 0 since K^{2} − H^{2} < 0, so Riemann-Roch and Severi’s theorem yield for the speciality h^{1}(O_{S}(1)) = 1. Now Clifford’s theorem gives h^{1}(O_{H}(2)) = h^{0}(O_{H}(K − H)) ≤ 2 for the generic hyperplane section H, whence h^{1}(O_{S}(2)) ≤ 3. On another side, Riemann-Roch and theorems 1.6 and 1.7 yield h^{1}(O_{S}(2)) ≥ χ(I_{S}(2)) = 2 and h^{1}(O_{S}(3)) ≥ χ(I_{S}(3)) = 2, while h^{1}(O_{S}(2)) ≥ h^{1}(O_{S}(3)) since (K − 2H)H < 0 and thus h^{1}(O_{H}(3)) = 0 for an irreducible H. It follows that the variety V ⊂ ǏP^{4} parametrizing hyperplane sections for which h^{1}(O_{H}(3)) > 0 is ruled in lines or contains at least a plane. Now S is in any case a regular surface since otherwise, K_{min}^{2} ≥ 12 by [Deb, Th.6.1] while HK_{min} ≤ 7, p_{a}(K_{min}) ≥ 13 and p_{g} = 6, which is absurd. Therefore an argument as in lemma 3.25. shows that there are only finitely many plane curves on S. In particular V cannot be ruled in lines since each line of the ruling would correspond to a IP^{2} cutting S along a plane curve. On the other side, if V contains a plane π, then the line L, base locus of the net of hyperplanes parametrized by π, needs to be contained in S and thus we have an exact sequence

0 ⟶ O_{C}(3H − L) ⟶ O_{H}(3H) ⟶ O_{L}(3H) ⟶ 0,

where C = H − L. The general element in |C| is irreducible since otherwise |C| would be composed with a basepoint free pencil of plane curves, thus 2p_{a}(C)−2 ≥ (H −L)(H −3L) which implies L^{2} ≥ 3, a contradiction.⊓ ⊔

Proposition 3.68. If S is a smooth surface of degree 11, with π = 14 and χ = 7, then S is a minimal regular surface of general type, contained in a cubic hypersurface. S is linked to a Veronese surface in the complete intersection of the cubic and a quintic hypersurface.

P roof. We have K^{2} = 10, so S is of general type. Riemann-Roch gives χ(O_{S}(1)) = 5, thus Severi’s theorem implies h^{1}(O_{S}(1)) = 0, since h^{2}(O_{S}(1)) = h^{0}(O_{S}(K − H)) = 0 by lemma 0.17. The cohomology of the exact sequence

0 ⟶ O_{S}(1) ⟶ O_{S}(2) ⟶ O_{H}(2) ⟶ 0

yields then h^{1}(O_{S}(2)) = h^{1}(O_{H}(2)) = h^{0}(O_{H}(K − H)). Since the general hyperplane section is not hyperelliptic, Clifford’s theorem implies h^{1}(O_{S}(2)) ≤ 2. On the other side, Riemann-Roch and (1.6) give h^{1}(O_{S}(2)) ≥ 1.

Claim. h^{1}(O_{S}(3)) = 0.

Proof of the claim. Assume the contrary. If h^{1}(O_{S}(2)) = 1, then there would exist a web of hyperplane sections H such that h^{1}(O_{H}(3)) ≠ 0, which is absurd because the general element in the web is smooth and O_{H}(3) is non-special. If h^{1}(O_{S}(2)) = 2, then S cannot be contained in a cubic hypersurface because theorem 1.7 implies that at most one of the two Hartshorne-Rao modules of S would be non-trivial in this case. Therefore Riemann-Roch gives h^{1}(O_{S}(3)) = 1 + h^{1}(I_{S}(3)). If h^{1}(O_{S}(3)) = 2, then there exists a quadric cone such that, for all hyperplanes H containing one of its rulings, h^{1}(O_{H}(3)) = 1. Each such line lies on S; otherwise the general hyperplane through it cuts out an integral curve and this is impossible ((K − 2H)H = −7 < 0). But then it would follow that S is ruled, which is again impossible. If h^{1}(O_{S}(3)) = 1, one uses Beilinson’s spectral sequence to obtain a contradiction as in proposition 3.60. ⊓ ⊔

Therefore h^{1}(O_{S}(3)) = 0 and thus h^{0}(I_{S}(3)) ≥ 1. Theorem 1.7 implies then h^{1}(O_{S}(2)) = 1, h^{1}(O_{S}(k)) = 0, for k ≥ 3, and h^{1}(I_{S}(k)) = 0 for all k. In particular, it follows that S can be linked in the complete intersection of a cubic and a quintic hypersurface to a Veronese surface. All the other claims are immediate.⊓ ⊔

Remark 3.69. Since the homogeneous ideal of the Veronese surfaces is generated by 7 cubics, we can link them (3, 5) to smooth surfaces as in the previous proposition.

Proposition 3.70. If S is a smooth surface of degree 11, with π = 14 and χ = 8, then S is a minimal regular surface of general type. S can be linked in the complete intersection of two quartic hypersurfaces to a Castelnuovo surface of degree 5, thus it is projectively Cohen-Macaulay.

P roof. Riemann-Roch and Severi’s theorem give h^{2}(O_{S}(1)) = h^{0}(O_{S}(K − H)) = 1 + h^{1}(O_{S}(1)) > 0, thus S is a minimal surface of general type. On the other side H(K−H) = 4 and p_{a}(K − H) = 0, so no part of |K − H| can move on S, thus h^{2}(O_{S}(1)) = 1 and h^{1}(O_{S}(1)) = 0. Now the cohomology of the exact sequence

0 ⟶ O_{S}(1) ⟶ O_{S}(2) ⟶ O_{H}(2) ⟶ 0

yields h^{1}(O_{H}(2)) = 1 + h^{1}(O_{S}(2)), while Clifford’s theorem gives h^{0}(O_{H}(K − H)) ≤ 2. Hence h^{1}(O_{S}(2)) ≤ 1, and the argument we used in (3.56) shows that h^{1}(O_{S}(k)) = 0, for all k ≥ 0. In particular, h^{1}(I_{S}(3)) = h^{0}(I_{S}(3)) = 0 (by theorem 1.6) and h^{0}(I_{S}(4)) ≥ 4. If we cut to a general plane Π in IP^{4}, we get a linear system |C| of plane quartic curves through 11 points, whose general member is irreducible. The projective dimension of |C| is at least 4, and an argument similar to that used in [Ra1] for the proof of (0.34) shows that |C| has only 11 base points. Thus the linear system of quartic hypersurfaces containing S has no base locus of codimension two outside S. By Bertini, we can link S in the complete intersection of two general quartic hypersurfaces to an irreducible surface Y ⊂ IP^{4}, with d = 5 and π = 2. The general hyperplane section H_{Y} of Y is an irreducible curve of degree 5 and arithmetic genus 2. Therefore H_{Y} is projectively normal, and so are Y and S. Hence Y is a Castelnuovo surface.⊓ ⊔

Remark 3.71. a) The homogeneous ideal of Castelnuovo surfaces is generated by a quadric and two cubic hypersurfaces, hence they can be linked (4, 4) to smooth surfaces as described in the above proposition.

b) The above proposition yields an easy proof for the uniqueness of the examples provided in [BSS] of smooth threefolds, with invariants d = 11 and π = 14. In particular, we obtain that any such threefold has Kodaira dimension 0 and χ(X) = 0.

4. Examples of smooth surfaces of degree 12

We construct in the sequel several examples of smooth, non-general type surfaces of degree 12 in IP^{4}. In [DES] the authors constructed a smooth, blown-up K3 surface S ⊂ IP^{4} with d = 12, π = 14, with one exceptional quartic and ten exceptional lines. We’ll provide here examples of smooth, regular, proper elliptic surfaces.

Recall first that for a smooth surface S of degree 12 in IP^{4} the double point formula reads

K^{2} = 47 − 5π + 6χ,

while Severi’s theorem and Riemann-Roch give

π = χ + 8 + h^{1}(O_{S}(H)) − h^{0}(O_{S}(K − H)).

Proposition 4.1. There exist smooth, regular, minimal proper elliptic surfaces S ⊂ IP^{4}, with d = 12, π = 13, χ = 3, and 10 skew 6-secant lines.

P roof. For construction we use the Eagon-Northcott approach. A promising Beilinson cohomology table is

i ↑

2

2  h^{i}(I_{S}(p))
4  5

⟶ p

so we may set F = 2O(−1) ⊕ 2Ω^{3}(3) and G = ˜ (ker ψ), where 4R(1)⟵^{ψ}15R is the minimal free presentation of the cohomology module H^{1}(I_{S}(∗ + 4)) = H^{1}(G(∗)), and try to construct the surface as the degeneracy locus of a general morphism φ ∈ Hom(F, G). However, for a general choice of the matrix ψ, the module M = coker ψ has a minimal free resolution of type

M ← 4R(1) ⟵^{ψ} 15R 15R(−1)
↖ ⊕
10R(−2) ↖ 30R(−3) ← 21R(−4) ← 5R(−5) ← 0

and thus Hom(Ω^{3}(3), G) = 0 in this case. What is needed is that the matrix defining ψ has at least two linear syzygies of second order. We’ll choose ψ carefully.

Let F be the Horrocks-Mumford bundle on IP^{4} (see [HM]). It is a stable rank 2 vector bundle with Chern classes c_{1} = −1, c_{2} = 4, and its H^{1}-cohomology module has a minimal free resolution of type (cf. [De1], [De2])

0 ← H^{1}(F (∗)) ← 5R ← 15R(−1) 10R(−2)
↖ ⊕
4R(−3)  2R(−3)
⊕ ↖ ⊕
15R(−4) 35R(−5) ← 20R(−6)
↖ 2R(−8) ← 0.

We provide a somewhat different construction of this module in chapter 7. From [HM], or just by looking to the above resolution since F (−1) is the cokernel of the unique morphism 0 → 2Ω^{3}(2)⟶^{θ}Syz_{1}(H^{1}(F (∗))), we have h^{0}(F) = h^{0}(F (1)) = 0 and h^{0}(F (2)) = 4.

Consider now a rank 3 vector bundle E on IP^{4} constructed as the extension

(4.2)  0 ⟶ F ⟶ E ⟶ O ⟶ 0

corresponding to a non-trivial element 0 ≠ ξ ∈ H^{1}(F) = Ext^{1}(O, F). E has Chern classes c_{1}(E) = −1, c_{2}(E) = 4 and c_{3}(E) = 0, and is stable because F is. Also h^{1}(E) = h^{1}(F)−1 = 4 by construction. We’ll assume in the sequel that the extension E comes from a generic element ξ ∈ H^{1}(F), meaning by this that ξ satisfies the following two conditions:

- the map C I ξ ⊗ H^{0}(O_{IP}^{4}(1)) → H^{1}(F (1)) induced by multiplication with linear forms is injective, while

- the similar natural map C I ξ ⊗ H^{0}(O_{IP}^{4}(2)) → H^{1}(F (2)) is surjective.

To see that such a choice is possible one either checks it for a random ξ with [Mac], or uses the invariance of H^{1}(F (∗)) under the group G = IH_{5} ⋊ ℤ_{2}, where IH_{5} is the Heisenberg group of level 5. Namely, using notations and facts mentioned in (7.4), if IP^{4} = IP(V) then we can find a basis e_{0}, …, e_{4} of V such that, under the Schroedinger representation of IH_{5} on IP^{4}, H^{1}(F) = V_{3}, H^{1}(F (1)) = 2V_{1}^{♯}, H^{1}(F (2)) = 2V_{0}^{♯}, while the multiplication map V^{∗} ⊗ H^{1}(F) → H^{1}(F (1)) is given by the projection on the second factor V^{∗} ⊗ V_{3} = 3V_{1} ⊕ 2V_{1}^{♯} → 2V_{1}^{♯}. One checks easily that ξ = ∑_{i=0}^{4}(−1)^{i}e_{i} ∈ H^{1}(F) has the desired properties, and thus deduces that there is a Zariski open subset of H^{1}(F) fulfilling the two conditions. With this general choice of ξ, the cohomology of the exact sequence (4.2) gives h^{0}(E(1)) = 0, h^{0}(E(2)) = 5, h^{1}(E(1)) = 5 and h^{1}(E(m)) = 0 for all m ≥ 2 or m ≤ −1. In conclusion, we deduce that M = H^{1}(E(∗)) is an artinian module with Hilbert function (4, 5) and with the desired syzygies

M ← 4R ⟵^{ψ} 15R(−1) 15R(−2)  2R(−3)
↖ ⊕ ⟵ ⊕
12R(−3)  30R(−4) ↖ 21R(−5) ← 5R(−6) ← 0

The two above linear syzygies of second order are inherited from those of H^{1}(F (∗)), and thus they involve too two proper Koszul complexes (see (7.5)).

