A Fact about Linear Spaces on Hypersurfaces

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 ⊂ Pk be a hypersurface of degree d > 1. For each integer m > 0, denote by Fm(X) the Fano scheme of m-planes in X, cf. [1]. There are a number of natural questions about Fm(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 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 Fm(X) is empty. Proof. Choose a system of homogeneous coordinates x0, . . . , xn on P such that Λ is given by xm+1 = · · · = xn = 0. Let F be a defining equation for X. Because Λ ⊂ X, F (x0, . . . , xm, 0, . . . , 0) = 0. Also, ∂F ∂xi (x0, . . . , xm, 0, . . . , 0) = 0, for i = 0, . . . ,m. Because d > 1, for i = 1, . . . , n−m, the homogeneous polynomial on Λ, ∂F ∂xm+i (x0, . . . , xm, 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 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 x0+ · · ·+xn = 0 is smooth and contains many m-planes, e.g., x0 + x1 = x2 + x3 = · · · = xn−1 + xn = 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.