Small Schemes and Varieties of Minimal Degree

by D. Eisenbud, M. Green, K. Hulek, and S. Popescu

Abstract: We prove that if X ⊂ ℙ^{r} is any 2-regular scheme (in the sense of Castelnuovo-Mumford) then X is small. This means that if L is a linear space and Y := L ∩ X is finite, then Y is linearly independent in the sense that the dimension of the linear span of Y is 1 + deg Y. The converse is true and well-known for finite schemes, but false in general. The main result of this paper is that the converse, “small implies 2-regular”, is also true for reduced schemes (algebraic sets). This is proven by means of a delicate geometric analysis, leading to a complete classification: we show that the components of a small algebraic set are varieties of minimal degree, meeting in a particularly simple way. From the classification one can show that if X ⊂ ℙ^{r} is 2-regular, then so is X_{red}, and so also is the projection of X from any point of X.

Our results extend the Del Pezzo-Bertini classification of varieties of minimal degree, the characterization of these as the varieties of regularity 2 by Eisenbud-Goto, and the construction of 2-regular square-free monomial ideals by Fröberg.

Throughout this paper we will work with projective schemes X ⊂ ℙ^{r} over an algebraically closed field k. The (Castelnuovo-Mumford) regularity of X ⊂ ℙ^{r} is a basic homological measure of the complexity of X and its embedding in ℙ^{r} that gives a bound for the degrees of the generators of the defining ideal I_{X} of X and for many other invariants. The only schemes of regularity 1 are the linear spaces; but no classification is known for projective schemes of regularity 2.

In this paper we prove a structure theorem for reduced 2-regular schemes, showing that their irreducible components are varieties of minimal degree and characterizing how these components can meet. We also show that the reduced structure on any 2-regular scheme is 2-regular, and thus we obtain a complete description of the reduced structures on 2-regular schemes. (Since a high Veronese re-embedding of any zero-dimensional scheme is 2-regular, one cannot hope to characterize the isomorphism types of all 2-regular non-reduced schemes.)

Before stating our results we review some basic notions. For any subscheme X ⊂ ℙ^{r} we write span(X) for the smallest linear subspace of ℙ^{r} containing X. Recall that every variety (≡ reduced irreducible scheme) X ⊂ ℙ^{r} satisfies the condition

(∗)  deg(X) ≥ 1 + codim(X, span(X))

The authors are grateful to BIRS, IPAM, MSRI, the DFG and the NSF for hospitality and support during the preparation of this work.

(see for instance Mumford [1976, Corollary 5.13]). We say that the variety X ⊂ ℙ^{r} has minimal degree (more precisely, minimal degree in its span) if equality holds. Surfaces of minimal degree were classified by Del Pezzo [1886], and the classification was extended to all dimensions by Bertini [1907] (see Eisenbud-Harris [1987] for a modern account):

Theorem 0.1 A projective variety of minimal degree in its span is either a linear space, a quadric hypersurface in a linear space, a rational normal scroll, or a cone over the Veronese surface in ℙ^{5}.

This classification was extended to equidimensional algebraic sets that are connected in codimension 1 – the ones for which “minimal degree” is a good notion – by Xambó [1981]. For more general algebraic sets it is not clear that there should exist any interesting generalization of the equality in (∗) above. Nevertheless, the notion of smallness is just such a generalization.

Varieties of minimal degree were characterized cohomologically by Eisenbud-Goto [1984], and their result offers a way to generalize the hypothesis of the Del Pezzo-Bertini Theorem to all projective schemes. To state their result recall that X ⊂ ℙ^{r} is said to have regularity d, or to be d-regular, in the sense of Castelnuovo-Mumford, if the ideal sheaf I_{X} satisfies H^{i}(I_{X}(d − i)) = 0 for all i > 0, or equivalently, if the j-th syzygies of the homogeneous ideal I_{X} are generated in degrees ≤ d + j for all j ≥ 0 (see Eisenbud-Goto [1984], or Eisenbud [2004] for a proof of the equivalence).

Theorem 0.2 A variety X ⊂ ℙ^{r} has minimal degree in its linear span if and only if X is 2-regular.

An old argument of Lazarsfeld (see for instance [2004]), recently refined by Sidman [2002], Caviglia [2003], Eisenbud-Green-Hulek-Popescu [2004] and others, shows that if X ⊂ ℙ^{r} is d-regular and Λ is a linear subspace such that X ∩ Λ has dimension 0, then X ∩ Λ is also d-regular. When d = 2 we can rephrase this geometrically:

We say that a finite scheme Y ⊂ ℙ^{r} is linearly independent if the dimension of the linear span of Y is 1 + deg Y. We say that a scheme X ⊂ ℙ^{r} is small if, for every linear subspace Λ ⊂ ℙ^{r} such that Y = Λ ∩ X is finite, the scheme Y is linearly independent. (An alternative definition of smallness by a more general property of intersections is given in Theorem 2.2.) Lazarsfeld’s argument gives:

Proposition 0.3 Any 2-regular scheme X ⊂ ℙ^{r} is small.

Our main results are that the converse holds in the reduced case, and that small reduced schemes have a simple inductive classification. To state the classification, we say that a sequence of closed subschemes X_{1}, …, X_{n} ⊂ ℙ^{r} is linearly joined if, for all i = 1, …, n − 1, we have

(X_{1} ∪ … ∪ X_{i}) ∩ X_{i+1} = span(X_{1} ∪ … ∪ X_{i}) ∩ span(X_{i+1}).

Theorem 0.4 Let X ⊂ ℙ^{r} be an algebraic set. The following conditions are equivalent:

(a) X is small.

(b) X is 2-regular.

(c) X = X_{1} ∪ … ∪ X_{n}, where X_{1}, …, X_{n} is a linearly joined sequence of varieties of minimal degree.

The implication (c) ⇒ (b) is easy (see Proposition 3.1) while (b) ⇒ (a) is a special case of Proposition 0.3 (see also Section 1 for details). Most of this paper is occupied with the proof of the implication (a) ⇒ (c), which requires a delicate geometric analysis of the notion of smallness in the style of classical projective geometry. One of the things that makes the argument subtle is the fact that the linearly joined property of a sequence of varieties is strongly dependent on the ordering, as the following example shows.

Example 0.5 Let L_{0} be a line in ℙ^{4}, and let L_{1}, L_{2}, L_{3} be 3 general lines that meet L_{0}. The union X = ⋃_{i=0}^{3} L_{i} is 2-regular; in fact it is connected in codimension 1, has minimal degree, and is a degeneration of a rational normal quartic curve. As required by Theorem 0.4, X can be written as the union of a linearly joined sequence of varieties L_{0}, L_{1}, L_{2}, L_{3}, which are trivially of minimal degree in their spans.

On the other hand, the reverse sequence L_{3}, L_{2}, L_{1}, L_{0} is not linearly joined. Indeed, the subset Y = L_{3} ∪ L_{2} ∪ L_{1} is not 2-regular: since Y meets the line L_{0} in three points, the ideal of Y requires a cubic generator. It is easy to check that there is no enumeration of the components of X as a linearly joined sequence for which the reverse sequence is linearly joined.

In the special case where X is a union of coordinate subspaces, the equivalence of parts (b) and (c) of Theorem 0.4 had been proved by Fröberg [1985, 1988] as an application of Stanley-Reisner theory. Unions of coordinate spaces correspond to simplicial complexes. Using earlier results of Dirac [1961] and Fulkerson-Gross [1965], Fröberg showed that a simplicial complex corresponds to a 2-regular set if and only if it is the clique complex of a chordal graph. (We reprove this and give a generalization in Eisenbud-Green-Hulek-Popescu [2004]. See also Herzog, Hibi, and Zheng [2003] for a related path to Dirac’s theorem.) The orderings described in part (c) of Theorem 0.4 are called perfect elimination orderings in this context. See Blair-Peyton [1993] for a survey.

Properties that are easy to check for algebraic sets satisfying one of the conditions of Theorem 0.4 may be quite obscure for sets satisfying another. Theorem 0.4 has a number of surprising algebraic and geometric consequences based on this observation:

Corollary 0.6 If X ⊂ ℙ^{r} is a 2-regular algebraic set, then the union of any two irreducible components of X is again 2-regular.

By Example 0.5 the same cannot be said of a union of three components.

Proof. By Theorem 0.4, the irreducible components of X are of minimal degree in their spans. and thus also 2-regular by Theorem 0.2. From Proposition 3.1 we see that any 2 components are again linearly joined. The result follows by applying Theorem 0.4 once again.

Corollary 0.7 Let X ⊂ ℙ^{r} be a 2-regular algebraic set. If p ∈ X is a point and π_{p} denotes the linear projection from p, then π_{p}(X) ⊂ ℙ^{r−1} is 2-regular.

By Theorem 0.4 a similar statement holds with “small” in place of “2-regular”.

Proof. If X_{1}, …, X_{n} is a linearly joined sequence of varieties of minimal degree, then π_{p}(X_{1}), …, π_{p}(X_{n}) is a linearly joined sequence of varieties of minimal degree. Theorem 0.4 completes the proof.

Corollary 0.8 If X ⊂ ℙ^{r} is 2-regular, then X_{red} ⊂ ℙ^{r} is also 2-regular.

Proof. By Eisenbud-Green-Hulek-Popescu [2004, Theorem 1.6] (see also Theorem 1.2) the scheme X ⊂ ℙ^{r} is small. It follows that X_{red} is also small (Proposition 2.1). By Theorem 0.4, X_{red} is 2-regular.

Corollary 0.9 Let X_{1}, …, X_{n} ⊂ ℙ^{r} be a collection of varieties of minimal degree in their spans. The union ⋃_{i} X_{i} is small if and only if each pair X_{i}, X_{j} is linearly joined and the union of the linear spans ⋃_{i} span(X_{i}) is small.