As we can check in an example, the dependency locus of two general sections in H^{0}(E(2)) is a smooth surface S ⊂ IP^{4} with d = 12, π = 13 and χ = 3:

(4.3)  0 ⟶ 2O ⟶ E(2) ⟶ I_{S}(5) ⟶ 0.

This description of S is equivalent to that as the degeneracy locus of a general morphism φ ∈ Hom(F, Syz_{1}(H^{1}(E(∗ + 1)))); we compute for S a minimal free resolution of type

3O(−5)
0 ← I_{S} ← ⊕
12O(−6) ↖ 30O(−7) ⟵ 21O(−8) ⟵ 5O(−9) ← 0.

Dualizing (4.3) we obtain

0 ⟶ O(−5) ⟶ E^{∨}(−2) ⟶ 2O ⟶ ω_{S} ⟶ 0

thus ω_{S} is globally generated, p_{g} = 2, and since the double point formula yields K^{2} = 0 we deduce that S is a minimal proper elliptic surface. Le Barz’s formula (0.11) gives N_{6} = 10, hence there are ten 6-secant lines to S because E(2) is globally generated outside a 3-codimensional set. ⊓ ⊔

Corollary 4.4. There exist smooth non-minimal K3 surfaces X ⊂ IP^{4} with d = 13, π = 16 which are embedded by a linear system

H = H_{min} − 7E_{0} − ∑_{i=1}^{10} E_{i}.

P roof. The minimal proper elliptic surface S ⊂ IP^{4} we’ve constructed above can be linked in the complete intersection of two quintic hypersurfaces to a surface X with invariants d = 13, π = 16, χ = 2 and a resolution of type

(4.5)  0 ⟶ E^{∨}(−2) ⟶ 4O ⟶ I_{X}(5) ⟶ 0.

Smoothness can be checked in an example. X is cut out by quintic hypersurfaces, hence there are no 6-secant lines. On the other hand, Le Barz’s formula gives N_{6} = 10, so there exist 10 exceptional lines on X, namely the 6-secant lines of S. Let now X_{1} denote the image of X under the adjunction map, and X_{2} denote the image of X_{1} under the adjunction map defined by |H_{1} + K_{1}|. From (0.13) we obtain the following invariants

X_{1} ⊂ IP^{16} H_{1}^{2} = 36  H_{1}K_{1} = 6  K_{1}^{2} = −1  π_{1} = 22
X_{2} ⊂ IP^{22} H_{2}^{2} = 47  H_{2}K_{2} = 5  K_{2}^{2} = −1 + b  π_{2} = 27,

where b is the number of (−1)-conics on X. Hodge index gives K_{2}^{2} = −1 + b ≤ 0, thus lemma 0.18 implies that X is either a K3 or a proper elliptic surface. Moreover, in case it is elliptic, X has a (−1) conic or a (−1) cubic and the proper transform of the canonical divisor on the minimal model is an elliptic curve of degree 4 or 5, while in case X is a K3 surface there is an exceptional rational septic curve on it. Now it is easily seen that in the case of an elliptic surface Kodaira’s formula for the canonical divisor implies that p_{2} = h^{0}(ω_{X}^{⊗2}) ≥ 2, so to prove the claim of the corollary we check that p_{2} = 1. Dualizing (4.5) we obtain a resolution of ω_{X}

0 ⟶ O(−5) ⟶ 4O ⟶ E(2) ⟶ ω_{X} ⟶ 0,

and thus one of the associated scandinavian complexes yields the resolution:

(4.6)  0 ⟶ 6O ⟶ 4E(2) ⟶ S^{2}(E)(4) ⟶ ω_{X}^{⊗2} ⟶ 0.

Splitting up (4.6) in short exact sequences and using the fact that h^{1}(E(2)) = 0 and h^{0}(E(2)) = 5 we obtain that p_{2} = h^{0}(S^{2}(E)(4)) − 14. Now, by (4.2), the bundle S^{2}(E) can be realized as an extension

0 ⟶ S^{2}(F) ⟶ S^{2}(E) ⟶ E ⟶ 0

and since h^{0}(S^{2}(F)(4)) = 0 and h^{1}(E(4)) = 0 we need to compute the rank of the cobord morphism H^{0}(E(4))⟶H^{1}(S^{2}(F)(4)). One checks in an example, via [Mac], that the kernel of the previous cobord is 15, thus p_{2} = 1 and X is a K3-surface of the claimed type. From (4.5) we obtain also a minimal free resolution of type

4O(−5)
0 ← I_{X} ← ⊕
5O(−6) ↖ 16O(−7) ← 10O(−8) ← 2O(−9) ← 0
⊓ ⊔

Proposition 4.7. There exist smooth, regular, proper elliptic surfaces S ⊂ IP^{4}, with invariants d = 12, π = 14, χ = 3 and embedded by one of the following linear systems

a)  H = H_{min} − 2E_{0} − ∑_{i=1}^{4} E_{i},

b)  H = H_{min} − ∑_{i=1}^{5} E_{i}.

P roof. A possible Beilinson cohomology table is

i ↑

2

3  2  h^{i}(I_{S}(p))
1  1

⟶ p Thus we may take F = 2O(−1)⊕3Ω^{3}(3) and G = ker ψ, for some epimorphism ψ : 2Ω^{2}(2)⊕ Ω^{1}(1) → O, and check for the degeneracy locus of a general morphism φ ∈ Hom(F, G). Identifying IP^{4} = IP(V), with V = span_{k}(e_{0}, …, e_{4}), the morphism ψ is induced by a triple (ψ_{11}, ψ_{12}, ψ_{2}), where ψ_{11}, ψ_{12} ∈ Λ^{2} V and ψ_{2} ∈ V. We check the various possibilities for ψ and see that only two of them lead to smooth surfaces. Namely, if we take α) ψ generic, then the corresponding vector bundle G has a minimal free resolution of type

0 ← G ← 25O(−1) 10O(−2)  O(−3)
↖ ⊕ ⟵ ⊕
4O(−3)  4O(−4) ↖ O(−5) ← 0,

and a general morphism φ ∈ Hom(F, G) gives a smooth surface S_{α} ⊂ IP^{4} with minimal free resolution

0 ← I_{S_{α}} ← 8O(−5) 7O(−6)  O(−7)
↖ ⊕ ⟵ ⊕
4O(−7)  4O(−8) ↖ O(−9) ← 0

while, if we take β) ψ distinguishing a plane, e.g., say ψ_{11} = 0, ψ_{12} = e_{0} ∧ e_{1} and ψ_{2} = e_{2}, then we obtain a vector bundle G with resolution

25O(−1)  12O(−2)  2O(−3)
0 ← G ← ⊕ ⟵ ⊕ ⟵ ⊕
2O(−2)  5O(−3)  4O(−4) ↖ O(−5) ← 0,

and the generic φ ∈ Hom(F, G) defines a smooth surface S_{β} ⊂ IP^{4} with syzygies

8O(−5)  9O(−6)  2O(−7)
0 ← I_{S_{β}} ← ⊕ ⟵ ⊕ ⟵ ⊕
2O(−6)  5O(−7)  4O(−8) ↖ O(−9) ← 0.

Smoothness can be checked in examples on a computer via [Mac]. It is easily seen, as in the proof of (3.18), that all other choices of ψ lead to singular surfaces or to determinantal loci which are not in the expected codimension. We determine next what type of surfaces we’ve constructed.

Let now S_{1} denote the image of S_{α} (or S_{β} resp.) under the adjunction map, and let S_{2} be the image of S_{1} under the adjunction map defined by |H_{1} + K_{1}|. From (0.13) we obtain the following invariants

S_{1} ⊂ IP^{16} H_{1}^{2} = 35  H_{1}K_{1} = 9  K_{1}^{2} = −5 + a  π_{1} = 23
S_{2} ⊂ IP^{24} H_{2}^{2} = 48 + a  H_{2}K_{2} = 4 + a  K_{2}^{2} = −5 + a + b  π_{2} = 27 + a,

where a is the number of (−1)-lines and b is the number of (−1)-conics on S_{α} (or S_{β} resp.).

In case α), the ideal I_{S_{α}} is generated by quintic hypersurfaces so S_{α} has no 6-secant lines and, since Le Barz’s formula (0.11) gives N_{6} = 4, there are 4 exceptional lines E_{1}, E_{2}, … E_{4} on S_{α}. Let S_{min} denote the minimal model of S_{α} and assume first that it is a surface of general type. Then S_{α} has at least two other exceptional curves F_{1} and F_{2} of degree ≥ 2, and thus there would exist a curve in |K_{min}−F_{1}| of degree ≤ HK_{min}−2 ≤ 4 and arithmetic genus p_{a}(K_{min}) ≥ 2, which is a contradiction. It follows that S_{α} is a non-minimal elliptic surface, and thus K ∼ K_{min} + ∑_{i=1}^{4} E_{i} + E_{0}, where E_{0} is a (−1) curve of degree ≥ 2. On the other side a curve in |K_{min} − E_{0}| has arithmetic genus one, so HE_{0} ≤ 3. We investigate first the case when HE_{0} = 3. Then a curve D ∈ |K_{min} − E_{0}| has degree 4 and arithmetic genus one, so it spans only a hyperplane in IP^{4}. The residual curve G ∼ H − D has degree 8 and genus 9. We check now that G lies on a quadric surface in IP^{3}. Namely, since Riemann-Roch gives χ(O_{G}(2H)) = 8 it is enough to check that h^{1}(O_{G}(2H)) ≤ 1. But this follows from the cohomology of the exact sequence

0 ⟶ O_{S}(H + D) ⟶ O_{S}(2H) ⟶ O_{G}(2H) ⟶ 0

since, in the constructed example, h^{1}(O_{S}(1)) = 3, h^{1}(O_{S}(2)) = 2 and h^{2}(O_{S}(H + D)) = h^{0}(O_{S}(K − D − H)) = 0, while the composite multiplication map

H^{1}(O_{S}(H))⟶^{D}H^{1}(O_{S}(H + D)) ⟶^{H−D}H^{1}(O_{S}(2H))

drops rank at most one on ǏP^{4}, for a general choice of the morphism φ ∈ Hom(F, G). Now the curve D lies on two quadrics, thus for the hyperplane section H cut out by the IP^{3} of D on S_{α}, we get h^{0}(I_{H}(4)) ≥ 2. This is a contradiction, since under the previous assumption of minimal cohomology table we have h^{0}(I_{H}(4)) ≤ h^{1}(I_{S}(3)) = 1, for all hyperplane sections H. Therefore E_{0} is an exceptional conic and S_{α} is a non-minimal elliptic surface embedded by a linear system of type a), as claimed in the statement of the proposition.

