FANO VARIETIES AND LINEAR SECTIONS OF HYPERSURFACES

JASON MICHAEL STARR

Abstract. When n satisfies an inequality which is almost best possible, we prove that the k-plane sections of every smooth, degree d, complex hypersurface in ℙ^{n} dominate the moduli space of degree d hypersurfaces in ℙ^{k}. As a corollary we prove that, for n sufficiently large, every smooth, degree d hypersurface in ℙ^{n} satisfies a version of “rational simple connectedness”.

1. Statement of results

In their article [2], Harris, Mazur and Pandharipande prove that for fixed integers d and k, there exists an integer n_{0} = n_{0}(d, k) such that for every n ≥ n_{0}, every smooth degree d hypersurface X in ℙ_{ℂ}^{n} has a number of good properties:

(i) The hypersurface is unirational.

(ii) The Fano variety of k-planes in X has the expected dimension.

(iii) The k-plane sections of the hypersurface dominate the moduli space of

degree d hypersurfaces in ℙ^{k}. It is this last property which we consider. To be precise, the statement is that the following rational transformation

Φ : 𝔾(k, n) −−→ ℙ^{N_{d}}//PGL_{k+1}

is dominant. Here 𝔾(k, n) is the Grassmannian parametrizing linear ℙ^{k}s in ℙ^{n}, ℙ^{N_{d}} is the parameter space for degree d hypersurface in ℙ^{k}, ℙ^{N_{d}}//PGL_{k+1} is the moduli space of semistable degree k hypersurface in ℙ^{k}, and Φ is the rational transformation sending a k-plane Λ to the moduli point of the hypersurface Λ ∩ X ⊂ Λ (assuming Λ ∩ X is a semistable degree k hypersurface in ℙ^{k}).

The bound n_{0}(d, k) is very large, roughly a d-fold iterated exponential. Our result is the following.

Theorem 1.1. Let X be a smooth degree d hypersurface in ℙ^{n}. The map Φ is dominant if

n ≥ \binom{d + k − 1}{k} + k − 1.

Question 1.2. For fixed d and k, what is the smallest integer n_{0} = n_{0}(d, k) such that for every n ≥ n_{0} and every smooth, degree d hypersurface in ℙ^{n}, the associated rational transformation Φ is dominant?

Theorem 1.1 is equvialent to the inequality

n_{0}(d, k) ≤ \binom{d + k − 1}{k} + k − 1.

Date: July 5, 2006.

If Φ is dominant, then the dimension of the domain is at least the dimension of the target, i.e.,

(k + 1)(n − k) = dim𝔾(k, n) ≥ dim(ℙ^{N_{d}}//PGL_{k+1}) = \binom{d + k}{k} − (k + 1)^{2}.

This is equivalent to the condition

n_{0}(d, k) ≥ \frac{1}{k + 1}\binom{d + k}{k} − 1.

As far as we know, this is the correct bound. The bound from Theorem 1.1 differs from this optimal bound by roughly a factor of k.

The main step in the proof is a result of some independent interest.

Proposition 1.3. Let X be a smooth degree d hypersurface in ℙ^{n}. Let F_{k}(X) be the Fano variety of k-planes in X. There exists an irreducible component I of F_{k}(X) of the expected dimension if

n ≥ \binom{d + k − 1}{k} + k.

Moreover, if

n = \binom{d + k − 1}{k} + k − 1

then there is a nonempty open subset U_{k−1} ⊂ F_{k−1}(X) such that for every [Λ_{k−1}] ∈ U_{k−1}, there exists no k-plane in X containing Λ_{k−1}.

Theorem 1.1 implies a result about rational curves on every smooth hypersurface of sufficiently small degree. The Kontsevich moduli space \overline{M}_{0,r}(X, e) parametrizes isomorphism classes of data (C, q_{1}, …, q_{r}, f) of a proper, connected, at-worst-nodal, arithmetic genus 0 curve C, an ordered collection q_{1}, …, q_{r} of distinct smooth points of C and a morphism f : C → X satisfying a stability condition. The space \overline{M}_{0,r}(X, e) is projective. There is an evaluation map

ev : \overline{M}_{0,r}(X, e) → X^{r}

sending a datum (C, q_{1}, …, q_{r}, f) to the ordered collection (f(q_{1}), …, f(q_{r})).