Of course a similar statement will hold for 2-regularity in place of smallness. That version is actually one of the key ingredients in the proof of Theorem 0.4.

Proof. Use Theorem 0.4 and Proposition 3.4

The plan of the paper is as follows. In Sections 1, 2 and 3 we establish basic properties of 2-regular sets, small projective schemes, and linearly joined sequences of projective schemes. Of particular interest is the Bézout type result, Theorem 2.2: If X ⊂ ℙ^{r} is a small subscheme and Λ ⊂ ℙ^{r} is any linear space, then the sum of the degrees of the irreducible components (reduced or not, but not embedded) of X ∩ Λ is bounded by codim(X ∩ Λ, Λ) + 1. The results in these sections are necessary for the proof of Theorem 0.4, which is carried out in Section 4.

The argument of Lazarsfeld showing that plane sections of X ∩ L of a 2-regular scheme X are small works even if X is not 2-regular, but only has an ideal generated by quadrics having only linear syzygies for at least dim L steps. With slightly stronger hypotheses one can prove a little more; for example that that the syzygies of X ∩ L come from syzygies of X by restriction. See Theorem 1.2, and Eisenbud-Green-Hulek-Popescu [2004].

It follows from Corollary 0.9 that the condition that an algebraic set X ⊂ ℙ^{r} be small (or 2-regular) has a “local” part, that the X_{i} of X should be of minimal degree in their spans, and pairwise linearly joined; and a “global”, or combinatorial part, that the subspace arrangement ⋃_{i} span(X_{i}) be small. In Section 5 we will study small subspace arrangements. In particular, we describe the orderings of subspaces that make them into a linearly joined sequence in terms of certain spanning forests of the intersection graph of Y.

Finally, in Section 6 we discuss how to find generators for the ideal of a reduced 2-regular projective scheme. In particular, we prove that any 2-regular union of linear spaces has ideal generated by products of linear forms. The proof of Theorem 6.1 also provides a free resolution for the ideal of a 2-regular algebraic set, and one can ensure that the first two terms of the resolution are minimal, obtaining a formula for the number of generators of the ideal. Some results along this line have also been obtained by Barile and Morales [2000, 2003].

We thank Aldo Conca, Harm Derksen, Mark Haiman, Jürgen Herzog, Bill Oxbury, Kristian Ranestad, Jerzy Weyman, and Sergey Yuzvinsky for useful discussions. Behind the scenes, the program Macaulay2 of Dan Grayson and Mike Stillman has been extremely useful to us in trying to understand the relations between the linear syzygies and geometry.

1 2-regular schemes

The regularity of the ideal sheaf of a closed subscheme X ⊂ ℙ^{r} is ≤ 1 if and only if X is defined by linear forms, so X is a linear subspace in this case.

By contrast, any finite scheme can be embedded as a 2-regular scheme. In fact, we see from the definition of regularity that a high Veronese re-embedding of any given embedding of a zero-dimensional scheme is 2-regular. In fact a complete characterization of 2-regular embeddings of zero-dimensional schemes is well-known:

Proposition 1.1 A zero-dimensional nondegenerate scheme X ⊂ ℙ^{r} is 2-regular if and only if deg X = 1 + dim span(X).

Proof. The cohomological condition for regularity is equivalent to saying that X imposes deg X independent conditions on linear forms, that is, the span of X has dimension deg X − 1.

We will often use the fact that a zero-dimensional plane section of a 2-regular scheme is again 2-regular. As described in the introduction, this follows from an argument of Lazarsfeld, and many others have studied it recently. Here is a version sufficient for our purposes in this paper. We say that an ideal I ⊂ S has 2-linear resolution for at least p steps if the i-th syzygies of I are generated in degrees ≤ i+1 for i = 0, …, p − 1. For example, if I contains no linear forms, this means that the minimal free resolution of I has the form

⋯ ⊕ S(−i)^{β_{p,i}}  S(−p − 1)^{β_{p−1}}  ⋯  ⊕ S(−2)^{β_{0}}  I  0.

We also say in this case that I satisfies property N_{2,p}; this is the terminology used in Eisenbud-Green-Hulek-Popescu [2004].

Theorem 1.2 Let S be the polynomial ring on r + 1 variables, and suppose that X ⊂ ℙ^{r} has homogeneous ideal I_{X} ⊂ S such that I_{X} is generated by quadrics and has linear resolution for at least p steps. If Λ ⊂ ℙ^{r} is a linear subspace with codim(Λ ∩ X, span(Λ ∩ X)) ≤ p − 1, then Λ ∩ X is 2-regular. In particular, any zero-dimensional plane section of a 2-regular scheme is 2-regular.

For an explicit proof see for example Eisenbud-Green-Hulek-Popescu [2004, Th 1.1]. Combining Proposition 1.1 and Theorem 1.2 we get as a corollary the Proposition 0.3 in the introduction:

Corollary 1.3 Any 2-regular closed scheme X ⊂ ℙ^{r} is small.

There is a geometric characterization of reduced 2-regular schemes of higher dimension in the Cohen-Macaulay case. By a result of Hartshorne [1962] (see Eisenbud [1995]), connectedness in codimension 1 is a necessary condition for Cohen-Macaulayness. It turns out that for reduced 2-regular algebraic sets they are equivalent.

Theorem 1.4 Let X ⊂ ℙ^{r} be an equidimensional projective scheme, and let L be the linear span of X. Suppose that X is reduced,and connected in codimension 1.

(a) deg X ≥ codim(X, L) + 1.

(b) X is 2-regular if and only if deg X = codim(X, L) + 1.

(c) If the equivalent conditions in part (b) hold, then the homogeneous coordinate ring of X is Cohen-Macaulay.

Proof. Part (a) of Theorem 1.4 is elementary: the proof works as in the irreducible case (see Hartshorne [1962]) using the connectedness hypothesis to guarantee that the plane section is nondegenerate. The rest is proven in Eisenbud-Goto [1984].

The following Corollary is Theorem 0.4 in the special case of irreducible or connected-in-codimension 1 algebraic sets:

Corollary 1.5 Let X ⊂ ℙ^{r} be a projective scheme, and let L be the linear span of X. Suppose that X is reduced, and connected in codimension 1. If X is small then X is 2-regular.

A well-known regularity criterion that is essentially due to Mumford [1966, Lecture 14] will play a central role. (See also the last section of Eisenbud [2004] for details.)

Theorem 1.6 A closed subscheme X ⊂ ℙ^{r} is 2-regular if and only if

(a) For some (respectively, any) hyperplane H ⊂ ℙ^{r}, defined by a linear form that is locally a nonzerodivisor on O_{X}, the scheme X ∩ H is 2-regular, and

(b) X is linearly normal; equivalently, the restriction map H^{0}(O_{ℙ^{r}}(1)) H^{0}(O_{X}(1)) is surjective.

We will say that a scheme X ⊂ ℙ^{r} is the direct sum of schemes Z_{i} ⊂ ℙ^{r} if, for each i,

span(Z_{i}) ∩ span(⋃_{j≠i} Z_{j}) = ∅.

In other words the underlying vector space of the span of X is the direct sum of the underlying vector spaces of the spans of the Z_{i}. If this holds, then most questions about X ⊂ ℙ^{r} can be reduced to questions about the Z_{i} ⊂ span(Z_{i}); for instance the next result will allow us to reduce questions about 2-regularity to the connected case.

Proposition 1.7 X ⊂ ℙ^{r} is 2-regular if and only if the connected components of X are 2-regular and X is the direct sum of its connected components.

Proof. The vanishing H^{i}(I_{X}(2 − i)) ≅ H^{i−1}(O_{X}(2 − i)) = 0, for all i ≥ 2, is equivalent to H^{i−1}(O_{Z}(2 − i)) = 0, for all i ≥ 2, and all connected components Z of X. On the other hand we may assume that X is nondegenerate. Then if X = Z_{1} ∪ Z_{2} is a disjoint union, the cohomology of the short exact sequence

0  I_{X}(1)  I_{Z_{1}}(1) ⊕ I_{Z_{2}}(1)  O_{ℙ^{r}}(1)  0

yields that H^{1}(I_{X}(1)) = 0 if and only if both H^{1}(I_{Z_{i}}(1)) = 0, for i = 1, 2, and H^{0}(I_{Z_{1}}(1)) ⊕ H^{0}(I_{Z_{2}}(1)) H^{0}(O_{ℙ^{r}}(1)) is an isomorphism.

2 Basic properties of small algebraic sets

The following remarks will be used frequently in the sequel. They follow at once from the definitions.

Proposition 2.1 Let X ⊂ ℙ^{r} be a small subscheme.

(a) Any plane section of X is small.

(b) X_{red} is small.

We could have defined smallness by a more general property of intersections. Recall that the geometric degree of a closed subscheme Y ⊂ ℙ^{r} is defined as the sum of the degrees of the isolated irreducible (not necessarily reduced) components of Y.

Theorem 2.2 If X ⊂ ℙ^{r} is a small scheme, and L ⊂ ℙ^{r} is a linear space, then the geometric degree of X ∩ L is at most 1 + codim(X ∩ L, L).

Does Theorem 2.2 hold with the arithmetic degree (sum of all the degrees of isolated and embedded components) in place of the geometric degree? See Bayer-Mumford [1993], Sturmfels-Trung-Vogel [1995], or Miyazaki-Vogel-Yanagawa [1997] for the definition and basic properties of such degrees.