In case β), it is easily seen that the distinguished plane Π = IP(span(e_{0}, e_{1}, e_{2})) meets S_{β} along a plane quintic curve C and the point P = IP(span(e_{2})) outside C. Therefore S_{β} has infinitely many 6-secant lines, namely all lines in Π going through P, and so Le Barz’s formula doesn’t apply in this case. In fact, using the explicit form of the syzygies of the ideal sheaf I_{S_{β}}, it is easily seen that these are all the 6-secant lines to S_{β}. Taking cohomology of the exact sequences

0 ⟶ I_{S_{β}}(k − 1) ⟶ I_{S_{β}}(k) ⟶ I_{H}(k) ⟶ 0  k = \overline{3, 4}

we observe that h^{1}(I_{H}(3)) = 3 for all hyperplane sections H of S_{β}, and that h^{1}(I_{H}(4)) = 1 if and only if P ∈ H. Therefore each hyperplane through P contains a plane π such that h^{1}(I_{π∩S_{β}}(4)) = 1. In particular, for the pencil of hyperplanes through Π there exists a quadric cone

Q = \{ det ( \begin{matrix}l & m \\ x_{3} & x_{4} \end{matrix} ) = 0 \},

where l and m are suitable linear forms, such that h^{1}(I_{π_{(λ:μ)}∩S_{β}}(4)) = 1 holds for all planes π_{(λ:μ)} = {μl + λm = μx_{3} + λx_{4} = 0} in one of the rulings of Q. Now the plane Π is obviously a member of the opposite ruling of Q, and thus residual to C in the complete intersection S_{β} ∩ Q there is a curve G ∼ 2H − C of degree 19 and arithmetic genus 23. On another side, residual to C, in the hyperplane sections through Π, there is a pencil |D| with base point P, of curves of degree 7 and genus 3. It follows that the curve G splits as G = G_{1} + G_{2}, where G_{1} is a union of plane curves contained in planes of the ruling π_{(λ:μ)}, while G_{2} is a curve of degree 14 which maps down via projection from the vertex of Q to a complete intersection of type (7, 7) on the quadric surface which is the base of the cone. It is easily checked that G_{1} splits as the union of 5 exceptional lines E_{1}, E_{2}, …, E_{5} on S_{β}. Therefore, on the first adjoint surface S_{1} we obtain K_{1}^{2} ≥ 0, and in fact, by Hodge index, the equality K_{1}^{2} = 0 holds. We argue further as in case α). If S_{β} would be a surface of general type then it contains further an exceptional curve E of degree ≥ 2 and thus a curve N ∈ |K_{min} − E| would have degree at most 5 and arithmetic genus at least 2. Thus the only possibility is K_{min}^{2} = 1, HE = 2 and HN = 5, p_{a}(N) = 2. If N spans only a IP^{3}, then the residual curve H − N would have degree 7 and arithmetic genus 8, which is impossible by lemma 0.34. Therefore N spans all of IP^{4} and necessarily splits as N = A + B, with A a plane quartic curve and B a line disjoint of it. But then A^{2} + B^{2} = N^{2} = 0, B^{2} ≤ −2 since B cannot be exceptional, while Hodge index yields A^{2} ≤ \frac{16}{12}, and thus we obtained a contradiction. It follows that S_{β} is this time a non-minimal elliptic surface embedded by a linear system of type b).⊓ ⊔

5. Examples of Enriques surfaces of degree 13

We construct in this chapter two examples of smooth, non-minimal Enriques surfaces S ⊂ IP^{4} with invariants d = 13, π = 16.

Lemma 5.1. Let S ⊂ IP^{4} be a smooth surface with d = 13, π = 16, p_{g} = q = 0. Then either:

a) S is rational, or

b) S is a non-minimal Enriques surface with 17 exceptional lines.

P roof. Adjunction and the double point formula give HK = 17 and K^{2} = −17. Let now S_{1} = φ_{H+K}(S) ⊂ IP^{15} be the image of S through the adjunction morphism. We obtain for its invariants

H_{1}^{2} = 30,  H_{1}K_{1} = 0,  π_{1} = 16,  K_{1}^{2} = −17 + a

where a is the number of (-1) lines on S. Hence, since H_{1}(nK_{1}) = 0 for all n, it follows that either p_{2} = 0 and S is rational by Castelnuovo’s criterion, or 2K_{1} ∼ 0 and S is an Enriques surface with 17 exceptional lines. ⊓ ⊔

Remark 5.2. Le Barz’s 6-secant formula gives N_{6} = N_{6}(13, 16, 1) = 17, thus a surface S which is cut out by quintic hypersurfaces is necessarily an Enriques surface of type b).

(5.3.) Constructions. We provide in the sequel a construction of smooth Enriques surfaces as in the above lemma. A possible Beilinson cohomology table is

i ↑

6  5  1  h^{i}(I_{S}(p))

1

⟶ p

Riemann-Roch gives also h^{2}(O_{S}(−1)) = h^{0}(O_{S}(H + K)) = π + χ − 1 = 16. Therefore, the above diagram suggests to construct I_{S}(4) as the cohomology of a quasi-monad

0 ⟶ 16O ⟶^{φ} T ⟶^{ψ} O ⟶ 0

where T is an appropriate sheafified syzygy module of M = ⊕_{m∈ℤ} H^{2}(I_{S}(m + 4)). An artinian module M with a minimal free resolution of the form

0 ← M ⟵ 6R(3) ⟵ 25R(2) 36R(1)  16R
α ↖ ⊕ ⟵ ⊕
R  10R(−1) ↖ 9R(−2)
β↖ R(−4) ← 0

would have the desired Hilbert function and would possibly fit our approach. To obtain a module with these syzygies it is easier to construct, as we did before, the k-dual module M^{∗}. Namely, we look for nine quadrics without common zeroes, having a single second order linear syzygy. Moreover, the second order syzygy needs to properly involve 5 of the first order syzygies since we want the induced morphism T ⟶^{ψ}O to be surjective. We can achieve this when, for instance, this part of the resolution comes from a truncated Koszul complex. Therefore we consider part of the Koszul complex

\begin{aligned}& & & & & & R(−2) \\ δ↑ & {↖ \atop }ε \\ 0 & ← & R & ← & 5R(−1) & ← & 10R(−2) & ←_{γ} & 10R(−3) \end{aligned} ,

a general projection δ and the composition ε = δ ∘ γ. Then a general morphism in

Hom(R(−4), ker ε)

will give by composing with γ the nine quadrics defining β^{∗}. Resolving β^{∗} gives α. We take now T = Syz_{2}(M) and φ, ψ those morphisms defined by the resolution of M. In fact, if we denote the linear part of α by α′ and put E = 16O_{IP}^{4}, F = ker(36O_{IP}^{4}(1)⟶^{α̃′}25O_{IP}^{4}(2)) = ker(T ⟶^{ψ}O) and φ̄ ∈ Hom(E, F) the morphism induced by φ ∈ Hom(16O, T ), then coker φ̄ = I_{S}(4) is the twisted ideal sheaf of a smooth surface S with the desired invariants. We compute also a minimal free resolution of type

5O(−5)
0 ← I_{S} ⟵ ⊕
O(−6) ↖ 10O(−7) ⟵ 6O(−8) ⟵ O(−9) ← 0

On the other hand, as we can check, S is cut out only by the quintics generating the homogeneous ideal I_{S} and therefore, by remark 5.2, S is a non-minimal Enriques surface with only 17 exceptional lines.

Remark 5.4 The above family was constructed in collaboration with Frank-Olaf Schreyer.

We describe in the sequel a variation of the above construction leading to a different family of Enriques surfaces. We start again with part of the Koszul complex

2R(−2)
δ↑  {↖ \atop }ε
0 ← R ← 5R(−1) ← 10R(−2) ←_{γ} 10R(−3)

and, this time, with a general projection δ on 2R(−2) and consider the composition ε = δ ∘ γ. A general morphism in Hom(R(−4), ker ε) will give by composing with γ eight quadrics cutting out a zero-dimensional scheme Z. We take as the nine quadrics defining β^{∗} the previous eight plus a general one avoiding the scheme Z. The resulting module M will have a minimal free resolution with the same Betti numbers. We put, as before, T = Syz_{2}(M) and consider the monad

0 ⟶ 16O ⟶^{φ} T ⟶^{ψ} O ⟶ 0

where φ and ψ are those induced by the resolution of M. Its cohomology is the twisted ideal sheaf of a smooth surface S with the same invariants as above, having a minimal free resolution of type

5O(−5)  O(−6)
0 ← I_{S} ⟵ ⊕ ⟵ ⊕
2O(−6)  10O(−7) ↖ 6O(−8) ⟵ O(−9) ← 0

We can check, as in the previous construction, that S is cut out only by the quintics generating the homogeneous ideal I_{S} and therefore, by remark 5.2, S is again a non-minimal Enriques surface with 17 exceptional lines. The extra linear syzygy of the quintics in I_{S} possess only 3 linearly independent components, which in turn define one of the exceptional lines of S.

Remarks 5.5.

a) An Enriques surface S belonging to one of the above two families is, by lemma 0.32, the unique minimal scheme in its even liaison class.

b) We were not able to find flat deformations from Enriques surfaces belonging to the first family to those of the second one, or to prove that the two constructed families reside in different components of the Hilbert scheme.

c) The existence, or non-existence of rational surfaces with these invariants is also completely open.

6. A K3 surface of degree 14

We construct in this chapter an example of a smooth non-minimal K3 surface of degree 14 in IP^{4}. Namely, we show

Proposition 6.1. There exist smooth, non-minimal K3 surfaces S = S_{min}(p_{1}, …, p_{15}) ⊂ IP^{4} with d = 14, π = 19, K^{2} = −15, and embedded via

H = H_{min} − 4E_{0} − ∑_{i=1}^{4} 2E_{i} − ∑_{j=5}^{14} E_{j},

where |H_{min}| is very ample on S_{min} of degree 56 and dimension 29.

P roof. We discuss first an approach as in chapter 1 via the Eagon-Northcott complex. A plausible Beilinson cohomology table for a surface with these invariants is

i ↑

1

7  7  3  h^{i}(I_{S}(p))

⟶ p

thus, in this case, everything is determined by M = ⊕_{m∈ℤ} H^{2}(I_{S}(m + 4)). We’ll look again to the dual module M^{∗} and we’ll assume that it is generated, as R−module, by Hom_{k}(H^{2}(I_{S}(3)), k). Then, by considering the Hilbert function of M^{∗}, we find that M^{∗} has a minimal free resolution of type

0 ⟵ M^{∗} ⟵ 3R(−1) ⟵^{ψ} 8R(−2) ⊕ mR(−3)

with m ≥ 0. The morphism ψ = (ψ_{1}, ψ_{2}) is given by a 3 × (8 + m) matrix with linear entries in ψ_{1} and quadratic entries in ψ_{2}. Also m > 0 if and only if ψ_{1} has at least three non-trivial linear syzygies. However, this doesn’t occur for a general choice of ψ_{1}; in this case the cokernel of ψ_{1} is an artinian graded module with Hilbert function (3, 7, 5). In order to obtain a module with the desired Hilbert function, it is necessary that the number of linear syzygies of ψ_{1} equals m + 2, and we’ll choose it correspondingly. The idea is to start with four planes P_{i} = \{l_{i1} = l_{i2} = 0\}, i = \overline{1, 4}, and to consider the direct sum of the four Koszul complexes built on { l_{i1}, l_{i2} }

4R ⟵^{α} 8R(−1) ⟵^{β} 4R(−2).

We take as ψ_{1} a morphism given by 3 general lines of α, i.e., ψ_{1} = γα(−1), with γ ∈ M_{3,4}(k) a random matrix, and as ψ_{2} a general 3 × 2 matrix with quadratic entries, because ψ_{1} has exactly 4 linear syzygies. M^{∗} = coker ψ is an artinian graded module with Hilbert function (3, 7, 7) and from (1.11) we can compute its minimal free resolution

0 ← M^{∗} ⟵ 3R(−1) 8R(−2) 4R(−3)
↖_{ψ} ⊕  ⊕
2R(−3) ↖ 5R(−4)
⊕
15R(−5) ↖ 38R(−6) ⟵ 28R(−7) ⟵ 7R(−8) ← 0

We dualize, and set according to proposition 1.15 F = Syz_{2}(M). To obtain the second bundle, we compare the syzygies of F with Beilinson’s spectral sequence for I_{S}. Namely, the E_{∞}-filtration yields an exact sequence

0 ⟶ O(−1) ⊕ (H^{0}(F) ⊗ O) ⟶ F ⟶ I_{S}(4) ⟶ 0.

Furthermore, h^{0}(F) = 15, and if we put E = O(−1) ⊕ 15O, one checks, via [Mac] in examples, that the degeneration locus of a φ ∈ Hom(E, F) = Hom(O(−1) ⊕ (H^{0}(F) ⊗ O), F) = H^{0}(F(1)) ⊕ Hom(H^{0}(F) ⊗ O, F), given by a general section and the natural evaluation map, is a smooth surface S with the desired numerical invariants and the desired cohomology. The minimal free resolution of the ideal sheaf of the surface is of type

4O(−5)  2O(−6)
0 ← I_{S} ⟵ ⊕ ⟵ ⊕
4O(−6)  8O(−7) ↖ 3O(−8) ← 0

so the homogeneous ideal is generated by 4 quintics and 4 sextics. Moreover, it follows from the construction of the module M^{∗} that the four quintics containing S intersect in

V ((I_{S})_{≤5}) = S ∪ ⋃_{i=1}^{4} P_{i},

and a closer look to the syzygies of M^{∗} shows that S cuts each plane P_{i} along a sextic curve. Hence each of the planes P_{i} contains an ∞^{2} of 6-secant lines, and in particular Le Barz’s formula doesn’t apply to this example.