Corollary 1.4. Let X be a smooth degree d hypersurface in ℙ^{n}. If

n ≥ \binom{d^{2} + d − 1}{d − 1} + d^{2} − 1

then for every integer e ≥ 2 there exists a canonically defined irreducible component M ⊂ \overline{M}_{0,2}(X, e) such that the evaluation morphism

ev : M → X × X

is dominant with rationally connected generic fiber, i.e., X satisfies a version of rational simple connectedness. Moreover X has a very twisting family of pointed lines, cf. [4, Def. 3.7].

This is proved in [4] assuming n satisfies a much weaker hypothesis

n ≥ d^{2}

but only for general hypersurfaces, not for every smooth hypersurface. The goal here is to find a stronger hypothesis on n that guarantees the theorem for every smooth hypersurface.

2. Flag Fano varieties

Naturally enough, the proof of Proposition 1.3 uses an induction on k. To set up the induction it is useful to consider not just k-planes in X, but flags of linear spaces

ℙ^{0} ⊂ ℙ^{1} ⊂ ℙ^{2} ⊂ ⋯ ⊂ ℙ^{k} ⊂ X.

The variety parametrizing such flags is the flag Fano variety of X. Also, although we are ultimately interested only in the case of a hypersurface in projective space, for the induction it is useful to allow a more general projective subvariety.

Let S be a scheme such that H^{0}(S, O_{S}) contains ℚ. Let E be a locally free O_{S}- module of rank n+1, and let X ⊂ ℙE be a closed subscheme such that the projection π : X → S is smooth and surjective of constant relative dimension dim(X/S). In other words, X is a family of smooth, dim(X/S)-dimensional subvarieties of ℙ^{n} parametrized by S.

Let 0 ≤ k ≤ n be an integer. Denote by Fl_{k}(E) the partial flag manifold representing the functor on S-schemes

T ↦ {(E_{1} ⊂ E_{2} ⊂ ⋯ ⊂ E_{k+1} ⊂ E_{T})|E_{i} locally free of rank i, i = 1, …, k + 1}.

For every 0 ≤ j ≤ k ≤ n, denote by ρ_{k}^{j} : Fl_{k}(E) → Fl_{j}(E) the obvious projection. The flag Fano variety is the locally closed subscheme Fl_{k}(X) ⊂ Fl_{k}(E) parametrizing flags such that ℙ(E_{k+1}) is contained in X. In particular, Fl_{0}(X) = X. Denote by ρ_{k}^{j} : Fl_{k}(X) → Fl_{j}(X) the restriction of ρ_{k}^{j}.

2.1. Smoothness. There are two elementary observations about the schemes Fl_{k}(X).

Lemma 2.1. [3, 1.1] There exists an open dense subset U ⊂ X such that U ×_{X} Fl_{1}(X) is smooth over U.

Lemma 2.2. Set S^{new} = U, the open subset from Lemma 2.1. Set E^{new} to be the universal rank n quotient bundle of π^{∗}E|_{U} so that ℙ(E^{new}) = U ×_{ℙ(E)} Fl_{1}(E) and set X^{new} = Fl_{1}(U). Then for every 0 ≤ k ≤ n − 1, Fl_{k}(X^{new}) = U ×_{X} Fl_{k+1}(X).

Proof. This is obvious. □

Proposition 2.3. There exists a sequence of open subschemes (U_{k} ⊂ Fl_{k}(X))_{0≤k≤n} satisfying the following conditions.

(i) The open subset U_{0} is dense in Fl_{0}(X), and for every 1 ≤ k ≤ n, U_{k} is dense in (ρ_{k}^{k−1})^{−1}(U_{k−1}).

(ii) For every 1 ≤ k ≤ n, ρ_{k}^{k−1} : (ρ_{k}^{k−1})^{−1}(U_{k−1}) → U_{k−1} is smooth.

Proof. Let U_{0} be the open subscheme from Lemma 2.1. By way of induction, assume k > 0 and the open subscheme U_{k−1} has been constructed. As in Lemma 2.2, replace S by U_{k−1}, replace E by the universal quotient bundle, and replace X by (ρ_{k}^{k−1})^{−1}(U_{k−1}). Now define U_{k} ⊂ (ρ_{k}^{k−1})^{−1}(U_{k−1}) to be the open subscheme from Lemma 2.1. □

2.2. Dimension. Using the Grothendieck-Riemann-Roch formula, it is possible to express the Chern classes of U ×_{X} F_{1}(X) in terms of the Chern classes of U. Iterating this leads, in particular, to a formula for the dimension of U_{k}. Denote by G_{1}, resp. G_{2}, the restriction to Fl_{1}(U) of E_{1}, resp. E_{2}. Denote by L the invertible sheaf

L := (G_{2}/G_{1})^{∨}.