Proof. By Proposition 2.1 we can assume L = ℙ^{r}. We do induction on r, the case r = 0 being obvious. Let Y be the union of the zero-dimensional components of X, so that X = Y ∪Z is the disjoint union of Y and a scheme Z whose components have positive dimension. If Y spans ℙ^{r}, then successively factoring out elements of the socle of O_{Y}, we see that Y contains a subscheme Y′ of length r spanning an (r − 1)- plane Λ ⊂ ℙ^{r}. If Z is non-empty then this plane meets Z nontrivially, and thus the sum of the degrees of the components of X ∩Λ is > r = dim Λ+1 ≥ codim(Λ∩X, Λ). Since X ∩Λ is small, this contradicts the inductive hypothesis and shows that Z = ∅. Therefore X = Y is finite, and the desired conclusion follows from the definition of smallness.

On the other hand, suppose that Y does not span ℙ^{r}. If Λ is a general hyperplane containing Y, then Λ meets all the irreducible components Z_{i} of Z in schemes Z_{i} ∩ X of the same degree as Z_{i}. Further, the intersection Z_{i} ∩ Z_{j} of distinct components has dimension strictly less than that of Z_{i} or Z_{j}, so the schemes Z_{i} ∩ Z_{j} ∩ Λ do not contain any component of Z ∩ Λ. It follows that the sum of the degrees of the components of Z ∩ Λ is equal to the corresponding sum for Z. By induction on r, the degree of Y ⊂ Λ plus the sum of the degrees of the components of Z ∩ Λ is bounded by 1 + codim(X ∩ Λ, Λ) = 1 + codim(X, ℙ^{r}). Thus the same bound holds for X.

We isolate a consequence of Theorem 2.2 for use in the proof of the Theorem 0.4:

Corollary 2.3 Suppose that Y, Z ⊂ ℙ^{s} are disjoint algebraic sets. If Y ∪Z is small then both Y and Z are small.

Proof. By symmetry it suffices to show that Y is small. Suppose that L is a linear space that meets Y in a finite scheme. Since L ∩ (Y ∪ Z) is the disjoint union of L ∩ Y and L ∩ Z, the geometric degree of L ∩ (Y ∪ Z) is at least as great as the length of L ∩ Y, so we are done by Theorem 2.2.

The conclusion of Theorem 2.2 may be interpreted as a Bézout type theorem in the case of small varieties. A special case of a result of Lazarsfeld says that if X ⊂ ℙ^{r} is a nondegenerate subvariety and Λ ⊂ ℙ^{r} is a linear subspace, then the geometric degree of X ∩ Λ is bounded above by deg(X) − codim(X, ℙ^{r}) + codim(X ∩ Λ, Λ) (see Fulton [1984], Example 12.3.5, which also states Lazarsfeld’s more general result, or Fulton-Lazarsfeld [1982]). In the case when X is a variety of minimal degree this yields Theorem 2.2.

Next we analyze the irreducible components, and the relative positions of pairs of irreducible components, of small sets. By Proposition 2.1 the same results would apply to the reduced irreducible components of any small projective scheme.

Proposition 2.4 Let X ⊂ ℙ^{r} be a small algebraic set.

(a) Any irreducible component X_{i} of X is a variety of minimal degree in its linear span; that is deg X_{i} = dim span(X_{i}) − dim X_{i} + 1.

(b) Any two irreducible components X_{i} and X_{j} of X are linearly joined; that is, X_{i} ∩ X_{j} = span(X_{i}) ∩ span(X_{j}).

Proof. Write X = ⋃_{i} X_{i}, where X_{i} are the irreducible components of X, and let L_{i} = span(X_{i}) denote the linear span of X_{i}. Any variety X_{i} in projective space satisfies deg X_{i} ≥ dim span(X_{i}) − dim X_{i} + 1, but in our case, X_{i} is a component of L_{i} ∩X, so the opposite inequality and thus part (a) follows by applying Theorem 2.2 to the linear space L_{i}.

For part (b), we first apply Theorem 2.2 to the linear space L = span(L_{i} ∪ L_{j}). By construction X_{i} and X_{j} are components of L ∩ X, so the geometric degree of L ∩ X is at least deg X_{i} + deg X_{j}, and so Theorem 2.2 yields

deg X_{i} + deg X_{j} ≤ dim L_{i} + dim L_{j} − dim(L_{i} ∩ L_{j}) − max(dim X_{i}, dim X_{j}) + 1.

On the other hand, by part (a), deg X_{i} = dim L_{i} − dim X_{i} + 1 and we deduce dim(L_{i} ∩ L_{j}) ≤ min(dim X_{i}, dim X_{j}) − 1.

Now suppose that X_{i} ∩ X_{j} ≠ L_{i} ∩ L_{j}. If dim L_{i} ∩ L_{j} > 0, then to simplify we may cut all the schemes concerned by a general hyperplane H. Because X_{i} is reduced, irreducible and of dimension > 1, the hyperplane section H ∩ X_{i} is again reduced and irreducible (by Bertini’s Theorem) and spans L_{i} ∩ H. The same holds for X_{j}. Continuing to take hyperplane sections, we may reduce to the case where the linear space L_{i} ∩ L_{j} is just a point p.

If now both X_{i} and X_{j} contain p, then X_{i} ∩ X_{j} = {p} = L_{i} ∩ L_{j} as desired. If however X_{i} does not contain p but X_{j} does, then L_{i} ∩ X contains a component through p and thus

deg(X ∩ L_{i}) > deg X_{i} ≥ dim L_{i} − dim X_{i} + 1 ≥ dim L_{i} − dim(X ∩ L_{i}) + 1,

contradicting Theorem 2.2. By symmetry, we may therefore assume that neither X_{i} nor X_{j} meets L_{i} ∩ L_{j} = {p}, and we must derive a contradiction.

Let Λ_{k} ⊂ L_{k}, for k = i, j, be general planes containing p and having dim Λ_{k} = codim(X_{k}, L_{k}). The scheme Λ_{k} ∩ X_{k} is zero-dimensional. Set Λ = span(Λ_{i} ∪ Λ_{j}). We have Λ ∩ L_{i} = Λ_{i} + (Λ_{j} ∩ L_{i}) = Λ_{i}, and similarly for Λ ∩ L_{j}, so the components of Λ_{k} ∩ X_{k}, k = i, j, are also components of Λ ∩ X. Thus the geometric degree of Λ ∩ X is at least deg X_{i} + deg X_{j}.

On the other hand, since Λ_{i} and Λ_{j} meet in a point, the dimension of Λ is dim Λ_{i} + dim Λ_{j} = codim(X_{i}, L_{i}) + codim(X_{j}, L_{j}) = deg X_{i} + deg X_{j} − 2, by part (a). But then Theorem 2.2 yields that the geometric degree of Λ ∩ X is at most dim Λ+1 = deg X_{i} +deg X_{j} −1, the desired contradiction. This concludes the proof of (b).

3 Linearly joined sequences of schemes

A first connection of the notions of smallness and 2-regularity with the notion of linearly joined sequence is provided by the following observations. A linearly joined sequence of schemes X, Y ⊂ ℙ^{r} is linearly joined in either order, so we simply say that the pair is linearly joined.

Proposition 3.1 Suppose X_{1}, X_{2} ⊂ ℙ^{r} are linearly joined, and set X = X_{1} ∪ X_{2}.

(a) X is small ⇒ X_{1} and X_{2} are both small.

(b) X is 2-regular ⇔ X_{1} and X_{2} are both 2-regular.

Proof. Part (a) is easy. To prove part (b), write L for the linear space X_{1} ∩ X_{2}. The result follows from the exact sequence

0  I_{X}  I_{X_{1}} ⊕ I_{X_{2}}  I_{L}  0.

(or see the more general statement in part (c) of Theorem 6.1).

Remark 3.2 In the reduced case the converse to part (a) follows from part (b) and Theorem 0.4.

As with 2-regularity, we can reduce questions about linearly joined sequences to the connected case.

Proposition 3.3 If X ⊂ ℙ^{r} is the union of a linearly joined sequence of irreducible schemes, then X is the direct sum of its connected components.

Proof. Let X_{1}, …, X_{n} be the linearly joined sequence of irreducible components of X. We do induction on n, the case n = 1 being trivial.

By induction we may assume that X′ = ⋃_{i=1}^{n−1} X_{i} is the direct sum of its connected components Z′_{1}, …, Z′_{s}. Since X_{n} ∩ X′ = span(X_{n}) ∩ span(X′) is a linear space, X_{n} can meet at most one of the Z′_{i}. If X_{n} does not meet any Z′_{i}, and thus forms a new connected component, then span(X_{n}) is disjoint from span(X′), as required. Thus we may assume that X_{n} meets a unique component, which we may as well call Z′_{s}, and the connected components of X are

Z_{1} = Z′_{1}, …, Z_{s−1} = Z′_{s−1}, and Z_{s} = Z′_{s} ∪ X_{n}.

If X were not the direct sum of the Z_{i}, then there would be a nontrivial dependence relation of the form ∑_{1}^{s−1} p_{i}+(p_{s}+q) = 0 where each p_{i} is a vector of homogeneous coordinates of a point in span(Z′_{i}), or the 0 vector, and q is a vector of homogeneous coordinates of a point in span(X_{n}). q is not the 0 vector since Z′_{1}, …, Z′_{s} are direct summands, so q must represent a point in span(X′)∩span(X_{n}) = X′∩X_{n} ⊂ Z′_{s}. Since X′ is a direct sum of the Z′_{i}, it follows that p_{1} = ⋯ = p_{s−1} = p_{s} + q = 0, as required.

Next we show that condition (b) of Proposition 2.4 is exactly the difference between saying that the sequence of schemes X_{1}, …, X_{n} is linearly joined and saying that the (same) sequence of their spans is linearly joined.