To determine the type of surface we constructed, we check in an example that S ∪ ⋃_{i=1}^{4} P_{i} is an arithmetically Cohen-Macaulay scheme of degree 18 and sectional genus 39, with syzygies of type

0 ← I_{S∪⋃_{i=1}^{4} P_{i}} ⟵ 4O(−5) 2O(−6)
↖ ⊕ ← 0
O(−8)

and, moreover, that the minors of the 4 × 2 submatrix 4O(−5)⟵^{α}2O(−6) vanish precisely along an exceptional quartic curve E_{0} on S. Furthermore, by computing equations for the divisor contracted by the adjunction mapping, we see that there are ten exceptional lines on the surface. Let now S_{1} denote the image of S under the adjunction map, and S_{2} denote the image of S_{1} under the map defined by |H_{1} + K_{1}|. From (0.13) we obtain the following invariants

S_{1} ⊂ IP^{19} H_{1}^{2} = 43  H_{1}K_{1} = 7  K_{1}^{2} = −5  π_{1} = 26
S_{2} ⊂ IP^{26} H_{2}^{2} = 52  H_{2}K_{2} = 2  K_{2}^{2} = −5 + b  π_{2} = 28,

where b is the number of (−1)-lines on S_{1}. But K_{2}^{2} = −5 + b ≥ −H_{2}K_{2} = −2, and there exists already an exceptional quartic curve E_{0} on S, so the only possibility is that b = 4 and K_{2} is a (−1) conic on S_{2}. As it turns out, S = S_{min}(p_{0}, …, p_{14}) is a minimal K3 surface blown up in 15 points and

H = H_{min} − 4E_{0} − ∑_{i=1}^{4} 2E_{i} − ∑_{j=5}^{14} E_{j},

where H_{min} is a very ample linear system on S_{min} = S_{4}, giving an embedding S_{min} ⊂ IP^{29} with deg S_{min} = 56.

We want further to recover an alternative liaison construction for the above K3 surface. Proposition 0.30 and remark 0.31 ensure that we can link (5, 5) the configuration S∪⋃_{i=1}^{4} P_{i} to a smooth surface Y of degree 7 and sectional genus 6. The cohomology of the liaison exact sequence

0 ⟶ O_{Y} (K_{Y}) ⟶ O_{Σ_{5,5}}(5) ⟶ O_{S∪⋃_{i=1}^{4} P_{i}}(5) ⟶ 0,

where Σ_{5,5} denotes the complete intersection of the two quintic hypersurfaces used in the linkage, gives p_{g}(Y) = 2 and q(Y) = 0 while the double point formula yields K_{Y}^{2} = 0. Surfaces with these invariants are classified in [Ok2] and are known to be arithmetically Cohen-Macaulay, minimal proper elliptic surfaces. Namely, |K_{Y} | is a pencil without base points of plane cubic curves; the planes spanned by its members being those in one ruling of the determinantal quadric defined by the linear syzygies in

0 ⟶ 2O(−5) ⟶ 2O(−4) ⊕ O(−2) ⟶ I_{Y} ⟶ 0.

Then liaison, once again, shows that Y cuts each plane P_{i} along a conic C_{i}, which is necessarily a section of the elliptic fibration, since there are no singular fibers and the fibration is by plane curves. In particular, this means that the rank of the Picard group of Y is at least 6, while the Picard number of a generic elliptic surface of degree 7 in IP^{4} is only 2 by [ElP2]. Therefore we‘ll have to choose in the sequel Y carefully in order to recover S via liaison from the scheme Y ∪ ⋃_{i=1}^{4} P_{i}.

(6.2.) Liaison construction. The above facts suggest us the following liaison method of construction for this family of K3 surfaces. Let P, P_{1}, P_{2}, P_{3}, P_{4} be five planes in general position in IP^{4} and denote by {p_{ij}} = P_{i} ∩ P_{j}, for 1 ≤ i < j ≤ 4, the mutual intersection points of the last four of them.

Lemma 6.3.

a) The homogeneous ideal I_{P ∪{p_{ij},1≤i<j≤4}} is generated by 3 quadrics and 4 cubics.

b) The three quadrics intersect along the plane P and a rational normal quartic curve Q, which is trisecant to P and goes through the points p_{ij}.

P roof. The first part follows from the cohomology of the residual intersection sequences

0 ⟶ I_{{p_{ij},i<j}}(m − 1) ⟶ I_{P ∪{p_{ij},i<j}}(m) ⟶ O_{IP}^{2}(m − 1) ⟶ 0

where m ∈ ℤ. For the second part observe that the plane P is linked in the complete intersection of two of the quadrics to a rational cubic scroll T. If H_{T} ∼ C_{0} + 2f, with C_{0}^{2} = −1, C_{0}f = 1 and f^{2} = 0, is the embedding of the scroll in IP^{4}, then P ∩T ∼ C_{0} +f is a conic and thus the third quadric cuts on T the rational normal quartic curve Q ∼ C_{0}+3f. Now P ∩ Q = Q(C_{0} + f) = 3 and the lemma follows.⊓ ⊔

We consider now a general quadric V ∈ H^{0}(I_{P ∪{p_{ij}}}(2)) and denote with C_{i} the conics V ∩ P_{i}, for i = \overline{1, 4}. They intersect pairwise in the points \{p_{ij}\} = C_{i} ∩ C_{j}, 1 ≤ i < j ≤ 4.

Lemma 6.4. There exists a unique rational normal quartic curve E_{0} which is contained in V, passes through the points p_{ij}, 1 ≤ i < j ≤ 4, and intersects the plane P in one point p.

P roof. The claim is closely related to a theorem by James, see [Ja], [Sem]. For the proof we use an idea of Semple [Sem], [SR]. Consider the rational map γ : IP^{4}^{−−− >}IP^{5} given by the quadrics through the rational normal quartic curve Q in (6.3). It is one to one onto a smooth hyperquadric Ω in IP^{5}, which we’ll identify in the sequel with the image of the grassmannian of lines in IP^{3} under the Plücker embedding. Let ĨP^{4} be the blowing up of IP^{4} along Q and denote by E the exceptional divisor and by γ̃ : ĨP^{4}⟶Ω ⊂ IP^{5} the induced morphism. Then the trisecant planes of Q are mapped through γ to the planes of one generating system, say α-planes, of the grassmannian Ω, while E is mapped by γ̃ onto a sextic threefold ruled in β-planes. Each of the β-planes corresponds to the normal directions in IP^{4} at points of Q. We remark also that quadric cones through Q are mapped via γ to special linear complexes, i.e., to tangent hyperplane sections of Ω. To fix notations, let now H ⊂ IP^{5} be the hyperplane corresponding to V and let β_{ij}, 1 ≤ i < j ≤ 4, be the β-planes corresponding to the points p_{ij}. Rational normal quartic curves which meet Q in six points are represented via γ by conics in which Ω is met by planes. Thus, in order to prove the lemma, all we need to check is that in H exists exactly one plane meeting all six lines H ∩ β_{ij} and not contained in the quadric cone H ∩ Ω. But this is clear since, by Schubert calculus, the Plücker embedding of the grassmannian of planes in IP^{4} has degree 5, while the planes of the cone H ∩ Ω describe via the same Plücker embedding the union of two conics. The rational quartic curve E_{0} represented by this unique plane meets P in one point because γ maps P in an α-plane contained in H.⊓ ⊔

Lemma 6.5. If T = P ∪⋃_{i=1}^{4} C_{i}, then its homogeneous ideal I_{T} is generated by 1 quadric, 2 cubic and 4 quartic hypersurfaces.

P roof. One uses again the residual exact sequences

0 ⟶ I_{∪_{i=1}^{4}C_{i}}(m − 1) ⟶ I_{T} (m) ⟶ I_{P ∪(∪_{i=1}^{4}C_{i}∩H)}(m) ⟶ 0

where H is a general hyperplane through P and m ∈ ℤ, together with the fact that I_{∪_{i=1}^{4}C_{i}} is generated by 1 quadric and 8 cubic hypersurfaces.⊓ ⊔

From the above lemma it follows that P can be linked in the complete intersection of V and a general quartic hypersurface W ∈ H^{0}(I_{T} (4)) to a smooth, minimal proper elliptic surface Y ⊂ IP^{4} with deg Y = 7, π(Y) = 6. Moreover, by construction, the conics C_{i} lie on Y.

Lemma 6.6.

a) On Y we have C_{i}^{2} = −3 and K_{Y} C_{i} = 1, so each conic C_{i} is a section of the elliptic fibration.

b) The planes P_{i} intersect Y exactly along the conics C_{i}.

P roof. In any case K_{Y} C_{i} ≥ 1 since there are no multiple fibers. On the other hand, we recall that the elliptic fibration is cut out on Y by the planes in one of the rulings of the cone V. Thus if K_{Y} C_{i} ≥ 2, then P_{i} would lie on V and this would contradict our choices. It follows that C_{i}^{2} = −3 and K_{Y} C_{i} = 1. Part b) is set theoretically clear by construction. The claim follows because residual to each conic C_{i} there is a pencil |H_{Y} − C_{i}| of curves of degree 5 and genus 2, without base points since (H_{Y} − C_{i})^{2} = 0.⊓ ⊔

We need in the sequel some classical facts of projective geometry.

Proposition (Segre) 6.7. With any four general planes P_{i}, i = \overline{1, 4}, there is associated a uniquely determined fifth plane P_{5}, such that all lines which meet the first four planes meet also the fifth.

P roof. As mentioned above, the Plücker embedding of the grassmannian of lines in IP^{4} has degree 5, thus the claim follows because the special linear complexes consist of lines meeting a given plane. See also [Seg] or [SR]. ⊓ ⊔

Corollary (Segre)[Seg] 6.8. The lines in IP^{4}, which meet the four initial planes, generate a cubic hypersurface X containing the five planes P_{i}, i = \overline{1, 5}, and having singularities (nodes) exactly at the ten points at which the planes meet in pairs.

P roof. We briefly recall the arguments in [Seg]. The first part of the claim follows from Bezout since H^{0}(I_{∪P_{i}}(3)) = 1 and from Schubert calculus in G(1, 4), since if l is a line meeting P_{4} at one point q, in which case it is contained in a hyperplane H through P_{4}, then there is one line through q which meets P_{1}, P_{2} and P_{3}, and there are two other lines meeting l and the three lines in which H cuts P_{1}, P_{2} and P_{3}, respectively. To check singularities, observe first that residual to a plane P_{i} in a general hyperplane section of X through it, there is a quadric surface containing 4 skew lines, thus smooth. Therefore X has only isolated singularities and an easy argument shows that these are exactly the ten points of pairwise intersection of the planes P_{i}.⊓ ⊔

A cubic threefold X ⊂ IP^{4} with the maximum number of ordinary double points, namely 10, is unique up to projective equivalence (cf. [Seg], [Ka]). It’s desingularization X̃ is isomorphic to IP^{3} blown-up in five points a_{1}, …, a_{5}, in general position. The morphism φ : X̃ → X ⊂ IP^{4} is given by the quadrics through the five points while the nodes are the images of the lines joining any two of the points a_{i}. We mention in the sequel some of the properties of this threefold (cf. [Seg], [SR], [Fi]).

The Segre cubic primal X has a symmetrical system of 15 planes, of which 5 correspond to the exceptional divisors over the points a_{i}, and 10 to the planes P_{ijk} = φ(span(a_{i}, a_{j}, a_{k})), for {i, j, k} ⊂ {1, 2, 3, 4, 5}. The symmetry of the the planes resides in the following properties:

- each plane contains four of the nodes,

- each plane is met in lines by 6 others, namely the plane corresponding to a_{i} by the planes P_{ijk}, for all {k, j} ⊂ {1, 2, 3, 4, 5} \ {i}, and the plane P_{ijk} by those corresponding to a_{i}, a_{j}, a_{k} and P_{α,β,γ}, with α ∈ {i, j, k} and {β, γ} = {1, 2, 3, 4, 5} \ {i, j, k}.