Denote by

π : ℙG_{2} → Fl_{1}(U),
σ : Fl_{1}(U) = ℙG_{1} → ℙG_{2},

and

f : ℙG_{2} → X

the obvious morphisms. In other words, ℙG_{2} is a family of ℙ^{1}s over Fl_{1}(U), σ is a marked point on each ℙ^{1}, and f is an embedding of each ℙ^{1} as a line in X. The formula for the Chern character of the vertical tangent bundle of ρ_{1}^{0} is,

ch(T_{Fl_{1}(U)/U}) = π_{∗}f^{∗}[(ch(T_{X/S}) − dim(X/S))Todd(O_{ℙE}(1)|_{X})] − ch(L) − 1.

Given a flag ℙ = (ℙ^{1} ⊂ ℙ^{2} ⊂ ⋯ ⊂ ℙ^{k} ⊂ ℙ^{n}) in U_{k}, the formula for the fiber dimension of ρ_{k}^{k−1} at ℙ is

dim(U_{k}/U_{k−1}) = ∑_{m=1}^{k} b_{k,m}⟨ch_{m}(T_{X/S}), ℙ^{m}⟩ − k − 1

where ch_{m}(E) is the m^{th} graded piece of the Chern character of E, and where the coefficients b_{k,m} are the unique rational numbers such that

\binom{x + k − 1}{k} = ∑_{m=1}^{k} \frac{b_{k,m}}{m!} x^{m}.

Now define the numbers a_{k,m} to be

a_{k,m} = ∑_{l=m}^{k} b_{l,m},

in other words,

∑_{m=1}^{k} \frac{a_{k,m}}{m!} x^{m} = ∑_{l=1}^{k} \binom{x + l − 1}{l}.

Then it follows from the previous formula that the dimension of U_{k} at ℙ equals

dim(U_{k}) = ∑_{m=1}^{k} a_{k,m}⟨ch_{m}(T_{X/S}), ℙ^{m}⟩ + dim(X) − k^{2}.

In a related direction, there is a class of complex projective varieties that is stable under the operation of replacing X by a general fiber of Fl_{1}(X) → X. Call a subvariety X of ℙ^{n} a quasi-complete-intersection of type

\underline{d} = (d_{1}, …, d_{c})

if there is a sequence

X = X_{c} ⊂ X_{c−1} ⊂ ⋯ ⊂ X_{1} ⊂ X_{0} = ℙ^{n}

such that each X_{k} is a Cartier divisor in X_{k−1} in the linear equivalence class of O_{ℙ^{n}}(d_{k})|_{X_{k−1}}. If X is a quasi-complete-intersection, then every fiber of U ×_{X} Fl_{1}(X) → U is also a quasi-complete-intersection in ℙ^{n−1} of type

(1, 2, …, d_{1}, 1, 2, …, d_{2}, …, 1, 2, …, d_{c}).

Iterating this, every (non-empty) fiber of (ρ_{k}^{k−1})^{−1}(U_{k−1}) → U_{k−1} is a quasicomplete-intersection in ℙ^{n−k} of dimension

N_{k}(n, \underline{d}) = n − k − ∑_{i=1}^{c} \binom{d_{i} + k − 1}{k}.

Since the m^{th} graded piece of the Chern character of T_{X} equals

ch_{m}(T_{X}) = (n + 1 − ∑_{i=1}^{c} d_{i}^{m})c_{1}(O(1))^{m}/m!

this agrees with the previous formula for the fiber dimension.

Corollary 2.4. Let X be a smooth quasi-complete-intersection of type \underline{d}. If the integer N_{k}(n, \underline{d}) is nonnegative, there exists an irreducible component I of Fl_{k}(X) having the expected dimension

dim(I) = ∑_{m=0}^{k} N_{m}(n, \underline{d}).

