A FACT ABOUT LINEAR SPACES ON HYPERSURFACES

JASON MICHAEL STARR

Abstract. A smooth, nondegenerate hypersurface in projective space contains no linear subvarieties of greater than half its dimension. It can contain linear subvarieties of half its dimension. This note proves that a smooth hypersurface of degree d ≥ 3 contains at most finitely many such subvarieties.

Let k be a field. Let X ⊂ ℙ_{k}^{n} be a hypersurface of degree d > 1. For each integer m > 0, denote by F_{m}(X) the Fano scheme of m-planes in X, cf. [1]. There are a number of natural questions about F_{m}(X): is this scheme nonempty, is this scheme connected, is this scheme irreducible, is this scheme reduced, is this scheme smooth, what is the dimension of this scheme, what is the degree of this scheme, etc.? For each of these questions, the answer is uniform for a generic hypersurface. More precisely, there is a non-empty open subset U of the parameter space of hypersurfaces, such that for every point in U the answer to the question is the same. For a generic hypersurface, the answer is often easy to find: The total space of the relative Fano scheme of the universal hypersurface is itself a projective bundle over the Grassmannian 𝔾(m, n), so if the question for a generic hypersurface can be reformulated as a question about the total space of the relative Fano scheme, it is easy to answer the question. However, much less is known if X is assumed to be smooth, but not generic.

There are a few easy results, such as the following.

prop-1

Proposition 0.1. Let X ⊂ ℙ^{n} be a hypersurface of degree d > 1 and let m be an integer such that 2m ≥ n. Every m-plane Λ ⊂ X intersects the singular locus of X. In particular, if X is smooth then F_{m}(X) is empty.

Proof. Choose a system of homogeneous coordinates x_{0}, …, x_{n} on ℙ^{n} such that Λ is given by x_{m+1} = ⋯ = x_{n} = 0. Let F be a defining equation for X. Because Λ ⊂ X, F (x_{0}, …, x_{m}, 0, …, 0) = 0. Also,

\frac{∂F}{∂x_{i}} (x_{0}, …, x_{m}, 0, …, 0) = 0,

for i = 0, …, m. Because d > 1, for i = 1, …, n − m, the homogeneous polynomial on Λ,

\frac{∂F}{∂x_{m+i}} (x_{0}, …, x_{m}, 0, …, 0),

is nonconstant. Since n − m ≤ m, these n − m nonconstant homogeneous polynomials have a common zero in Λ. By the Jacobian criterion, this is a singular point of X. □

Remark 0.2. This also follows easily from the Lefschetz hyperplane theorem.

What happens if n = 2m + 1? If m > 1 or if m = 1 and d > 3, then a generic hypersurface X ⊂ ℙ^{n} contains no m-plane. However there do exist smooth hypersurfaces containing an m-plane. For instance, if char(k) does not divide d then the Fermat hypersurface x_{0}^{d} + ⋯ + x_{n}^{d} = 0 is smooth and contains many m-planes, e.g., x_{0} + x_{1} = x_{2} + x_{3} = ⋯ = x_{n−1} + x_{n} = 0 when d is odd. However, if d ≥ 3, a smooth hypersurface cannot contain a positive-dimensional family of m-planes. This was proved independently by Olivier Debarre, using a different argument.

Date: July 19, 2005.

The setup is as follows. Let n = 2m+1. Let X ⊂ ℙ^{n} be a hypersurface of degree d. Let Λ_{1}, Λ_{2} ⊂ X be m-planes. Denote by Z the intersection Λ_{1} ∩ Λ_{2}. This is either empty or else an r-plane for some integer r. If Z is empty, define r to be −1.

Denote by X_{sm} ⊂ X the smooth locus of X, i.e., the maximal open subscheme that is smooth. Denote Λ_{i,sm} = Λ_{i} ∩ X_{sm} for i = 1, 2. There are Chow classes,

[Λ_{i,sm}] ∈ A_{m}(X_{sm}), i = 1, 2.

Because X_{sm} is smooth, the intersection product [Λ_{1,sm}] · [Λ_{2,sm}] ∈ A_{0}(X_{sm}) is defined.

lem-2 Lemma 0.3. If Z is contained in X_{sm}, then the degree of [Λ_{1,sm}] · [Λ_{2,sm}] is (1 − (1 − d)^{r+1})/d.

Proof. If r = −1, i.e., if Z is empty, this is obvious. Therefore suppose that Z is an r-plane for some r ≥ 0. By the excess intersection formula, the class [Λ_{1,sm}] · [Λ_{2,sm}] is the pushforward from Z of the refined intersection product, (Λ_{1,sm} · Λ_{2,sm})^{Z}. And by [2, Prop. 9.1.1], the refined intersection product is,

(Λ_{1,sm} · Λ_{2,sm})^{Z} = {c(N_{Λ_{1,sm}/X_{sm}})/c(N_{Z/Λ_{2,sm}}) ∩ [Z]}_{r} .

Denote H = c_{1}(O_{Z}(1)). The normal bundle of Z in Λ_{1,sm} is O_{Z}(1)^{m−r}. The restriction to Z of the normal bundle of Λ_{2,sm} in ℙ^{n} is O_{Z}(1)^{n−m} = O_{Z}(1)^{m+1}. And the restriction to Z of the normal bundle of X in ℙ^{n} is O_{Z}(d). Therefore,

c(N_{Λ_{1,sm}/X_{sm}})/c(N_{Z/Λ_{2,sm}}) = \frac{(1 + H)^{m+1}}{(1 + H)^{m−r}(1 + dH)} = \frac{(1 + H)^{r+1}}{1 + dH} .