Proposition 3.4 Let X_{1}, …, X_{n} ⊂ ℙ^{r} be a sequence of closed subschemes and, for each i, let L_{i} denote the linear span of X_{i}. The sequence X_{1}, …, X_{n} is linearly joined if and only if the sequence L_{1}, …, L_{n} is linearly joined and each pair X_{i}, X_{j} is linearly joined, for all i ≠ j.

Proof. First suppose that the sequence X_{1}, …, X_{n} is linearly joined. It follows at once that L_{1}, …, L_{n} is linearly joined. By induction on n we may suppose that all pairs X_{i}, X_{j} are linearly joined for i, j < n, and it suffices to show that L_{j} ∩ L_{n} ⊂ X_{j} ∩ X_{n} for each j < n. Since the sequence X_{1}, …, X_{n} is linearly joined we have

⋃_{j<n}(X_{j} ∩ X_{n}) = ( ⋃_{j<n} X_{j}) ∩ X_{n} = span( ⋃_{j<n} X_{j}) ∩ L_{n}.

Since this is a linear space it must be contained in one of the X_{i} ∩X_{n} for some i < n. In particular, L_{j} ∩ L_{n} ⊂ X_{i} ∩ X_{n}, for all j < n, and thus we see that L_{j} ∩ L_{n} ⊂ X_{n}. Since, by induction, the pairs X_{i}, X_{j} are linearly joined we also have

L_{j} ∩ L_{n} = L_{j} ∩ L_{n} ∩ X_{i} ∩ X_{n} ⊂ L_{j} ∩ X_{i} ⊂ X_{j},

completing the argument.

Conversely, suppose that the X_{i} are pairwise linearly joined and L_{1}, …, L_{n} is a linearly joined sequence. By induction, we may assume that X_{1}, …, X_{n−1} is linearly joined sequence. But

span( ⋃_{i<n} X_{i}) ∩ L_{n} = span( ⋃_{i<n} L_{i}) ∩ L_{n} = ( ⋃_{i<n} L_{i}) ∩ L_{n} = L_{j} ∩ L_{n}

for some j < n. Since the pair X_{j}, X_{n} is linearly joined we also have L_{j} ∩ L_{n} = X_{j} ∩ X_{n}, so span(⋃_{i<n} X_{i}) ∩ L_{n} = (⋃_{i<n} X_{i}) ∩ X_{n} as required.

4 Proof of Theorem 0.4

We already have the tools to dispose of two implications in Theorem 0.4 easily. Theorem 0.2, together with Proposition 3.1 gives the implication (c) ⇒ (b). On the other hand, Theorem 1.2 (proved in Eisenbud-Green-Hulek-Popescu [2004]) includes the implication (b) ⇒ (a) as a special case.

The last implication, (a) ⇒ (c), will occupy us for the rest of this section. Here is an outline: We first prove (a) ⇒ (c) in the case where each X_{i} is a linear space, then we use this case to prove the implication (b) ⇒ (c) in general. Finally we will use the implication (b) ⇒ (c) to prove (a) ⇒ (c).

Proof of (a) ⇒ (c) for unions of planes:

To finish the proof of Theorem 0.4 in the case where X is a union of planes, we will project from a general point on a carefully chosen component of X, using the following results.

Proposition 4.1 Let V_{i} ⊂ V, i = 1, …, m, be distinct linear subspaces, not contained in one another, such that any m − 1 of them span the ambient space V. Then there exist vectors u_{i} ∈ V_{i} \ ⋃_{j≠i} V_{j} such that the collection u_{1}, …, u_{m} is linearly dependent.

Proof. Since the subspaces V_{i} are distinct and not contained in one another Z_{i} = (∪_{j≠i}V_{j}) ∩ V_{i} is a union of finitely many proper linear subspaces of V_{i}, for all i = 1, …, m. In particular Z_{i} ≠ V_{i}. Let W denote the kernel of the natural summation map

φ : V_{1} ⊕ V_{2} ⊕ ⋯ ⊕ V_{m} ^{(v_{1},...,v_{m})↦∑_{i=1}^{m} v_{i}} V.

Observe that if π_{i} : V_{1} ⊕ V_{2} ⊕ ⋯ ⊕ V_{m} → V_{i} denotes the projection on the i-th factor, then π_{i}(W) = V_{i} by our hypothesis. It follows that π_{i}^{−1}(Z_{i}) ≠ W for all i. Since the ground field is infinite, there exists a vector w = (v_{1}, …, v_{m}) ∈ W such that π_{i}(w) ∉ Z_{i} for all i = 1, …, m. In other words, v_{i} ∈ V_{i} \ (∪_{j≠i}V_{j}), i = 1, …, m and v_{1} + ⋯ + v_{m} = 0.

Proposition 4.2 Let X = ⋃_{i=1}^{t} X_{i} ⊂ ℙ^{r} be a small union of closed subschemes. If Λ_{i} ⊂ X_{i} is a linear subspace for each i such that Λ_{i} does not meet any X_{j} for j ≠ i, then the Λ_{i} are linearly independent and their linear span meets each X_{j} precisely in Λ_{j}.

Proof. Let Λ be the span of the Λ_{i}. If the Λ_{i} were dependent, then there would be a set of points p_{i} ∈ Λ_{i} that were dependent. Similarly if Λ ∩ X_{j} ≠ Λ_{j} then there would be a set of points p_{i} ∈ Λ_{i}, such that the span of the p_{i} met X_{j} outside Λ_{j}. Consequently, to prove either statement, we may assume that Λ_{i} = {p_{i}}.

Since p_{i} ∉ X_{j} for j ≠ i, there is a component X′_{i} of Λ ∩ X containing p_{i} but none of the p_{j}. By Theorem 2.2, the sum of the degrees of the X′_{i} is at most codim(X ∩ Λ, Λ) + 1 ≤ dim(Λ) + 1. Since Λ is spanned by the points p_{i} ∈ X′_{i}, the dimension of Λ is at most one less than the number of components X′_{i}. By Theorem 2.2 we conclude that each X′_{i} = {p_{i}}, the p_{i} are linearly independent, and there are no other points in X ∩ Λ.

Corollary 4.3 If X = ⋃_{i} X_{i} ⊂ ℙ^{r} is a small union of linear subspaces, then there exists an i such that X_{i} is not in the linear span of the ⋃_{j≠i} X_{j}.

Proof. If each component X_{i} were contained in the span of the others, then passing to affine cones we would have a system of subspaces satisfying the hypothesis of Proposition 4.1. The set of points corresponding to the vectors in the conclusion of Proposition 4.1 would violate Proposition 4.2.

Proposition 4.4 Let X = ⋃_{i=1}^{n} X_{i} ⊂ ℙ^{r} be a small union of linear subspaces. If p ∈ X_{1} is a point that is not on the span of ⋃_{i=2}^{n} X_{i}, then the image of X under linear projection from p, π_{p} : ℙ^{r} ........ ℙ^{r−1}, is small.

Proof. By hypothesis there is a hyperplane H ⊂ ℙ^{r} that contains ⋃_{i=2}^{n} X_{i} but does not contain p. By Proposition 2.1 the union of linear spaces X′ = (H∩X)_{red} is small, and the projection π_{p} induces a linear isomorphism X′ ⊂ H π_{p}(X) ⊂ ℙ^{r−1}, proving that π_{p}(X) ⊂ ℙ^{r−1} is also small.

Proposition 4.5 Let X = ⋃_{i=1}^{n} X_{i} ⊂ ℙ^{r} be a small union of linear spaces, and let p ∈ X_{1} be a point not in the linear span of the union of X_{2}, …, X_{n}. If, for some i, the schemes π_{p}(X_{i}) and π_{p}(⋃_{j≠i} X_{j}) are linearly joined, then X_{i} and ⋃_{j≠i} X_{j} are linearly joined.

Proof. Let Z = ⋃_{j≠i} X_{i}. We must show that if q ∈ X_{i} ∩ span(Z) then q ∈ Z. Note that p cannot be in both X_{i} and span(Z), so p ≠ q, and the projection π_{p}(q) is defined.

First suppose i = 1. We have π_{p}(q) ∈ π_{p}(X_{1}∩span(Z)) = π_{p}(X_{1})∩π_{p}(span(Z)) because p ∈ X_{1}. Moreover, π_{p}(span(Z)) = span(π_{p}(Z)), and thus, because π_{p}(X_{1}) and π_{p}(Z) are linearly joined,

π_{p}(q) ∈ π_{p}(X_{1}) ∩ span(π_{p}(Z)) = π_{p}(X_{1}) ∩ π_{p}(Z).

In particular, π_{p}(q) ∈ π_{p}(Z). As q ∈ span(Z), and π_{p} is an isomorphism on span(Z), we get q ∈ Z ∩ X_{1} as required.

Next suppose i ≠ 1. Because p ∈ span(Z), we may argue as before and obtain

π_{p}(q) ∈ π_{p}(X_{i}) ∩ π_{p}(span(Z)) = π_{p}(X_{i}) ∩ π_{p}(Z).

Thus the line span(p, q) meets Z in a point q′ and meets X_{i} in a point q′′. If q = q′, then q ∈ Z, and we are done. Thus we may as well assume that q ≠ q′, which implies that p ∈ span(q, q′). By hypothesis p ∉ span(⋃_{i≠1} X_{i}), so at least one of the points q, q′ must be in X_{1}. Since p ∈ X_{1}, this means that both q, q′ ∈ X_{1} ⊂ Z; in particular q ∈ Z as required.

Conclusion of the Proof of (a) ⇒ (c) for a union of linear spaces. Again, let X = ⋃_{i=1}^{n} X_{i} ⊂ ℙ^{r} be a small union of linear spaces. We do induction on the dimension r of the ambient projective space. After renumbering the components we may assume by Corollary 4.3 that there exists a point p ∈ X_{1} that is not in the linear span of ⋃_{i≥2} X_{i}. By Proposition 4.4, the projection π_{p}(X) ⊂ ℙ^{r−1} is also small.