Proof. Of course we define I to be the closure of any connected component of U_{k}. The issue is whether or not U_{k} is empty. By construction U_{k} is not empty if for every m = 1, …, k the morphism ρ_{m}^{m−1} is surjective. By the argument above every fiber of ρ_{m}^{m−1} is an iterated intersection in ℙ^{n−m} of pseudo-divisors (in the sense of [1, Def. 2.2.1]) in the linear equivalence class of an ample divisor. Thus the fiber is nonempty if the number of pseudo-divisors is ≤ n − m. This follows from the hypothesis that N_{k}(n, \underline{d}) ≥ 0. □

3. Proofs

Proof of Proposition 1.3. The first part follows from Corollary 2.4. For the second part, observe that if N_{k}(n, d) = −1, then N_{k−1}(n, d) is nonnegative. Therefore, by the first part, the open subset U_{k−1} from Proposition 2.3 is nonempty. Since (ρ_{k}^{k−1})^{−1}(U_{k−1}) → U_{k−1} is smooth of the expected dimension, and since the expected dimension is negative, (ρ_{k}^{k−1})^{−1}(U_{k−1}) is empty. In other words, for every [Λ_{k−1}] ∈ U_{k−1}, there exists no k-plane in X containing Λ_{k−1}. □

Proof of Theorem 1.1. Let (H_{k,n}, e) be the universal pair of a scheme H_{k,n} and a closed immersion of H_{k,n}-schemes

(pr_{H}, e) : H_{k,n} × ℙ^{k} → H_{k,n} × ℙ^{n}

whose restriction to each fiber {h} × ℙ^{k} is a linear embedding. In other words, H_{k,n} is the open subset of ℙHom(ℂ^{k+1}, ℂ^{n+1}) parametrizing injective matrices. Of course there is a natural action of PGL_{k+1} on H_{k,n}, and the quotient is the Grassmannian 𝔾(k, n). Denote by F̃_{k}(X) the inverse image of F_{k}(X) in H_{k,n}, i.e., F̃_{k}(X) parametrizes linear embeddings of ℙ^{k} into X.

Let F be a defining equation for the hypersurface X. Then e^{∗}F is a global section of e^{∗}O_{ℙ^{n}}(d). By definition, this is canonically isomorphic to pr_{ℙ^{k}}^{∗}O_{ℙ^{k}}(d). Therefore e^{∗}F determines a regular morphism

Φ̃ : H_{k,n} → H^{0}(ℙ^{k}, O_{ℙ^{k}}(d)).

Denote by V the open subset of H_{k,n} of points whose fiber dimension equals

dimH_{k,n} − dimH^{0}(ℙ^{k}, O_{ℙ^{k}}(d)).

The rational transformation Φ is dominant if and only if Φ̃ is dominant. And the morphism Φ̃ is dominant if and only if V is nonempty.

The scheme F̃_{k}(X) is the fiber Φ̃^{−1}(0). If

n ≥ \binom{d + k − 1}{k} + k

then Proposition 1.3 implies there exists an irreducible component I of F_{k}(X) of the expected dimension. Thus the inverse image Ĩ in H_{k,n} is an irreducible component of F̃_{k}(X) of the expected dimension, or what is equivalent, the expected codimension. But the expected codimension is precisely

h^{0}(ℙ^{k}, O_{ℙ^{k}}(d)) = \binom{d + k}{k}.

Thus, the generic point of Ĩ is contained in V, i.e., V is not empty.

This only leaves the case when

n = \binom{d + k − 1}{k} + k − 1.

The argument is very similar. Let y be a linear coordinate on ℙ^{k}, and let G̃_{k}(X) be the closed subscheme of H_{k,d} where e^{∗}F is a multiple of y^{d}. In other words, G̃_{k}(X) parametrizes linear embeddings of ℙ^{k} into ℙ^{n} whose intersection with X contains d𝕍(y). There is a projection morphism G̃_{k}(X) → F_{k−1}(X) associating to the linear embedding the (k − 1)-plane

Λ_{k−1} = Image(𝕍(y)).

Denote by G_{k}(X) the image of G̃_{k}(X) under the obvious morphism

G̃_{k}(X) → F_{k−1}(ℙ^{n}) × F_{k}(ℙ^{n}).