We’ll assume in the sequel that we’ve chosen the desingularization morphism φ such that the planes P_{i}, i = \overline{1, 4}, correspond to the exceptional divisors over a_{i}. Let now Z = Y ∪ ⋃_{i=1}^{4} P_{i}. It is a local complete intersection scheme, except for the points P_{ij} which are Cohen-Macaulay of the type described in (0.31), and has invariants deg Z = 11, π(Z) = 10, χ = 3, q = 0. We remark here that Hodge index implies that there is no smooth surface in IP^{4} with these invariants. By computing syzygies one shows that Z has a resolution of type

0 ⟶ 2O(−1) ⊕ (H^{0}(G) ⊗ O) ⟶ G ⟶ I_{Z}(4) ⟶ 0

with G = Syz_{1}(M^{∗})(3), where M^{∗} is the graded artinian module we constructed above. In particular the homogeneous ideal I_{Z} is generated by 3 quintic and 15 sextic hypersurfaces, and thus we can link Z in the complete intersection of two quintics to a surface S with d = 14, π = 19, χ = 2, q = 0. One checks in examples via [Mac] that S is smooth.

Remark 6.9. By liaison, each plane P_{i}, i = \overline{1, 4}, intersects S along a sextic curve D_{i}, thus each of them contains an ∞^{2} of 6-secant lines.

Lemma 6.10.

a) E_{0} is an exceptional quartic on S.

b) The planes P_{ijk}, with {i, j, k} ⊂ {1, 2, 3, 4}, cut the surface S along four exceptional conics.

P roof. The rational normal curve E_{0} is contained in V and intersects W in a scheme of length 16, of which one point is on P. Thus, for general choices, E_{0} cuts Z = Y ∪ ⋃_{i=1}^{4} P_{i} along a scheme of length 15 + 6 = 21 and, by Bezout, lies on all quintic hypersurfaces containing Z, whence on S. We show now that, say P_{123} cuts S along a conic; the other cases being similar. Observe first that P_{123} cuts P_{1}, P_{2} and P_{3} along the lines pairwise joining the points p_{12}, p_{13}, p_{23}, while P_{4} and P_{5} meet both this plane at the node v_{45} corresponding to the line through a_{4} and a_{5}. For general choices P_{123} meets Y in a scheme of length 7: p_{12}, p_{13}, p_{23} and four extra points. Let E_{4} denote the unique conic through these four points and the node v_{45}. It is easily seen that E_{4} is a 11-secant conic to the configuration Z, so by Bezout it necessarily lies on S.

By linkage K_{S} + ((Y ∪ ⋃_{i=1}^{4} P_{i}) ∩ S)_{S} ∼ 5H_{S}. On the other hand (Y ∩ S)_{S} ∼ 5H_{Y} − K_{Y} − ∑_{i=1}^{4} C_{i}, which is a curve of degree 24 and arithmetic genus 37, and the quartic E_{0} lie both on the quadric cone V. It follows that K_{S} + ((⋃_{i=1}^{4} P_{i}) ∩ S)_{S} ∼ 3H_{S} + E_{0}, thus K_{S} + ∑_{i=1}^{4} D_{i} ∼ (X ∩ S)_{S} + E_{0}. Since H(K − E_{0} − ∑_{i=1}^{4} E_{i}) = 22 − 12 = 10 and K^{2} = −15 we deduce easily that E_{i}, i = \overline{0, 4}, are exceptional curves on S.⊓ ⊔

Adjunction and the above lemma show also that S must have 10 exceptional lines, thus it is a non-minimal K3 surface embedded by

H = H_{min} − 4E_{0} − ∑_{i=1}^{4} 2E_{i} − ∑_{j=5}^{14} E_{j}.

Corollary 6.11. The Segre cubic primal X intersects S along the union of 10 exceptional lines, 4 exceptional conics and 4 plane sextic curves.

A similar liaison construction gives also the following

Proposition 6.12. There exist smooth, non-minimal general type surfaces S ⊂ IP^{4} with invariants d = 15, π = 22, p_{g} = 3, q = 0, K^{2} = −6, and with 9 exceptional lines.

P roof. We start this time with a Castelnuovo surface Y ⊂ IP^{4}, i.e., with a smooth, arithmetically Cohen-Macaulay, rational surface with d = 5, π = 2 and K^{2} = 1 (see [Be] or [Ok1]). Y is linked to a plane in the complete intersection of a hyperquadric and a cubic hypersurface, and can be represented via the adjunction map as IF_{1} blown up in 7 general points, thus it is embedded in IP^{4} by

H_{Y} = 4l − 2E_{0} − ∑_{i=1}^{7} E_{i}.

Consider now the following conics on Y:

C_{0} = 3l − 2E_{0} − ∑_{i=1}^{6} E_{i}
C_{1} = 2l − E_{0} − E_{7} − E_{1} − E_{3} − E_{5}
C_{2} = 2l − E_{0} − E_{7} − E_{1} − E_{4} − E_{6}
C_{3} = 2l − E_{0} − E_{7} − E_{2} − E_{4} − E_{5}
C_{4} = 2l − E_{0} − E_{7} − E_{2} − E_{3} − E_{6}.

They intersect pairwise in one point and the planes they span, denoted in the sequel by P_{i}, for i ∈ {0, 4}, intersect the Castelnuovo surface Y only along the conics C_{i}. The scheme Z = Y ∪ ⋃_{i=0}^{4} P_{i} is regular, of degree 10 and sectional genus 7, and has a minimal free resolution of type

5O(−5)
0 ← I_{Z} ⟵ ⊕
10O(−6) ↖ 34O(−7) ⟵ 27O(−8) ⟵ 7O(−9) ← 0.

The five quintics in the ideal intersect along Z and the union of 9 skew lines. In fact, if, according to (6.7), Q_{i} denotes the unique Segre cubic hypersurface containing the planes P_{i+1}, P_{i+2}, P_{i+3} and P_{i+4}, for i ∈ ℤ_{5}, then one checks that Q_{1}, Q_{2}, Q_{3} and Q_{4} each contain 6 skew lines which are 6-secant to the configuration Z, while Q_{0} contains only 5 such lines. On the other hand, the five Segre cubics Q_{i} cut out an elliptic quintic scroll T ⊂ IP^{4} (see [Seg], [SR, Th.XXXIII, p.278]); each of the planes P_{i} intersecting it along a cubic curve, section of the ruling. It follows that there are exactly 5 rulings of the scroll which are 6-secant to Z, and thus altogether 9 skew lines with this property. The scheme Z can be linked in the complete intersection of two quintic hypersurfaces to a surface S with the desired invariants, having the above 9 lines as exceptional curves. We compute the cohomology table

i ↑

3

7  8  4  h^{i}(I_{S}(p))

⟶ p

and a minimal free resolution of type

2O(−5)
0 ← I_{S} ← ⊕
7O(−6) ↖ 12O(−7) ⟵ 4O(−8) ← 0.

Finally, we remark that each of the planes P_{i} intersects S along a sextic curve, and thus contains an ∞^{2} of 6-secant lines to S.⊓ ⊔

Remark 6.13. It follows from lemma 0.32 that S and Z are minimal elements in their even liaison classes.

7. Construction of smooth surfaces with d=15, π=21, χ=0

First examples of surfaces with these invariants are due to Ellingsrud and Peskine; see [Au2]. Namely, for construction one starts with a minimal abelian surface A ⊂ IP^{4} with d = 10, π = 6. Any such surface comes as the zero-set of a section of the (twisted) Horrocks-Mumford bundle E_{HM}(3), and thus it is Heisenberg invariant (with respect to IH_{5}, see below for notations) and lies on a net of Heisenberg invariant quintic hypersurfaces. The base locus of the net is made of 25 skew lines and one shows that A can be linked in the complete intersection of two such quintics to a smooth abelian surface S with d = 15, π = 21, having the 25 lines as (−1) curves, see [Au2]. A short discussion about a construction using the Eagon-Northcott complex method can be found in [DES].

In the sequel we recall the determinantal construction of this example from a different point of view and an upshot of our approach will be the construction of a new abelian surface lying on only one quintic hypersurface. A rather striking fact, which was the starting point of this investigation, is the existence of a degeneration of this abelian surface which is scheme-theoretically the first infinitesimal neighborhood of an elliptic quintic scroll. This chapter owes very much to the discussions I had with the authors of [ADHPR] during the preparation of that paper.

Concerning the possible invariants of such surfaces we mention the following

Lemma 7.1. Let S ⊂ IP^{4} be a smooth surface with d = 15, π = 21, χ = 0. Then either:

a) S is a ruled surface over an elliptic curve, or

b) S is a non-minimal bielliptic surface embedded by

H = H_{min} − ∑_{i=1}^{25} E_{i},

or

c) S is a non-minimal abelian surface embedded by

H = H_{min} − ∑_{i=1}^{25} E_{i}.

P roof. Adjunction and the double point formula give HK = 25 and K^{2} = −25. Therefore, when p_{g} ≥ 1, it follows that S has exactly 25 exceptional lines and the canonical class on the minimal model needs to be trivial, whence b). Assume now p_{g} = 0 and let S_{1} = φ_{H+K}(S) ⊂ IP^{19} be the image of S through the adjunction morphism. The invariants of S are

H_{1}^{2} = 40,  H_{1}K_{1} = 0,  π_{1} = 21,  K_{1}^{2} = −25 + a

where a is the number of (-1) lines on S. In particular H_{1}(nK_{1}) = 0 for all n, so if κ(S) ≥ 0 then some multiple of K_{1} is trivial, a = 25 and S is a bielliptic surface as claimed in c), otherwise S needs to be ruled over an elliptic curve.⊓ ⊔ Remark 7.2.

a) I know nothing about the existence of ruled surfaces with the above invariants.

b) Examples of smooth non-minimal bielliptic surfaces as in lemma 7.1, b) have been constructed in [ADHPR].

b) Le Barz’s 6-secant formula gives N_{6} = N_{6}(15, 21, 0) = 25 so an abelian or a bielliptic surface as in the lemma has either none or infinitely many 6-secant lines.

7.3. We recall briefly some classical facts about the Heisenberg group IH_{5}. Let, as usual, IP^{4} = IP(V), with V =<e_{0}, e_{1}, …, e_{4}> and define IH_{5} ⊂ SL(V) as the subgroup generated by σ and τ, where

σ(e_{i}) = e_{i−1}, τ(e_{i}) = ξ^{i}e_{i}

for all i ∈ ℤ_{5} and ξ = e^{\frac{2πi}{5}} . In fact [σ, τ] = ξ, so IH_{5} is a central extension

0 ⟶ ℤ_{5} ⟶ IH_{5} ⟶ ℤ_{5} × ℤ_{5} ⟶ 0.

The representation IH_{5} ⊂ GL(V) described above is called the Schrödinger representation and will be denoted in the sequel as V_{0}. Let θ, with θ(ξ) = ξ^{2}, be the generator of Gal(ℚ(ξ) : ℚ) and denote by V_{i} the composition IH_{5}⟶^{θ^{i}}IH_{5}⟶^{V_{0}}GL(V). Then V_{0}, V_{1}, V_{2}, V_{3} together with the 25 characters of ℤ_{5} × ℤ_{5} form a basis of irreducible representations of IH_{5} [HM]. Let now ι ∈ N_{IH_{5}|SL_{5}(C I)} be the standard Heisenberg involution ι(e_{i}) = e_{−i} and let G = IH_{5} ⋊ ℤ_{2} be the subgroup generated by IH_{5} and ι in the normalizer N_{IH_{5}|SL_{5}(C I)}. Let also V^{+}, resp. V^{−} be the eigenspaces corresponding to the eigenvalues 1, resp. −1 of the Heisenberg involution ι. As usual, we denote IP_{+}^{2} = IP(V^{+}) and IP_{−}^{1} = IP(V^{−}). We will make use in the sequel of the representation theory of G. For convenience, we list here its character table [HM], [Ma]:

Z_{α}  C_{m,n}  C_{p}
1  1  1  I
5θ^{i}(α)  0  θ^{i}(ξ^{p})  V_{i}
1  1  −1  S
5θ^{i}(α)  0  −θ^{i}(ξ^{p})  V_{i}^{♯}
2  ξ^{sn+tm} + ξ^{−sn−tm}  0  Z_{s,t}

where the conjugacy classes of G are

Z_{α} = { α }, with α ∈ Z(IH_{5}) = ℤ_{5} = { 1, ξ, …, ξ^{4} } (5 items)

C_{m,n} = { ξ^{2mn+p}σ^{m}τ^{n} , ξ^{2mn+p}σ^{−m}τ^{−n} | p ∈ ℤ_{5} } (12 items)

and

C_{p} = { ξ^{2mn+p}σ^{m}τ^{n}ι | m, n ∈ ℤ_{5} } (5 items).