By induction on the dimension of the ambient space, there is a permutation of {1, …, n}, such that π_{p}(X_{σ(t+1)}) is linearly joined to ⋃_{i=1}^{t} π_{p}(X_{σ(i)}) for t = 1, …, n − 1. If 1 ∉ {σ(1), …, σ(t + 1)}, then π_{p} is an isomorphism on the linear span of ⋃_{i=1}^{t+1} X_{σ(i)}, so X_{t+1} is linearly joined to ⋃_{i=1}^{t} X_{σ(i)}. If on the other hand 1 ∈ {σ(1), …, σ(t + 1)}, then Proposition 4.5 yields the same conclusion.

Proof of (b) ⇒ (c) in the general case. If X is 2-regular, then by Proposition 2.4 all irreducible components X_{i} of X are varieties of minimal degree in their spans, and they are pairwise linearly joined. To prove (c) it suffices, by Proposition 3.4, to show that the union of the linear spans of the X_{i} can be arranged in a linearly joined sequence. The next result shows that this union is 2-regular, and thus reduces the proof to the case we have already treated. The proof will use an induction, and for this we need to know that taking hyperplane sections commutes with taking spans in certain cases.

Lemma 4.6 Let Y ⊂ ℙ^{s} be a scheme, and H the hyperplane defined by a linear form h. Suppose h is sufficiently general that it is locally a nonzerodivisor on O_{Y}. If h^{0}(O_{Y}) = 1, then span(H ∩ Y) = H ∩ span(Y).

Proof. The diagram

\begin{matrix}0 & \overline{} & H^{0}(O_{ℙ^{s}}) & \overline{\begin{matrix}h \\ \end{matrix}} & H^{0}(O_{ℙ^{s}}(1)) & \overline{\begin{matrix}ρ \\ \end{matrix}} & H^{0}(O_{H}(1)) & 0 \\ 0 & \overline{} & H^{0}(O_{Y}) & & \overline{\begin{matrix}h \\ \end{matrix}} & H^{0}(O_{Y} (1)) & & \overline{} & H^{0}(O_{Y ∩H}(1)) & & \overline{} & ⋯ \end{matrix}

has exact rows, and the left-hand vertical map is surjective by the connectedness of Y. By the snake lemma, the restriction ρ induces the isomorphism H^{0}(I_{Y} (1)) ≅ H^{0}(I_{Y ∩H,H}(1)). It follows that span(H ∩ Y) = H ∩ span(Y).

Theorem 4.7 Let X = ⋃_{i} X_{i} ⊂ ℙ^{r} be a closed subscheme with irreducible components X_{i} that are Cohen-Macaulay and 2-regular. Let L_{i} = span(X_{i}) and set Y = ⋃_{i} L_{i}. If X_{i} ∩ X_{j} = L_{i} ∩ L_{j} for each i ≠ j, then X is 2-regular if and only if Y is 2-regular.

Proof. We will show that conditions (a) and (b) of Theorem 1.6 hold for X if and only if they hold for Y. (a): Suppose that X is 2-regular. By Proposition 1.7, X is the direct sum of its connected components, which are also 2-regular. In particular, if X has any zerodimensional components, they form a direct summand and we may drop them. Thus we may assume that every component of X has dimension ≥ 1.

Let H ⊂ ℙ^{r} be a general hyperplane, so that that H ∩ X is 2-regular. Since X_{i} has dimension ≥ 1 and is Cohen-Macaulay, h^{0}(O_{X_{i}}) = 1. By Lemma 4.6, H ∩ L_{i} is the span of H ∩ X_{i}.

By Proposition 1.7 again, H ∩X is the direct sum of its connected components, which are also 2-regular, and it suffices to show that the same is true of H ∩ Y. The connected components of H ∩ Y have the form ⋃_{i∈I} H ∩ L_{i} where I is a minimal set of indices i such that

dim[⋃_{i∈I} L_{i} ∩ ⋃_{j∉I} L_{j}] ≤ 0.

By hypothesis L_{i}∩L_{j} = X_{i}∩X_{j}, for all i ≠ j, so ⋃_{i∈I} H ∩X_{i} is a union of connected components of H ∩X (extra components appear when there are 1-dimensional components of X). Since such a union is a direct summand of H ∩ X, it follows that H ∩ Y is a direct sum of its connected components.

Each 1-dimensional irreducible component X_{i} of X contributes a zerodimensional scheme spanning a single connected component H ∩ L_{i} of H ∩ Y that is a single linear space, and is thus 2-regular. The union of the higherdimensional irreducible components of X contributes the union of a subset of the connected components of H ∩ X, which is thus 2-regular. But for an irreducible component X_{i} of dimension at least 2, H ∩X_{i} is again irreducible and spans H ∩L_{i}, by Lemma 4.6. Thus we may use induction, and deduce that H ∩ Y is 2-regular as required.

The case where H ∩ Y is 2-regular is similar. Since it will not be required for the proof of Theorem 0.4, and is a consequence of that result in the case where X is reduced, we omit the details. (b): Consider the diagram

[figure: 0; H^{0}(O_{Y} (1)); ⨁_{i} H^{0}(O_{L_{i}}(1)); ⨁_{i<j} H^{0}(O_{L_{i}∩L_{j}}(1)); 0; H^{0}(O_{X}(1)); ρ_{Y/X}; ⨁_{i} H^{0}(O_{X_{i}}(1)); ⊕ρ_{L_{i}/X_{i}}; ⨁_{i<j} H^{0}(O_{X_{i}∩X_{j}}(1)); ^{⊕ρ_{L_{i}∩L_{j}/X_{i}∩X_{j}} ‖}]

where the vertical maps are restrictions. The maps ρ_{L_{i}/X_{i}} are isomorphisms because the X_{i}’s are linearly normal and each span L_{i}. Since X_{i} ∩ X_{j} = L_{i} ∩ L_{j} the right hand side vertical map is an equality. Thus the map ρ_{Y/X} is also an isomorphism, so X and Y are either both linearly normal in ℙ^{r} or both not.

Proof of Theorem 0.4 continued. We complete the proof of Theorem 0.4 by proving (a) ⇒ (b), using induction on dim X. A general hyperplane section of X is reduced by Bertini’s Theorem, and by Proposition 2.1, it is small. By induction, it is 2regular. Thus by Theorem 1.6, it is enough to show that X is linearly normal.

If X were not linearly normal, we could write X as a linear projection of a linearly normal variety Y in some larger projective space ℙ^{s} = ℙ(H^{0}(O_{X}(1)) from a linear space M ⊂ ℙ^{s} that is disjoint from Y in such a way that the projection is an isomorphism on Y, but M is contained in the linear span of Y. We first want to show that Y ∪ M is small. For this we will need the following basic fact about geometric degree:

Proposition 4.8 Suppose that Y ⊂ ℙ^{s} is a scheme, and M ⊂ ℙ^{s} is a linear space disjoint from Y. Suppose that the linear projection from M, denoted π_{M} : ℙ^{s} .......... ℙ^{r}, induces an isomorphism Y → π_{M}(Y). If L ⊂ ℙ^{s} is a linear space containing M, then π_{M}(L) ∩ π_{M}(Y) = π_{M}(L ∩ Y) ≅ L ∩ Y, and the geometric degree of L ∩ Y ⊂ ℙ^{s} is the same as the geometric degree of π_{M}(L) ∩ π_{M}(Y).

Proof. First note that because M ⊂ L the map π_{M}(Y ∩ L) → π_{M}(Y) ∩ π_{M}(L) is set-theoretically surjective, so it is enough to check the result locally at a point q ∈ Y ∩ L.

Suppose that W is the space of linear forms vanishing on M, and that V ⊂ W is the space of linear forms vanishing on L. We may identify W with H^{0}(O_{ℙ^{r}}(1)), and under this identification, V is identified with the space of linear forms on ℙ^{r} that vanishes on π_{M}(L). With this understanding, the first claim in the statement of the theorem may be written, locally at q, as

O_{Y ∩L,q} ≅ O_{Y,q}/V O_{Y,q} ≅ O_{π_{M} (Y),π_{M} (q)}/V O_{π_{M} (Y),π_{M} (q)},

which holds because π_{M} induces an isomorphism O_{Y,q} ≅ O_{π_{M} (Y),π_{M} (q)}.

The statement about geometric degrees follows because all the isomorphisms preserve linear forms.

Proposition 4.9 Suppose that Y ⊂ ℙ^{s} is an algebraic set, and that M ⊂ ℙ^{s} is a linear space, disjoint from Y, such that the linear projection π_{M} induces an isomorphism Y → π_{M}(Y). If π_{M}(Y) is small, then so is Y ∪ M.

Proof. Suppose that Λ ⊂ ℙ^{s} is a plane that meets X = Y ∪ M in a finite set. We must show that deg(X ∩ Λ) ≤ 1 + dim(Λ). Let L = span(Λ ∪ M). As Y ∩ Λ is a plane section of Y ∩ L, the degree of Y ∩ Λ is bounded by the geometric degree of Y ∩ L. Proposition 4.8 implies that this geometric degree is the same as the geometric degree of π_{M}(Y ∩ L) = π_{M}(Y) ∩ π_{M}(L) = π_{M}(Y) ∩ π_{M}(Λ). Since π_{M}(Y) is small by hypothesis, this is bounded by dim(π_{M}(Λ)) + 1. Combining this inequality with the obvious deg(M ∩ Λ) + dim(π_{M}(Λ)) = dim(Λ), we deduce that deg(X ∩ Λ) ≤ 1 + dim(Λ) as required.