Expanding this out gives,

(∑_{i=0}^{r+1} \binom{r + 1}{i}H^{i}) (∑_{j=0}^{∞}(−1)^{j}d^{j}H^{j}) .

In particular, the coefficient of H^{r} is,

∑_{i=0}^{r} \binom{r + 1}{i}(−1)^{r−i}d^{r−i} = \frac{−1}{d} ∑_{i=0}^{r} \binom{r + 1}{i}(−1)^{r+1−i}d^{r+1−i}
= (1 − (1 − d)^{r+1})/d.

□ prop-3

Proposition 0.4. If d ≥ 3, if Λ_{1} and Λ_{2} are distinct, and if at least one of Λ_{1}, Λ_{2} is contained in X_{sm}, then [Λ_{1,sm}] is not numerically equivalent to [Λ_{2,sm}].

Proof. Let Λ_{1} be contained in X_{sm}. By Lemma 0.3,

deg([Λ_{1}] · [Λ_{1}]) = (1 − (1 − d)^{m+1})/d.

Also, Z = Λ_{1} ∩ Λ_{2} is contained in X_{sm}. So by Lemma 0.3,

deg([Λ_{1}] · [Λ_{2,sm}]) = (1 − (1 − d)^{r+1})/d.

Because d − 1 ≥ 2 and r < m, we have (d − 1)^{r+1} < (d − 1)^{m+1}. Therefore (1 − (1 − d)^{r+1})/d ≠ (1 − (1 − d)^{m+1})/d, and so [Λ_{1}] is not numerically equivalent to [Λ_{2,sm}]. □

cor-4 Corollary 0.5 (Debarre). There are only finitely many m-planes contained in X_{sm}.

Proof. By Proposition 0.4, distinct m-planes contained in X_{sm} are not algebraically equivalent. Therefore every irreducible component of F_{m}(X) that contains a point parametrizing an m-plane in X_{sm} is just a point. Because F_{m}(X) is quasi-compact, the number of these irreducible components is finite. □

rmk-5 Remark 0.6. Debarre’s proof shows more than Corollary 0.5: for any m-plane Λ contained in X_{sm}, h^{0}(Λ, N_{Λ/X}) = 0. It follows that each such point is a connected component of F_{m}(X), and that F_{m}(X) is reduced at this point.

A natural question is, what is the maximal number of m-planes contained in a smooth hypersurface of degree d in ℙ^{2m+1}? There is a naive upper bound that grows as d^{(m+1)^{2}}, but this is too large. The Fermat hypersurface contains C_{m}d^{m+1} distinct m-planes, where C_{m} = (2m + 1)(2m − 1) · · · · · 3 · 1. Joe Harris points out that for m = 1, the degree of the flecnodal curve gives an upper bound of 11d^{2} − 24d.

More generally, define the parabolic locus P (X) ⊂ X to be the set of points p ∈ X such that there is a line L ⊂ ℙ^{2m+1} that has contact of order 3m + 1 with X at p. This is the pushforward in X of a subscheme in ℙ(T_{X}) that is the zero locus of a section of a locally free sheaf. If X is generic, this section is a regular section. Then a Chern class computation gives that the degree of P (X) is a polynomial p_{m}(d) of degree m + 1 in d whose leading term is,

(\frac{(3m + 1)!}{2} − 1) d^{m+1}.

Therefore, for arbitrary X, deg(P_{m}(X)) ≤ p_{m}(d), where P_{m}(X) is the m-cycle of all m-dimensional irreducible components of P (X) (weighted by multiplicity).

Of course every m-plane Λ is contained in P (X). It is not clear that every m-plane is contained in P_{m}(X), i.e., that Λ is an irreducible component of P (X). And, indeed, this fails if d ≤ 3m. For d ≥ 3m, it may be true that every m-plane is an irreducible component of P (X).

In the special case that m = 1, P (X) is a curve, the parabolic curve, for all d ≥ 3. Therefore, the number of lines in a smooth surface of degree d ≥ 3 in ℙ^{3} is at most 11d^{2} − 24d. Note this gives the correct answer for d = 3. For d = 4 this gives the wrong answer; Segre proved the maximal number of lines on a quartic surface is 64, cf. [3]. In fact, by a more involved analysis, Segre proved that the number of lines on a smooth surface of degree d ≥ 3 is at most 11d^{2} − 28d + 12. The Fermat surface contains 3d^{2} lines. The true maximum is probably strictly between 3d^{2} and 11d^{2} − 28d + 12.

References

[1] A. B. Altman and S. L. Kleiman. Foundations of the theory of Fano schemes. Compositio Math., 34(1):3–47, 1977.

[2] W. Fulton. Intersection theory, volume 2 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1984.

[3] B. Segre. The maximum number of lines lying on a quartic surface. Quarterly J. Math. Oxford Ser., 14:86–96, 1943. Historical Jrl., Ann Arbor MI 0.

Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139

   E-mail address: jstarr@math.mit.edu