There are 12 different irreducible representations Z_{s,t} and we denote by Z their direct sum. We recall also the following formulae from [Ma]:

V_{i} ⊗ V_{i} = 3V_{i+1} ⊕ 2V_{i+1}^{♯},  V_{i} ⊗ V_{i+1} = 3V_{i+3} ⊕ 2V_{i+3}^{♯},  V_{i} ⊗ V_{i+2} = I ⊕ Z

(V_{i})^{∗} = V_{i+2},  (V_{i}^{♯})^{∗} = V_{i+2}^{♯},  V_{i} ⊗ S = V_{i}^{♯},  V_{i} ⊗ Z = 12V_{i} ⊕ 12V_{i}^{♯}
where i ∈ ℤ_{4}.

For the exterior powers the following isomorphisms hold

Λ^{2} V_{0} = 2V_{1}^{♯},  Λ^{3} V_{0} = 2V_{3}^{♯},  Λ^{4} V_{0} = V_{2}

hence the Koszul complex, as complex of G modules, looks

0← k ← R ← V_{2} ⊗ R(−1) ← 2V_{3}^{♯} ⊗ R(−2) ← 2V_{1}^{♯} ⊗ R(−3) ← V_{0} ⊗ R(−4) ← R(−5) ←0.

Finally, the well-known formula for the characters of the symmetric powers

χ_{S^{m}V} = ∑_{i=1}^{5} (−1)^{i+1}χ_{S^{m−i}V ⊗Λ^{i} V}

yields

H^{0}(IP^{4}, O_{IP}^{4}(1)) = V_{2},  H^{0}(IP^{4}, O_{IP}^{4}(2)) = 3V_{3},  H^{0}(IP^{4}, O_{IP}^{4}(3)) = 5V_{1} ⊕ 2V_{1}^{♯}
H^{0}(IP^{4}, O_{IP}^{4}(4)) = 10V_{0} ⊕ 4V_{0}^{♯}, H^{0}(IP^{4}, O_{IP}^{4}(5)) = 6I ⊕ 5Z.

7.4. The H^{1}-module of the Horrocks-Mumford bundle E_{HM} is a well understood module with Hilbert function (5, 10, 10, 2). We construct in the sequel other Heisenberg invariant, graded artinian modules M with Hilbert function of type (5, 10, 10, a), with a ≥ 0, and we make use for that of the double complex described in chapter 1.

Namely, assume that M is normalized such that M_{n} = 0 for n < 0 and that M_{0} ≅ V_{3} as G-representation spaces. Assume also that M_{0} generates M as R-module. The previous formulae yields that, in terms of G-modules, the top part of the double complex in theorem 1.11 looks like

↓  ↓  ↓  ↓
V_{3} ⊗_{k} R ⟵^{1I_{M_{0}} ⊗d_{1}} (3V_{1} ⊕ 2V_{1}^{♯}) ⊗_{k} R(−1) ⟵ (4V_{0} ⊕ 6V_{0}^{♯}) ⊗_{k} R(−2) ⟵ (2S ⊕ 2Z) ⊗_{k} ..
↓  ↓^{(δ_{1})_{0}⊗1I_{R(−1)}}  ↓^{(δ_{2})_{0}⊗1I_{R(−2)}}  ↓
0  ⟵  M_{1} ⊗_{k} R(−1)  ⟵ M_{1} ⊗_{k} V_{2} ⊗_{k} R(−2) ⟵ M_{1} ⊗_{k} 2V_{3} ⊗_{k} ..
↓  ↓  ↓^{(δ_{1})_{1}⊗1I_{R(−2)}}  ↓
0  ⟵  0  ⟵  M_{2} ⊗_{k} R(−2)  ⟵ M_{2} ⊗_{k} V_{2} ⊗_{k} ..
↓  ↓  ↓  ↓

The compatibility of the multiplication map (δ_{1})_{0} ⊗ 1I_{R(−1)} with the action of G on M implies that we have roughly speaking exactly three choices, as G-module, for the 10dimensional vector space M_{1}: a) M_{1} = 2V_{1}, or

b) M_{1} = 2V_{1}^{♯}, or

c) M_{1} = V_{1} ⊕ V_{1}^{♯},

We’ll discuss each case separately. The presentation matrix σ_{1} of M which comes out from the homology spectral sequence of the double complex can be easily recovered in all cases. Namely, fix first a splitting M_{0} ⊗_{k} V_{2} = ⊕_{i=0}^{4} L_{i} with L_{i} ≅ V_{1} for i = \overline{0, 2} and L_{j} ≅ V_{1}^{♯} for j = \overline{3, 4}. The general IH_{5} invariant subspace of M_{0} ⊗_{k} V_{2} can be displayed as L^{a} = a_{0}L_{0} + a_{1}L_{1} + ⋯ + a_{4}L_{4} for some a = (a_{0} : a_{1} : …: a_{4}) ∈ IP^{4}, where the above sum means component-wise addition of the bases with the prescribed a_{j} as coefficients. The restriction of 1I_{M_{0}} ⊗ d_{1} to L^{a} ⊗_{k} R(−1) is given by the 5 × 5 matrix with linear entries

Φ_{a} = ( \begin{matrix}x_{0}a_{0} & x_{3}(a_{1} − a_{4}) & x_{1}(a_{2} − a_{3}) & x_{4}(a_{2} + a_{3}) & x_{2}(a_{1} + a_{4}) \\ x_{3}(a_{1} + a_{4}) & x_{1}a_{0} & x_{4}(a_{1} − a_{4}) & x_{2}(a_{2} − a_{3}) & x_{0}(a_{2} + a_{3}) \\ x_{1}(a_{2} + a_{3}) & x_{4}(a_{1} + a_{4}) & x_{2}a_{0} & x_{0}(a_{1} − a_{4}) & x_{3}(a_{2} − a_{3}) \\ x_{4}(a_{2} − a_{3}) & x_{2}(a_{2} + a_{3}) & x_{0}(a_{1} + a_{4}) & x_{3}a_{0} & x_{1}(a_{1} − a_{4}) \\ x_{2}(a_{1} − a_{4}) & x_{0}(a_{2} − a_{3}) & x_{3}(a_{2} + a_{3}) & x_{1}(a_{1} + a_{4}) & x_{4}a_{0} \end{matrix} )

Therefore, introducing R.Moore’s matrices M_{y}(x) = (x_{3i+3j}y_{i−j})_{i,j}, with i, j ∈ ℤ_{5}, Φ_{a} can be expressed as

Φ_{a} = M_{z}(x)

where z ∈ IP^{4} has the following components:

z_{0} = a_{0}, z_{1} = a_{1} + a_{4}, z_{2} = a_{2} + a_{3}, z_{3} = a_{2} − a_{3}, z_{4} = a_{1} − a_{4}.

It follows that the restriction of 1I_{M_{0}} ⊗ d_{1} to a L^{a} of type V_{1}^{♯} corresponds to a matrix Φ_{a} = M_{z}(x) with z ∈ IP_{+}^{2}, while the restriction to an L^{a} of type V_{1}^{♯} corresponds to a matrix Φ_{a} = M_{z}(x) with z ∈ IP_{−}^{1}. To summarize, in case a)

σ_{1} = (M_{z}(x) | M_{z_{1}}(x) | M_{z_{2}}(x))

with z ∈ IP_{+}^{2}, z_{1} = (0, 1, 0, 0, −1) ∈ IP_{−}^{1}, z_{2} = (0, 0, 1, −1, 0) ∈ IP_{−}^{1}, while in case b)

σ_{1} = (M_{z_{0}}(x) | M_{z_{1}}(x) | M_{z_{2}}(x))

with z_{i} ∈ IP_{+}^{2}, i = \overline{0, 2} and thus, without loss of generality, we can choose say z_{0} = (1, 0, 0, 0, 0), z_{1} = (0, 1, 0, 0, 1) and z_{2} = (0, 0, 1, 1, 0), and in case c)

σ_{1} = (M_{z_{1}}(x) | M_{z_{2}}(x) | M_{z}(x))

with z_{i} ∈ IP_{+}^{2}, i = \overline{1, 2} and z ∈ IP_{−}^{1}.

We look first to the case b). The vertical differential (δ_{3})_{0} ⊗ 1I_{R(−1)} : (2S ⊕ 2Z) ⟶ M_{1} ⊗_{k} 2V_{3} = 4I ⊕ 4Z has a two-dimensional kernel. Since the two copies of S involve two Koszul complexes in the top row of the double complex it follows that the differential (δ_{3})_{0} ⊗ 1I_{R(−1)} : 4V_{0} ⊕ 6V_{0}^{♯} ⟶ M_{0} ⊗_{k} V_{2} = 4V_{0} ⊕ 6V_{0}^{♯} has a 10-dimensional kernel of type 2V_{0}^{♯}. One checks easily that M_{2} ≅ 2V_{0}^{♯} and M_{3} ≅ 2S, while M_{k} = 0 for k ≥ 4. Therefore M has Hilbert function (5, 10, 10, 2) and a minimal free resolution of type

M ⟵ 5R ⟵^{σ_{1}} 15R(−1) 10R(−2)
↖ ⊕
4R(−3)  2R(−3)
⊕ ↖ ⊕
15R(−4) 35R(−5) ⟵ 20R(−6)
↖ 2R(−8) ⟵ 0

This module is precisely (up to a twist) the H^{1}-module of the Horrocks-Mumford bundle. See also [De2] for a presentation of this module and [DES] for the Eagon-Northcott construction using the syzygy bundles of this module of the abelian surface in degree 10 and its (5, 5) linked abelian surface of degree 15.

In case c), for a generic projection on a subspace of type V_{1} ⊕ V_{1}^{♯}, the vertical differential (δ_{2})_{0} ⊗1I_{R(−1)} : 4V_{0} ⊕6V_{0}^{♯} ⟶ M_{0} ⊗_{k} V_{2} = 5V_{0} ⊕5V_{0}^{♯} has only a 5-dimensional kernel. It is easy to check that the general such module will have Hilbert function (5, 10, 5). However, special projections give modules with Hilbert function (5, 10, 10, 1), or (5, 10, 10), namely the first type being the H^{1}-module of the ideal sheaf of a union of two elliptic quintic scrolls meeting along an elliptic normal curve in IP^{4} (construction due to Frank-Olaf Schreyer). Since we’ll make no use of this we don’t insist on further details.