Proof of Theorem 0.4 continued. From the fact that Y ∪ M is small, it follows by Corollary 2.3 that Y is small. As Y is linearly normal by construction, we see from induction on the dimension, Proposition 2.1 (a), and Theorem 1.6 that Y is 2regular. We want to show next that Y actually coincides with X (i.e. M is empty), and as a first step we show the following result which interesting on its own.

Proposition 4.10 Let Y ⊂ ℙ^{s} be a 2-regular algebraic set. If p ∈ ℙ^{s} is a point in the span of Y, then there is a plane L containing p such that L ∩ Y is finite and L ∩ Y spans L.

Proof. Using the implication (b) ⇒ (c) of Theorem 0.4, we see that Y is the union of a linearly joined sequence of irreducible varieties of minimal degree Y_{1}, …, Y_{n}. By Proposition 2.4 (a) and Theorem 1.4 (c), the homogeneous coordinate rings of the irreducible components of Y are Cohen-Macaulay. We will prove, by induction on the dimension of Y, a more general result: if Y ⊂ ℙ^{s} is the union of a sequence of linearly joined irreducible schemes Y_{1}, … Y_{n} such that each homogeneous coordinate ring S_{Y_{i}} is Cohen-Macaulay, and no (Y_{i})_{red} is contained in another, then for any point p ∈ span(Y), there exists a plane L containing p such that L ∩ Y is finite and L ∩ Y spans L.

By Proposition 3.3 the problem reduces to the case where Y is connected. If Y is supported at a point the assertion is obvious. Otherwise, each component of Y must have dimension ≥ 1. Under this circumstance, we show that h^{0}(O_{Y}) = 1. Since S_{Y_{i}} has depth at least 2 we have ⨁_{m} H^{0}(O_{Y_{i}}(m)) ≅ S_{Y_{i}}, so h^{0}(O_{Y_{i}}) = 1 for every i. In general, if A, B ⊂ ℙ^{s} are schemes with h^{0}(O_{A}) = h^{0}(O_{B}) = 1, and if A ∩ B ≠ ∅, then taking the cohomology of the short exact sequence

0  O_{A∪B}  O_{A} ⊕ O_{B}  O_{A∩B}  0

gives that h^{0}(O_{A∪B}) = 1 also. In particular, h^{0}(O_{Y}) = 1 as claimed.

Let H be a general hyperplane containing p, defined by a linear form x ∈ H^{0}(O_{ℙ^{s}}(1)). Since H cuts properly every component Y_{i} of Y, S_{H∩Y_{i}} = S_{Y_{i}}/(x) is again Cohen-Macaulay. It follows from the definitions that the sequence H ∩ Y_{1}, …, H ∩ Y_{n} is linearly joined. Since Y is connected and reduces we see that h^{0}(O_{Y}) = 1. By Lemma 4.6 H is spanned by H ∩ Y, and we are done.

Corollary 4.11 If Y is a 2-regular algebraic set, and Y′ is a scheme disjoint from Y that meets the span of Y, then Y ∪ Y′ is not small.

Proof. By Proposition 4.10 there is a linear space L that contains a point of Y′ and meets Y in a finite scheme that spans L ∩ Y. Whatever the dimension of L ∩ Y′ this violates the conclusion of Theorem 2.2, so Y ∪ Y′ is not small.

Conclusion of the proof of Theorem 0.4. We may apply Corollary 4.11 to the case of the small schemes Y ∪ M from Proposition 4.9 and Y to conclude that M is empty! That is, the original small scheme X was linearly normal, and thus 2-regular. This finishes the proof of (a) ⇒ (b), and with it the proof of Theorem 0.4.

Remark 4.12 The conclusion of Proposition 4.10 seems to be true for a wide class of schemes Y. We are grateful to Harm Derksen for pointing out that it does not hold for arbitrary schemes, however: Let M_{1} ⊂ M_{2} ⊂ ℙ^{4} be a line contained in a 2-plane in ℙ^{4}, and let p_{1}, p_{2} be general points of M_{2}. Let Y = M_{1} ∪ P_{1} ∪ P_{2}, where P_{i} is a double point with (P_{i})_{red} = p_{i} and general tangent vector. It is immediate that Y spans ℙ^{4}. However, if q is a general point of ℙ^{4}, then q ∉ span(M_{1} ∪ P_{1}) ∪ span(M_{1} ∪ P_{2}) ∪ span(P_{1} ∪ P_{2}) and so any linear space L containing q and spanned by L ∩ Y must contain at least p_{1}, p_{2}, and a point of M_{1} which is not collinear with p_{1}, p_{2}. It follows easily that such an L is all of ℙ^{4}, and thus L ∩ Y is not finite.

On the other hand, it seems that the conclusion of Proposition 4.10 might hold for all reduced schemes. Kristian Ranestad has proven this in characteristic 0. We are grateful to him for allowing us to include it here.

Proposition 4.13 Let X ⊂ ℙ^{s} be a reduced scheme over an algebraically closed field of characteristic zero. If p is a point in the linear span of X, then there is a linear subspace L ⊂ ℙ^{s} containing p such that L ∩ X is finite and contains a set of distinct points that spans L.

Proof. We may as well assume that X spans ℙ^{s}. If p ∈ X or X is finite there is nothing to prove. So let X_{0} be a positive dimensional irreducible component of X and let L_{0} be the linear span of X_{0}. Since L_{0} has dimension r > 0, we may choose distinct points x_{0}, …, x_{r} ∈ X_{0}, and x_{r+1}, …, x_{s} ∈ X \ X_{0}, such that x_{0}, …, x_{s} span ℙ^{s}.

If L ⊂ ℙ^{s} is a linear space of dimension < s such that L is spanned by some set of distinct points of X and p ∈ L, then we may replace X by (L∩X)_{red} and we are done by induction on s. Thus for example we are done if p ∈ L_{1} = span{x_{r+1}, …, x_{s}}. Therefore we may assume that p ∉ L_{1}, so the plane L_{2} = span(p, L_{1}) meets L_{0} in a unique point q. If q ∈ X, then since L_{2} = span{q, x_{r+1}, …, x_{s}} and r ≥ 1, we are again done by induction.

Thus we may suppose q ∉ X. It follows that the system of hyperplanes containing L_{2} has no basepoints on X_{0}. Since the the base field is of characteristic zero it follows that a general hyperplane H containing L_{2} meets X_{0} in a reduced set. Since X_{0} is reduced and irreducible, Lemma 4.6 shows that H∩X_{0} spans H∩L_{0}. Therefore we may find distinct points y_{1}, …, y_{r} ∈ H ∩ X_{0} such that y_{1}, …, y_{r}, x_{r+1}, …, x_{s} span H. Since p ∈ H again we are done by induction on s, and the proposition follows.

5 Small subspace arrangements

In this section we will study the combinatorics of the condition for union of linear subspaces to be small. Recall that a subgraph F of a graph G is called a forest if F is a disjoint union of trees (acyclic connected graphs). A leaf of F is a vertex of F connected to at most one other vertex of F. A forest F ⊂ G spans G if F contains all the vertices of G and any two vertices connected by a path in G are connected by a path in F. We say that an ordering of the vertices of G is compatible with a spanning forest F ⊂ G if the smallest vertex in every connected component is a leaf, and the ordering restricts to the natural ordering on the vertices of any path starting from that leaf.

By a subspace arrangement we mean a finite union of incomparable linear subspaces in a projective space, say Y = ⋃_{i=1}^{n} L_{i} ⊂ ℙ^{r}. We generally do not distinguish between the set of subspaces and their union. To a subspace arrangement Y we associate the weighted graph G_{Y}, whose vertices are the subspaces L_{i} of Y, and whose edges join the pairs of subspaces with non-empty intersection. We define the weight of the edge (L_{i}, L_{j}) to be 1 + dim(L_{i} ∩ L_{j}), and the weight of a subgraph is the sum of the weights of its edges. We will be interested in the spanning forests of maximal weight in G_{Y}. To simplify the notation, we give the vertex L_{i} weight 1 + dim(L_{i}) and, for any graph G with edge and vertex weights we define the weighted Euler characteristic χ_{w}(G) to be the sum of the weights of the vertices minus the sum of the weights of the edges. Thus a spanning forest F for G_{Y} has maximal weight if and only if it has minimal χ_{w}(F).

The main result of this section says that smallness and linearly joined sequences of components of Y are features of certain spanning forests in G_{Y}.

Theorem 5.1 Let Y ⊂ ℙ^{r} be a subspace arrangement.

(a) Y is small if and only if the weighted graph G_{Y} has a spanning forest F with χ_{w}(F) = 1 + dim(span(Y)), the smallest possible value.

(b) If Y is small, then an ordering of the components of Y makes them into a linearly joined sequence if and only if it is compatible with some spanning forest satisfying the equality above.

We begin with an elementary result that shows the above given value of χ_{w} is the smallest possible and explains the connection with linearly joined sequences.

Lemma 5.2 Let Y = ⋃_{i=1}^{n} L_{i} ⊂ ℙ^{r} be a subspace arrangement, and let F be a spanning forest of G_{Y}.

(a) Suppose that L_{n} is a leaf of F. If Y′ and F′ are obtained from Y and F by removing L_{n}, then

χ_{w}(F) ≥ χ_{w}(F′) + [dim(span(Y)) − dim(span(Y′))],

with equality if and only if L_{n} is linearly joined with Y′ and either L_{n} ∩ Y′ is empty or L_{n} ∩ Y′ = L_{n} ∩ L_{j}, where L_{n} is connected to L_{j} in F.

(b) χ_{w}(F) ≥ 1 + dim(span(Y)).

Proof of Lemma 5.2. Part (a) is elementary, and part (b) follows from part (a) by induction on the number of components.

Proof of Theorem 5.1. We prove parts (a) and (b) together. First suppose that F is a spanning forest with χ_{w}(F) = 1 + dim(span(Y)), and L_{1}, …, L_{n} is a compatible ordering of the subspaces in Y.

It follows that L_{n} is a leaf of F. Let Y′ and F′ be obtained by deleting L_{n} from Y and F respectively. By Lemma 5.2 we have

1 + dim(span(Y′)) ≤ χ_{w}(F′)
≤ χ_{w}(F) − [dim(span(Y)) − dim(span(Y′))]
= 1 + dim(span(Y′))

so all the equalities hold. Thus L_{n} is linearly joined to Y′ and, by induction on the number of components, L_{1}, …, L_{n} is a linearly joined sequence and Y′ is small. Using Theorem 0.4 it follows from Proposition 3.1 that Y is small.

Now suppose that Y is small. By Theorem 0.4 (c) we may order the components of Y to form a linearly joined sequence L_{1}, …, L_{n}. It suffices to find a spanning forest with which this ordering is compatible. We do induction on the number n of components of Y, the case n = 1 being trivial.

Since the sequence L_{1}, …, L_{n−1} is linearly joined, Y′ = ⋃_{i=1}^{n−1} L_{i} is also small. By induction, G_{Y′} contains a spanning forest F′ with χ_{w}(F′) = 1 + dim(span(Y′)) such that the ordering L_{1}, …, L_{n−1} is compatible with F′. Since Y′ ∩ L_{n} = span(Y′) ∩ L_{n} is a linear space, and the ground field is infinite, Y′ ∩ L_{n} is either empty or it is equal to L_{j} ∩ L_{n} for some j < n. In the first case L_{n} is a connected component of G_{Y}, so adjoining it to F′ we get a spanning forest F of G_{Y} In the second case, we may adjoin L_{n} to F′ and connect it to L_{j}, obtaining a new spanning forest. In either case we get equality in the formula of Lemma 5.2 (a), and the given order is compatible with the forest F.

Theorem 5.1 makes it interesting to understand better which forests have minimal weighted Euler characteristic. If Y = ⋃ L_{i} is a subspace arrangement, we say that a forest F ⊂ G_{Y} satisfies the clique-intersection property if whenever L_{1}, …, L_{j} form a path in F we have

L_{1} ∩ L_{j} = L_{1} ∩ L_{2} ∩ ⋯ ∩ L_{j}.

Proposition 5.3 Let Y = ⋃_{i=1}^{n} L_{i} ⊂ ℙ^{r} be a small subspace arrangement with intersection graph G_{Y}. A spanning forest F ⊂ G_{Y} has χ_{w}(F) = 1 + dim(span(Y)) (or equivalently, maximal weight) if and only if F satisfies the clique-intersection property.

Proof. Prim’s algorithm [1957] (see also Graham-Hell [1985]) shows that a spanning forest F in a weighted connected graph G has minimal χ_{w}(F) if and only if, for each edge (x, y) of G, the path in T joining x to y consists of edges (each) of weight ≥ the weight of (x, y) (see also Tarjan [1983, pp. 71–72].) In particular, a spanning forest T ⊂ G_{Y} satisfying the clique-intersection property must have minimal weighted Euler characteristic. Since Y is small, Lemma 5.2 shows this is 1 + dim(span(Y)).

On the other hand, suppose that F is any spanning forest with χ_{w}(F) = 1 + dim(span(Y)) and L_{1}, …, L_{j} are the spaces along a path in F. By Theorem 5.1 the spaces L_{1}, …, L_{j} form a part of a linearly joined sequence L_{−t}, …, L_{0}, L_{1}, …, L_{j}, L_{j+1}, …, L_{s} involving all the components of Y. It follows that all of the L_{i} ∩ L_{j}, for i < j are contained in one L_{h} ∩ L_{j}, with h < j. By Prim’s algorithm, dim(L_{h} ∩ L_{j}) ≤ dim(L_{j−1} ∩ L_{j}) so all the L_{i} ∩ L_{j} are contained in L_{j−1} ∩ L_{j}. Thus by induction on j we get

\begin{aligned}L_{1} ∩ L_{j} & = L_{1} ∩ L_{j−1} ∩ L_{j} \\ & = L_{1} ∩ L_{2} ∩ … ∩ L_{j−1} ∩ L_{j} \end{aligned}

as required.

Another way of finding an ordering of the components of a small subspace arrangement Y as a linearly joined sequence is the following: select vertices of G_{Y} inductively by choosing, at each step, an unselected vertex i_{k} which has the maximal number of adjacent vertices among the vertices already selected. For a proof see Tarjan-Yannakakis [1984].

The union of all spanning forests of maximal weight of G_{Y} is the subgraph H_{Y} ⊂ G_{Y} with the same vertices as the intersection graph G_{Y}, but where L_{i} and L_{j} are joined by an edge only when L_{i}∩L_{j} disconnects Y. Indeed, by Proposition 5.3 a maximal weight spanning forest F of G_{Y} satisfies the clique-intersection property which easily implies that F is actually a spanning forest in H_{Y}. Conversely, if L_{i}∩L_{j} ≠ ∅ disconnects Y and L_{i} = L_{k_{0}}, L_{k_{1}}, …, L_{k_{r}} = L_{j} is the path joining L_{i} and L_{j} in a maximal weight spanning forest F of G_{Y}, then necessarily L_{k_{m}} ∩ L_{k_{m+1}} = L_{i} ∩ L_{j}, for some m < r and the forest F′ obtained by replacing in F the edge (L_{k_{m}}, L_{k_{m+1}}) by (L_{i}, L_{j}) is also of maximal weight.

6 Equations and syzygies of reduced 2-regular schemes

Next we show how to find generators for the ideal of a reduced 2-regular projective scheme using property (c) of Theorem 0.4. Consider a closed subscheme X = X′ ∪ X′′ ⊂ ℙ^{r}, and set L′ = span(X′), L′′ = span(X′′). In general it is difficult to find generators for the intersection of two ideals, but if X′ and X′′ are linearly joined (that is, X′ ∩ X′′ = L′ ∩ L′′) then we can give minimal generators and a (non-minimal) free resolution of I_{X} = I_{X′∪X′′} = I_{X′} ∩ I_{X′′} explicitly from minimal generators and free resolutions for I_{X′/L′} and I_{X′′/L′′}. Our result extends results of Barile and Morales [2000, 2003].

For simplicity we will suppose throughout that L′ ∪ L′′ spans the whole ambient space ℙ^{r}, and leave the reader the easy task to adapt Theorem 6.1 below to the degenerate case. We write μ(I) for the minimal number of generators of a homogeneous ideal I, and reg(I) for its regularity.

Theorem 6.1 Let X = X′ ∪ X′′ ⊂ ℙ^{r} be a nondegenerate closed scheme that is the union of two subschemes X′ and X′′ with linear spans L′ and L′′, respectively. Suppose that X′ and X′′ are linearly joined along L = L′ ∩ L′′. (a)

I_{X} = Ĩ_{X′,L′} + Ĩ_{X′′,L′′} + I_{L′} · I_{L′′},

where Ĩ_{X′,L′} is any ideal that vanishes on L′′ and restricts on L′ to the ideal I_{X′,L′} of X′ in L′, and similarly for Ĩ_{X′′,L′′}.

(b) μ(I_{X}) = μ(I_{X′,L′}) + μ(I_{X′′,L′′}) + μ(I_{L′})μ(I_{L′′}).

(c) reg(I_{X}) = max {2, reg(I_{X′}), reg(I_{X′′})}.

The simplest way to construct an ideal Ĩ_{X′,L′} as required in part (a) is to choose coordinates in ℙ^{r} so that x_{0}, …, x_{i} are coordinates for L′ while the linear space where x_{0}, …, x_{i} vanish is a subspace of L′′. Let I_{X′,L′} be the ideal of X′ in the homogeneous coordinate ring of L′. If we write generators for I_{X′,L′} in terms of the coordinates x_{0}, …, x_{i}, then we may take the same expressions as generators for Ĩ_{X′,L′}.

Geometrically, this construction amounts to choosing a subspace L′′_{0} ⊂ L′′ complementary to L = L′ ∩ L′′, and taking Ĩ_{X′,L′} to be the ideal of the cone with base X′ and vertex L′′_{0}. However, this is not the only choice possible: we may perturb the generators given above by any elements of I_{L′} ∩ I_{L′′}. This may even change the codimension of Ĩ_{X′,L′}. Note that the generators of I_{X′,L′} have degree at least 2. Since (I_{L′} ∩ I_{L′′})_{≥2} = (I_{L′} · I_{L′′})_{≥2} such a perturbation will not, as it must not, change the given form of I_{X}.

Proof of Theorem 6.1. Since I_{X} = I_{X′} ∩ I_{X′′} and I_{X′} + I_{X′′} = I_{L} there is an exact sequence

0  I_{X}  I_{X′} ⊕ I_{X′′}  I_{X′∩X′′}  0.

Since we have assumed that X ⊂ span(L′ ∪ L′′) = ℙ^{r} is nondegenerate, the space of linear forms vanishing on L is the direct sum of the spaces of linear forms vanishing on L′ and L′′. If we choose minimal free modules F′, F′′ and surjections F′ → I_{L′} and F′′ → I_{L′′} we get a minimal surjection F′ ⊕ F′′ → I_{L}, and we may write a minimal free resolution of I_{L} in the form

∧(F′ ⊕ F′′)_{≥1} :  ⋯  ∧^{2} (F′ ⊕ F′′)  F ⊕ F′  I_{L}  0.