In case a), for a general projection on 2V_{1}, the differential (δ_{2})_{0} ⊗ 1I_{R(−1)} : 4V_{0} ⊕ 6V_{0}^{♯} ⟶ M_{0} ⊗_{k} V_{2} = 6V_{0} ⊕ 4V_{0}^{♯} has a 10-dimensional kernel isomorphic with 2V_{0}^{♯}. Its cokernel is M_{2}, whence M_{2} ≅ 2V_{0}. A similar argument shows that M_{3} ≅ 2I and M_{k} = 0 for k ≥ 4. Therefore, M has in this case again Hilbert function (5, 10, 10, 2), but this time a different minimal free resolution. Namely, one gets

M ⟵ 5R ⟵^{σ_{1}} 15R(−1) 10R(−2)
↖ ⊕
2R(−3)
⊕
15R(−4) ↖ 35R(−5) ⟵ 20R(−6)
↖ 2R(−8) ⟵ 0

We use the syzygy bundles of the dual module to construct a smooth, non-minimal abelian surface of degree 15 which is not coming via a (5, 5) linkage from the Horrocks-Mumford torus. A plausible cohomology table for an abelian surface S of degree 15, sectional genus 21 in IP^{4} is the following:

i ↑

1

2 10 10 5  h^{i}(I_{S}(p))

⟶ p

Let now E = O_{IP}^{4}(−1)⊕(2V_{2}⊕V_{2}^{♯})⊗_{k}O_{IP}^{4} = O_{IP}^{4}(−1)⊕15O_{IP}^{4} and F = S⊗_{k}Syz_{2}(M^{∗})(1). One checks in examples that the degeneracy locus of a general morphism φ ∈ Hom(E, F) is a smooth surface S ⊂ IP^{4} with d = 15, π = 21, p_{g} = 1, q = 2, K^{2} = −25 which, by lemma 7.1, is thus a non-minimal abelian surface with 25 exceptional lines. The minimal free resolution of the ideal sheaf is of the form

I ⊗_{k} O(−5)
0 ⟵ I_{S} ⟵  ⊕
2V_{2} ⊗_{k} O(−6) ↖ (2V_{3} ⊕ V_{3}^{♯}) ⊗_{k} O(−7) ⟵ V_{1}^{♯} ⊗_{k} O(−8) ⟵ 0

or, numerically, of type

O(−5)
0 ⟵ I_{S} ⟵ ⊕
10O(−6) ↖ 15O(−7) ⟵ 5O(−8) ⟵ 0

We’ll focus in the sequel on the properties of the new abelian surface of degree 15 and of the unique quintic hypersurface on which it lies.

To fix notations, assume as above the presentation matrix of M to be given by σ_{1} = (M_{z}(x) | M_{z_{1}}(x) | M_{z_{2}}(x)), where z = (a_{0}, a_{1}, a_{2}, a_{2}, a_{1}) ∈ IP_{+}^{2}, z_{1} = (0, 1, 0, 0, −1) ∈ IP_{−}^{1} and z_{2} = (0, 0, 1, −1, 0) ∈ IP_{−}^{1}. In this setting the 10 linear syzygies of σ_{1} are given by

σ_{21} = ( \begin{matrix}L_{y_{1}^{−}}(x) & L_{y_{2}^{−}}(x) \\ L_{y_{11}}(x) & L_{y_{21}}(x) \\ L_{y_{12}}(x) & L_{y_{22}}(x) \end{matrix} ) ,

where L_{y}(x) = (x_{2i−j}y_{i−j})_{i,j} with i, j ∈ ℤ_{5}, and the parameters have the following values

y_{1}^{−} = (0, 1, 0, 0, −1),  y_{11} = (2a_{2}, −a_{1}, 0, 0, −a_{1}), y_{12} = (0, −a_{2}, a_{0}, a_{0}, −a_{2}),
y_{2}^{−} = (0, 0, 1, −1, 0),  y_{21} = (0, −a_{0}, a_{1}, a_{1}, −a_{0}), y_{22} = (2a_{1}, 0, −a_{2}, −a_{2}, 0).

However, the two quadratic syzygies involve only the last ten columns of σ_{1}. If a_{0} = 0, then M = coker σ_{1} is not artinian. If a_{0} ≠ 0, which we’ll assume in the sequel, they are given by the 15 × 2 matrix

σ_{22} = ( \begin{matrix}0 & 0 \\ (x_{i+2}x_{i+3})_{i∈ℤ_{5}} & (x_{i}^{2})_{i∈ℤ_{5}} \\ (x_{i}^{2})_{i∈ℤ_{5}} & (−x_{i+1}x_{i+4})_{i∈ℤ_{5}} \end{matrix} ) .

Define now G as the kernel of the morphism ψ induced by the matrix with blocks (σ_{21}, σ_{22})

15O(4) ⟵^{ψ} 10O(3) ⊕ 2O(2) ⟵ G ⟵ 0.  (∗)

For a general choice of the parameter z ∈ IP_{+}^{2}, G is a rank 2 reflexive sheaf because ψ has rank 10 (generically): it has rank smaller or equal 10 being defined by a part of the syzygies of σ_{1}, whereas it is easily seen that the submatrix of σ_{21} given by

A = ( \begin{matrix}L_{y_{11}}(x) & L_{y_{21}}(x) \\ L_{y_{12}}(x) & L_{y_{22}}(x) \end{matrix} )

has a non-trivial determinant. In fact ψ has rank 10 outside codimension 3 since its restriction to a general IP^{2}, say {x_{0} = x_{1} = 0} for a_{2} ≠ 0 or {x_{0} = x_{2} = 0} for a_{1} ≠ 0, is an epimorphism on the restriction of Syz_{1}(M)(5) to this plane. It follows that G has first Chern class c_{1} = −1, and thus dualizing the exact sequence (∗) we obtain the minimal free resolution

2O(−3)
0 ⟵ G ⟵ ⊕
10O(−4) ↖^{ψ^{T}} 15O(−5) ⟵^{σ_{1}^{T}} 5O(−6) ⟵ 0.

In particular G is a stable rank 2 reflexive sheaf with Chern classes c_{1} = −1, c_{2} = 9, c_{3} = 25 and c_{4} = 50. Observe now that the morphism ψ drops rank on the line E_{00} = {x_{1} − x_{4} = x_{2} − x_{3} = a_{0}x_{0} + 2a_{1}x_{1} + 2a_{2}x_{2} = 0} ⊂ IP_{+}^{2}, whence by Heisenberg invariance also on its translates E_{ij} = σ^{i}τ^{j}E_{00}, for i, j ∈ ℤ_{5} by the group IH_{5}. Therefore E_{ij} ⊂ Sing(G), and since Porteus’ formula yields c_{3}(G) = [Sing(G)] = deg Sing(G) (see also [Ok3]) we see that the singular locus of the sheaf G consists of the 25 lines E_{ij}, i, j ∈ ℤ_{5}. Now h^{0}(G(3)) = 2 and, by construction, the zero locus of section s ∈ H^{0}(G(3)) is a surface S of the type described before, thus a smooth abelian surface of degree 15 for a general choice of the parameter z ∈ IP_{+}^{2} and of the section s:

0 ⟶ O ⟶ G(3) ⟶ I_{S}(5) ⟶ 0.

Lemma 7.5. The lines E_{ij}, i, j ∈ ℤ_{5}, are the 25 exceptional lines of S.

P roof. Clear since Sing(G) is contained in the zero locus of a section, while a line on an abelian surface is necessarily exceptional.⊓ ⊔

One checks easily that G(3) is generated by sections outside the the lines E_{ij}. Therefore, for each fixed value of the parameter z ∈ IP_{+}^{2}, we obtain a pencil of abelian surfaces S_{λμ} = {λs_{1} + μs_{2} = 0} ⊂ IP^{4}, where {s_{1}, s_{2}} is a base of H^{0}(G(3)), all lying on the unique quintic hypersurface V = {s_{1} ∧ s_{2} = 0} ⊂ IP^{4} and sharing the base locus of the pencil, namely the (−1) lines E_{ij}. To compute the equation of the quintic hypersurface we remark that from

( M_{z_{1}}(x) M_{z_{2}}(x) ) · A = −M_{z}(x) · ( L_{y_{1}^{−}}(x) L_{y_{2}^{−}}(x) )

it follows easily that detM_{z}(x) | detA and thus finally that

V = {detA/detM_{z}(x) = 0}.

A more geometrical way to describe the quintic hypersurface V is given by the following

Lemma 7.6.

a) For a general choice of the parameter z = (a_{0}, a_{1}, a_{2}, a_{2}, a_{1}) ∈ IP_{+}^{2}, there exists an unique quintic hypersurface V ⊂ IP^{4} containing the configuration ∪_{i,j∈ℤ_{5}}E_{ij}. Furthermore, V is Heisenberg invariant.

b) For a general choice of the parameter z on the modular conic C = {a_{0}^{2} + 4a_{1}a_{2} = a_{1} − a_{4} = a_{2} − a_{3} = 0} ⊂ IP_{+}^{2}, the lines E_{ij}, i, j ∈ ℤ_{5}, are rulings of an elliptic quintic scroll X, and thus are contained in five independent Segre cubics. Moreover, in this case, the homogeneous ideals I_{X} and I_{∪E_{ij}} coincide in degrees less or equal to 6.

P roof. We use the idea in [Au2, 2.2]. The group IH_{5} acts as ℤ_{5} × ℤ_{5} = IH_{5}/Z(IH_{5}) on IP^{4} and so H^{0}(O_{IP}^{4}(5)) = 6V_{0,0} ⊕_{(r,s)≠(0,0)} 5V_{r,s}, where V_{r,s} are the characters of the group ℤ_{5} × ℤ_{5} =< σ > × < τ >. On the other side, by construction, H^{0}(O_{∪E_{ij}}(5)) is six times the regular representation, so, by Schur’s lemma, the restriction morphism ρ : H^{0}(O_{IP}^{4}(5)) → H^{0}(O_{∪E_{ij}}(5)) decomposes as ρ = ⊕ρ_{r,s}, where ρ_{0,0} : 6V_{0,0} → 6V_{0,0}, while ρ_{r,s} : 5V_{r,s} → 6V_{r,s} for (r, s) ≠ (0, 0). As a consequence H^{0}(I_{∪E_{ij}}(5)) = ker ρ = ⊕_{r,s} ker ρ_{r,s}. Thus, in order to prove a), we check that the mappings ρ_{r,s} are injective for (r, s) ≠ (0, 0), while ker ρ_{0,0} is one dimensional. These are open conditions on IP_{+}^{2}, so it suffices to check them in a special case. Namely, we choose a_{0} = 1, a_{1} = a_{2} = 0 and make the computations using explicit bases:

B_{r,s} = {∑_{i=0}^{4} ξ^{−ri} ∏_{j∈ℤ_{5}}^{4} x_{i+j}^{m_{j}} | ∑_{j∈ℤ_{5}} jm_{j} ≡ s mod 5, ∑_{j∈ℤ_{5}} m_{j} = 5}

for 5V_{r,s}, when (r, s) ≠ (0, 0), and the standard basis of H^{0}(O_{IP}^{4}(5))^{IH^{5}}

γ_{0} = x_{0}x_{1}x_{2}x_{3}x_{4},  γ_{1} = ∑_{i∈ℤ_{5}} x_{i}x_{i+2}^{2}x_{i+3}^{2},  γ_{2} = ∑_{i∈ℤ_{5}} x_{i}^{3}x_{i+2}x_{i+3},
γ_{3} = ∑_{i∈ℤ_{5}} x_{i}^{3}x_{i+1}x_{i+4},  γ_{4} = ∑_{i∈ℤ_{5}} x_{i}x_{i+1}^{2}x_{i+4}^{2},  γ_{5} = ∑_{i∈ℤ_{5}} x_{i}^{5} − 5x_{0}x_{1}x_{2}x_{3}x_{4}.

This is straightforward and we’ll omit the details. We remark also that for this choice of the parameters V = {γ_{0} = x_{0}x_{1}x_{2}x_{3}x_{4} = 0}. For part b), we observe that in this case E_{00} is tangent at z to the conic section invariant under the icosahedral group 𝔄_{5} on IP_{+}^{2}: C = {x_{0}^{2} + 4x_{1}x_{2} = x_{1} − x_{4} = x_{2} − x_{3} = 0}. This conic can be naturally identified with the modular curve of level 5, which is in a 1:1 correspondence with the Heisenberg equivariantly embedded normal elliptic curves in IP^{4} (see [BHM]). Under this identification E_{00} corresponds to the line spanned by two non-trivial 2-torsion points, say τ_{1} and τ_{1} + τ_{2}, on the elliptic curve E defined as the scheme-theoretic intersection of Q_{i} = a_{0}x_{i}^{2} +2a_{1}x_{i+2}x_{i+3} +2a_{2}x_{i+1}x_{i+4} = 0, i ∈ ℤ_{5}. Then (cf. [BHM])

X = ⋃_{p∈E} \overline{p, p + τ_{2}} ⊂ IP^{4}

is a Heisenberg equivariantly embedded, elliptic quintic scroll containing the 25 rulings E_{ij}. X is embedded by a linear system numerically equivalent to C_{0} + 2f, where f is the class of a ruling while C_{0} is the class of a section with C_{0}^{2} = −e = 1. Therefore, if there would exist a sextic hypersurface containing ∪_{i,j∈ℤ_{5}}E_{ij} and not containing the scroll, then its intersection with X would contain residual to the rulings E_{ij} an effective divisor whose numerical class is 6C_{0} − 13f, which is absurd.⊓ ⊔

Part b) in the previous lemma shows that, for a choice of the parameter z on the conic C, the pencil of surfaces S_{λμ} has the scroll X as base component because I_{S_{λμ}} has generators in degrees 5 and 6. Moreover, it follows easily that in this case the quintic V is the trisecant variety of X (see [ADHPR] for its equation ), thus it is singular and has multiplicity 3 along the scroll. Each S_{λμ} is set-theoretically cut out by the quintic and two sextic hypersurfaces, so one checks easily that S_{λμ} contains residual to the scroll a surface T_{λμ} of degree 10 and sectional genus 6, which meets the scroll X along a section of degree 15. The general T_{λμ} are smooth minimal abelian surfaces of degree 10, isogeneous to a product. A special member in the family is the first infinitesimal neighborhood of the elliptic quintic scroll X.

Remark 7.7. For z ∈ IP_{+}^{2} general, V coincides with the unique quintic hypersurface containing the abelian surface S. In terms of the above basis for H^{0}(O_{IP}^{4}(5))^{IH^{5}}, V has the equation

\begin{matrix}(a_{0}^{5} − 8a_{1}^{5} − 8a_{2}^{5} + 15a_{0}^{3}a_{1}a_{2})γ_{0} + (a_{0}^{4}a_{1} + 8a_{1}^{3}a_{2}^{2} − 4a_{0}a_{2}^{4})γ_{1} + (a_{0}^{3}a_{2}^{2} − 2a_{0}^{2}a_{1}^{3} − 4a_{0}a_{1}a_{2}^{3})γ_{2} \\ +(a_{0}^{3}a_{1}^{2} − 4a_{0}a_{1}^{3}a_{2} − 2a_{0}^{2}a_{2}^{3})γ_{3} + (a_{0}^{4}a_{2} + 8a_{1}^{2}a_{2}^{3} − 4a_{0}a_{1}^{4})γ_{4} + a_{0}^{3}a_{1}a_{2}γ_{5} = 0 \end{matrix}

P roof. First part is clear from (7.5) and (7.6), while for the second it is enough to check that

\begin{matrix}(a_{0}^{5} − 8a_{1}^{5} − 8a_{2}^{5} + 15a_{0}^{3}a_{1}a_{2})γ_{0} + (a_{0}^{4}a_{1} + 8a_{1}^{3}a_{2}^{2} − 4a_{0}a_{2}^{4})γ_{1} + (a_{0}^{3}a_{2}^{2} − 2a_{0}^{2}a_{1}^{3} − 4a_{0}a_{1}a_{2}^{3})γ_{2} \\ +(a_{0}^{3}a_{1}^{2} − 4a_{0}a_{1}^{3}a_{2} − 2a_{0}^{2}a_{2}^{3})γ_{3} + (a_{0}^{4}a_{2} + 8a_{1}^{2}a_{2}^{3} − 4a_{0}a_{1}^{4})γ_{4} + a_{0}^{3}a_{1}a_{2}γ_{5} ∈ ker ρ_{0,0} \end{matrix}

⊓ ⊔

We fix from now on a general z ∈ IP_{+}^{2}.

Proposition 7.8. The quintic V has 100 ordinary double points as singularities, four of them on each line E_{ij}.

P roof. By computing, e.g., with [Mac], a standard basis for the jacobian ideal one checks that the singular locus of V is supported on ∪_{i,j∈ℤ_{5}}E_{ij}. Moreover, if f denotes the above equation of V then

\frac{∂f}{∂x_{2}} − \frac{∂f}{∂x_{3}} ∈ I_{IP}_{+}^{2}  a_{0} (\frac{∂f}{∂x_{2}} + \frac{∂f}{∂x_{3}} ) − 2a_{2} \frac{∂f}{∂x_{0}} ∈ I_{E_{00}}

\frac{∂f}{∂x_{1}} − \frac{∂f}{∂x_{4}} ∈ I_{IP}_{+}^{2}  a_{1} (\frac{∂f}{∂x_{2}} + \frac{∂f}{∂x_{3}} ) − a_{2} \frac{∂f}{∂x_{1}} ∈ I_{E_{00}}

a_{1} \frac{∂f}{∂x_{0}} − a_{0} \frac{∂f}{∂x_{1}} ∈ I_{E_{00}}

thus, for a_{0} ≠ 0, the singularities of V on E_{00} are defined by \frac{∂f}{∂x_{0}} . Restricted to E_{00} \frac{∂f}{∂x_{0}} has simple roots, and one checks in fact that, for general choices, these points are A_{1} singularities on V. Therefore, by symmetry, V has 100 nodes as singularities. ⊓ ⊔

Let Z denote the singular locus of V and let V̂ be a small resolution of the hypersurface; i.e., V̂ is smooth and a singular point p ∈ Z is replaced by a IP^{1}, denoted by L_{p}. We recall that the defect of V is the rank of the subspace spanned by {L_{p}}_{p∈Z} in H_{2}(V̂, ℚ) (cf. [We]). Also, by [Sch], it can be computed as

defect (V) = h^{1}(I_{Z}(5)).

We can write Z = ∪_{g∈ℤ_{5}×ℤ_{5}} ∪_{i=1}^{4} g(p_{i}), where p_{1}, p_{2}, p_{3} and p_{4} are the singularities of V on E_{00}, and thus split the natural restriction morphism ρ : H^{0}(O_{IP}^{4}(5)) → H^{0}(O_{Z}) as ρ = ⊕ρ_{r,s}, with ρ_{0,0} : 6V_{0,0} → 4V_{0,0} and ρ_{r,s} : 5V_{r,s} → 4V_{r,s} for (r, s) ≠ (0, 0). By (7.6) and (7.8) the mapping IP^{4}^{−−− >}IP^{5} defined by 6V_{0,0} sends the points p_{1}, p_{2}, p_{3} and p_{4} to four distinct points spanning a IP^{4} ⊂ IP^{5}. Hence ker ρ_{0,0}, which comes from the set of hyperplanes through this IP^{3}, is 2-dimensional. As in [Au2, 2.2] we obtain:

Remark 7.9.

a) defect (V) = 1

b) Pic (V̂) ≅ ℤ^{2} , and e(V̂) = 200 − 2(♯ nodes) = 0

c) If Ĥ is the pullback of a hyperplane section on V and Ŝ the pullback of the abelian surface then {Ĥ, Ŝ} is a basis for Pic (V̂)

d) dim_{C I}H^{1}(V̂, Θ_{V}) = h^{1}(Ω2 ; _{V}) = dim_{C I}(H^{0}(I_{Z}(5))/< x_{i} \frac{∂f}{∂x_{j}} >) = 2, thus IP_{+}^{2} is the whole moduli space of these quintic hypersurfaces.

P roof. Part b) follows from the fact that the Picard group of a small resolution of a nodal hypersurface in IP^{4} is torsion free by Lefschetz’ theorem, while its rank is the defect +1 (cf. [We]). For d), one observes that ⊕_{(r,s)≠(0,0)} ker ρ_{r,s} is contained in < x_{i} \frac{∂f}{∂x_{j}} | 0 ≤ i, j ≤ 4 >, so the claim follows from Griffith’ residues.⊓ ⊔

Remark 7.10. We recall that, in contrast with (7.9), the general Horrocks-Mumford quintic hypersurfaces have also 100 nodes, but have bigger defect, namely 3.

We mention in the sequel one further interesting degeneration of the abelian surface we’ve described above. Namely, for a_{0} = 1, a_{1} = a_{2} = 0 the construction yields a pencil of (singular) surfaces S_{λ,μ} ⊂ IP^{4}, (λ, μ) ∈ IP^{1}, which all lie on the degenerated quintic V_{0} = {γ_{0} = x_{0}x_{1}x_{2}x_{3}x_{4} = 0}. Restricting the resolutions to each of the hyperplanes H_{i} = {x_{i} = 0} we see that

S_{λ,μ} = ⋃_{i∈ℤ_{5}} S_{i}(λ, μ),

where S_{i}(λ, μ) ⊂ H_{i} are the cubic surfaces defined by

S_{i}(λ, μ) = {x_{i} = λ(x_{i+1}^{2}x_{i+3} − x_{i+2}x_{i+4}^{2}) + μ(x_{i+1}x_{i+2}^{2} − x_{i+3}^{2}x_{i+4}) = 0}.

Fix now a general (λ, μ) ∈ IP^{1} and let S_{i} = S_{i}(λ, μ), for i ∈ ℤ_{5}.

Proposition 7.11.

a) S_{i} is a smooth Del Pezzo surface in the hyperplane H_{i}. It is invariant under the action of τ, whereas σ(S_{i}) = S_{i−1}, i ∈ ℤ_{5}.

b) Any two surfaces S_{i} and S_{j} meet along a smooth conic and a point outside it. There are altogether five such points, namely these being the vertices p_{i} = IP(e_{i}), i ∈ ℤ_{5}, of the standard simplex in C I^{5}. Through each point p_{i} pass exactly four such Del Pezzo surfaces.

c) Consider the complete pentagon with vertices p_{i}, i ∈ ℤ_{5}, i.e., C_{∞} = ∪_{i∈ℤ_{5}}L_{i}, with L_{i} = {x_{i+2} = x_{i+3} = x_{i+4} = 0} for i ∈ ℤ_{5}, and C_{0} = ∪_{i∈ℤ_{5}}L′_{i}, with L′_{i} = {x_{i+1} = x_{i+3} = x_{i+4} = 0}. Then

S ∩ H_{i} = S_{i} ∪ L′_{i+1} ∪ L_{i+2} ∪ L_{i+3} ∪ L′_{i+4}.

Furthermore, these four lines are exceptional on all S_{j} with j ∈ ℤ_{5} \ {i}.

P roof. The claims are straightforward using the explicit equations of S_{i}, so we omit the calculations.⊓ ⊔

Observe now that the exceptional lines of the abelian surface degenerate in our situation to E_{00} = {x_{0} = x_{1} − x_{4} = x_{2} − x_{3} = 0} and E_{ij} = σ^{i}τ^{j}E_{00}, with i, j ∈ ℤ_{5}. One checks easily that E_{i0}, E_{i1}, E_{i2}, E_{i3} and E_{i4} are (−1) lines on the Del Pezzo surface S_{i}. As a consequence we obtain the following geometric characterization of our configuration:

Proposition 7.12. For each i ∈ ℤ_{5}, the lines E_{i0}, E_{i1}, E_{i2}, E_{i3}, E_{i4} and L_{i+3}, L′_{i+1} can be completed to a Schläfli’s double six configuration of lines in the hyperplane H_{i}, which then determines the Del Pezzo surface S_{i} as the unique cubic surface in H_{i} containing the given double six.

P roof. The intersection patterns are clear from the explicit description of the configuration, and the claim follows from [H-CV, §25].⊓ ⊔

For comparison, we mention here a rather similar degeneration for the abelian surfaces of degree 15 which lie on three quintic hypersurfaces and thus come via liaison from degree 10. Namely, we start with the zero-scheme Y_{αβ} of a special section of the twisted Horrocks-Mumford bundle E_{HM}(3), which is the union of 5 quadric surfaces [HM]:

Y_{αβ} = ⋃_{i∈ℤ_{5}}{x_{i} = αx_{i+2}x_{i+3} + βx_{i+1}x_{i+4} = 0}.

It lies on the quintic V_{0} and on two other independent quintic hypersurfaces, and thus can be linked on V_{0} to a configuration

X_{λμ} = ⋃_{i∈ℤ_{5}} X_{i}(λ, μ)

where this time

X_{i}(λ, μ) = {x_{i} = λ(x_{i+1}^{2}x_{i+3} + x_{i+2}x_{i+4}^{2}) + μ(x_{i+1}x_{i+2}^{2} + x_{i+3}^{2}x_{i+4}) = 0},

the parameters λ, μ being functions of α, β and the linkage.

The configuration X_{λμ} lies now on three quintic hypersurfaces, nevertheless has similar symmetry properties to those listed for S_{λ,μ} in proposition 7.11. Finally, there is an analogue of the claim (7.12), the only difference being that the role of the E_{ij} is taken here by the Horrocks-Mumford lines L_{ij}: L_{00} = IP_{−}^{1} and L_{ij} = σ^{i}τ^{j}L_{00}.

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