Let Ĩ_{X′,L′} be any ideal that vanishes on L′′ and restricts on L′ to the ideal of X′ in L′, and similarly for Ĩ_{X′′,L′′}. Let G′_{1} and G′′_{1} be free modules that map minimally onto Ĩ_{X′,L′} and Ĩ_{X′′,L′′} respectively, and set H′_{1} = F′ ⊕ G′_{1} and H′′_{1} = F′′ ⊕ G′′_{1}. Because Ĩ_{X′,L′} reduces modulo I_{L′} to I_{X′,L′} the induced map H′_{1} → I_{X′} is surjective, and similarly for H′′_{1} → I_{X′′}. Choose now a minimal free resolution

H′ :  ⋯  H′_{2}  H′_{1}  I_{X′}  0.

The Koszul complex on the generators of I_{L′} may be written as (∧F′)_{≥1}. Considering degrees, we see that each term ∧^{i}F′ is a summand of the corresponding term H′_{i} of H′, and we may write H′_{i} = ∧^{i}F′ ⊕ G′_{i} for a suitable free module G′_{i}. Define H′′, ∧F′′ and G′′_{i} similarly, and let a : H′_{1} ⊕ H′′_{1} → I_{X′} ⊕ I_{X′′} be the projection.

We will choose a map of complexes ϕ : H′ ⊕ H′′ → ∧(F′ ⊕ F′′)_{≥1} lifting the natural sum map I_{X′} ⊕ I_{X′′} → I_{L}:

⋮  ⋮

[figure: (G′_{2} ⊕ ∧^{2}F′) ⊕ (G′′_{2} ⊕ ∧^{2}F′′); ϕ_{2}; ∧^{2}F′ ⊕ (F′ ⊗ F′′) ⊕ ∧^{2}F′′]

[figure: (G′_{1} ⊕ F′) ⊕ (G′′_{1} ⊕ F′′); ϕ_{1}; F′ ⊕ F′′; δ]

[figure: I_{X′} ⊕ I_{X′′}; a; I_{L}.; b]

We may make this choice so that the restriction of ϕ to the subcomplex (∧F′ ⊕ ∧F′′)_{≥1} ⊂ H is the canonical injection into ∧(F′ ⊕ F′′)_{≥1}.

The mapping cone of ϕ is a complex

⋯ [∧^{2}F′ ⊕ (F′ ⊗ F′′) ⊕ ∧^{2}F′′] ⊕ [(G′_{1} ⊕ F′) ⊕ (G′′_{1} ⊕ F′′)] ^{δ⊕ϕ_{1}} F′ ⊕ F′′

whose only homology is I_{X}, occurring at H′_{2} ⊕ H′′_{2}.

Since the map ϕ_{1} is surjective and ϕ_{2} maps ∧^{2}F′⊕∧^{2}F′′ onto the corresponding summands of ∧^{2}(F′ ⊕ F′′) = ∧^{2}F′ ⊕ (F′ ⊗ F′′) ⊕ ∧^{2}F′′, we see that I_{X} has a free resolution beginning with

⋯ (G′_{2}⊕∧^{2}F′)⊕(G′′_{2}⊕∧^{2}F′′)⊕∧^{3}(F′⊕F′′)  F′⊗F′′⊕ker(ϕ_{1})  I_{X}  0.

The ideal I_{X} is embedded in I_{X′} ⊕I_{X′′} as the diagonal. Thus the image of F′⊗F′′ in I_{X} is computed by lifting δ along ϕ_{1} and then composing with the map H′_{1} ⊕ H′′_{1} → I_{X′} ⊕ I_{X′′}, which lands in the kernel of the projection I_{X′} ⊕ I_{X′′} I_{X′∩X′′} = I_{L}. Because of our choice of ϕ, the image of this composite is I_{L′} · I_{L′′}.

Because Ĩ_{X′,L′} ⊂ I_{L′′} we may choose ϕ_{1}|_{G′_{1}} to be a map with image contained in F′′. We similarly choose ϕ_{1}|_{G′′_{1}} to map into F′. It follows that ker(ϕ_{1}) is the direct sum of ker(ϕ_{1}|_{G′_{1}⊕F′′}) = Transpose(1, −ϕ_{1}|_{G′_{1}})(G′_{1}) and a corresponding copy of G′′_{1}. Since I_{X} is embedded diagonally in I_{X′} ⊕I_{X′′} the image of Transpose(1, −ϕ_{1}|_{G′_{1}})(G′_{1}) is Ĩ_{X′,L′}. The corresponding result for Ĩ_{X′′,L′′} follows symmetrically, proving the formula (a).

The resolution of I_{X} that we have just constructed is not in general minimal. However, the syzygies corresponding to elements of G′_{2} ⊕ G′′_{2} must involve the elements of G′_{1} ⊕ G′′_{1}. As these elements have degrees ≥ 2, the generators of G′_{2} and G′′_{2} must have degrees ≥ 3. Thus the map from (F′ ⊗ F′′) to S sends the generators minimally onto the generators of I_{L′} ·I_{L′′}. Moreover, the generators of G′_{1} map onto forms that minimally generate the ideal of X′ in its span, and similarly for X′′. If there were a dependence relation of the form

f′ + f′′ + p = 0

where

f′ ∈ Im(G′_{1}), f′′ ∈ Im(G′′_{1}) and p ∈ Im(F′ ⊗ F′′),

then working modulo the equations of L′ we see that f′ = 0 and similarly f′′ = 0. It follows that p = 0 as well, establishing part (b), the desired relation on minimal numbers of generators.

Statement (c) on the regularity follows at once from the form of the above (not necessarily minimal) resolution.

Corollary 6.2 Let X = ⋃_{i=1}^{n} L_{i} ⊂ ℙ^{r} be a nondegenerate 2-regular union of linear subspaces, linearly joined in that order as in Theorem 0.4. If, for each i ∈ {2, …, n}, L′_{i} is a linear complement in L_{i} for L_{r_{i}} ∩L_{i}, where span(L_{1}, …, L_{i−1})∩L_{i} = L_{r_{i}} ∩L_{i} with 1 ≤ r_{i} < i, then

I_{X} = ∑_{j=2}^{n} I_{span(L_{j},L_{j+1},...,L_{n})}I_{span(L_{1},...,L_{j−1},L′_{j+1},...,L′_{n})}

= ∑_{j=2}^{n} I_{span(L_{j},L′_{j+1},...,L′_{n})}I_{span(L_{1},...,L_{j−1},L′_{j+1},...,L′_{n}).}

Proof. The case n = 2 is immediate. By induction on n we may assume that the equations of X′ = L_{1} ∪ … ∪ L_{n−1} are given by a similar formula with n − 1 in place of n. Set X′′ = L_{n}. In the expression for I_{X} in Theorem 6.1, the ideal Ĩ_{X′′/L′′} may be taken to be 0. We may choose Ĩ_{X′,L′} to be the ideal of the cone with base X′ and vertex L′_{n}, as in the remark after Theorem 6.1. Since taking the ideals of cones commutes with sums and products in an appropriate sense, and the cone over a span of a collection of linear spaces is obtained by taking the span with the vertex, we arrive at the given formula.

Remark 6.3 By the above choice of L′_{i} we have, for each i < j, that

span(L_{i}, L′_{i+1}, …, L′_{j}) = span(L_{i}, L_{i+1}, …, L_{j}).

Applying this with i = 1, j = n we see that L_{1}, L′_{2}, …, L′_{n} span ℙ^{r}. Moreover, since

span(L_{1}, …, L_{j}) ∩ L_{j+1} = L_{r_{j}} ∩ L_{j+1}

by hypothesis, we see that L′_{j+1} is disjoint from span(L_{1}, …, L_{j}). Thus if we think of ℙ^{r} as lines in a vector space V, then V is the direct sum of spaces corresponding to L_{1}, L′_{2}, …, L′_{n}. It follows that we may choose variables x_{i} so that for each j the space span(L′_{j+1}, …, L′_{n}) is defined by the vanishing of an initial segment x_{0}, …, x_{i_{j}}, and this set of variables are coordinates on span(L_{1}, …, L_{j}). If Y ⊂ span(L_{1}, …, L_{j}) is a subvariety, then the cone with base Y and vertex span(L′_{j+1} ∪ … ∪ L′_{n}) is defined by the equations of Y in span(L_{1}, …, L_{j}) written in these coordinates. Moreover, the cone over this variety with vertex L′_{j+2} is given by the same equations, and so on.

Corollary 6.4 The homogeneous ideal I ⊂ S of any 2-regular union of linear spaces is generated by products of pairs of distinct (independent) linear forms.

Remark 6.5 Motivated by the structure in Corollary 6.4 for the homogeneous ideal of a 2-regular union of linear spaces one may believe that such an ideal can always be obtained from the squarefree monomial ideal of a 2-regular union of coordinate subspaces in a larger space by factoring out a sequence of linear forms that is a regular sequence on every component and any nonempty mutual intersection of irreducible components. Unfortunately this is false as the following example shows: Let X ⊂ ℙ^{6} consist of a ℙ^{3} and three general ℙ^{2}’s sticking out of it, each meeting the ℙ^{3} in a line, say denoted as L_{i}. The set X is small, but the existence a squarefree monomial ideal “inflation” as above would imply that the homogeneous ideal of the union of the three (general) lines L_{1}, L_{2} and L_{3} in ℙ^{3} would also be generated by products of pairs of linear forms, which is not the case.

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Author Addresses:

David Eisenbud Department of Mathematics, University of California at Berkeley, Berkeley CA 94720

de@msri.org

Mark Green Department of Mathematics, University of California at Los Angeles, Los Angeles CA 90095-1555

mlg@math.ucla.edu

Klaus Hulek Institut für Mathematik, Universität Hannover, D-30060 Hannover, Germany

hulek@math.uni-hannover.de

Sorin Popescu Department of Mathematics, SUNY at Stony Brook, Stony Brook, NY 11794-3651

sorin@math.sunysb.edu