Recall that for a quasi-complete-intersection X, the fiber of F_{1}(X) → X is an interated intersection of ample pseudo-divisors in projective space. By a very similar argument, every fiber of G_{k}(X) → F_{k−1}(X) is an iterated intersection of ample pseudo-divisors in the projective space ℙ^{n}/Λ_{k−1} ≅ ℙ^{n−k}. Moreover, the fiber of Fl_{k}(X) → Fl_{k−1}(X) (for any extension of Λ_{k−1} to a flag in Fl_{k−1}(X)) is an ample pseudo-divisor in G_{k}(X). By the second part of Proposition 1.3, there exists a nonempty open subset U_{k−1} ⊂ Λ_{k−1} such that for every Λ_{k−1} ∈ U_{k−1} this ample pseudo-divisor is empty. Therefore the fiber in G_{k}(X) is finite or empty. But the equation

n = \binom{d + k − 1}{k} + k − 1

implies the expected dimension of the fiber is 0. Since an intersection of ample pseudo-divisors is nonempty if the expected dimension is nonnegative, the fiber of G_{k}(X) → F_{k−1}(X) is not empty and has the expected dimension 0. Since U_{k−1} has the expected dimension, the open set U_{k−1} ×_{F_{k−1}(X)} G̃_{k}(X) is nonempty and has the expected dimension. Thus it has the expected codimension. Therefore a generic point of this nonempty open set is in V, i.e., V is not empty. □

Proof of Corollary 1.4. Let M_{e} be an irreducible component of \overline{M}_{0,0}(X, e) not entirely contained in the boundary Δ. Then for every integer r ≥ 0 there exists a unique irreducible component M_{e,r} of \overline{M}_{0,r}(X, e) whose image in \overline{M}_{0,0}(X, e) equals M_{e}. Before defining the irreducible component M of \overline{M}_{0,2}(X, e), we will first inductively define an irreducible component M_{e} of \overline{M}_{0,0}(X, e) which is not entirely contained in the boundary Δ and such that the evaluation morphism

ev : M_{e,1} → X

is surjective. Then we define M to be M_{e,2}.

Let U denote the open subset of \overline{M}_{0,1}(X, 1) where the evaluation morphism

ev : \overline{M}_{0,1}(X, 1) → X

is smooth, i.e., U parametrizes free pointed lines. By [3, 1.1], U contains every general fiber of ev. By the argument in Subsection 2.2 (or any number of other references), a general fiber of ev is connected if d ≤ n − 2. Therefore U ×_{X} U is irreducible. There is an obvious morphism U ×_{X} U → \overline{M}_{0,0}(X, 2). By elementary deformation theory, the morphism is unramified and \overline{M}_{0,0}(X, 2) is smooth at every point of the image. Therefore there is a unique irreducible component M_{2} of \overline{M}_{0,0}(X, 2) containing the image of U ×_{X} U. Because U → X is dominant, M_{2} → X is also dominant.

By way of induction assume e ≥ 3 and M_{e−1} is given. Form the fiber product M_{e−1,1} ×_{X} U. As above this is irreducible, and there is an unramified morphism

M_{e−1,1} ×_{X} U → \overline{M}_{0,0}(X, e)

whose image is in the smooth locus. Therefore there exists a unique irreducible component M_{e} of \overline{M}_{0,0}(X, e) containing the image of M_{e−1,1} ×_{X} U. Because M_{e−1,1} → X is dominant, M_{e,1} → X is also dominant. This finishes the inductive construction of the irreducible components M_{e}, and thus also of M_{e,2}.

It remains to prove that

ev : M_{e,2} → X × X

is dominant with rationally connected generic fiber. The article [4] gives an inductive argument for proving this. To carry out the induction, one needs two results: the base of the induction and an important component of the induction argument. Set k to be d^{2}. For a general degree d hypersurface Y in ℙ^{k}, [4, Prop. 4.6, Prop. 10.1] prove the two results for Y. By Theorem 1.1, since

n ≥ \binom{d + k − 1}{k} + k − 1,

for a general ℙ^{k} ⊂ ℙ^{n} the intersection Y = ℙ^{k} ∩X is a general degree d hypersurface in ℙ^{k}. Thus the two results hold for Y. As is clear from the proofs of [4, Prop. 4.6, Prop. 10.1], the results for Y imply the corresponding results for X. □

References

[1] 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.

[2] J. Harris, B. Mazur, and R. Pandharipande. Hypersurfaces of low degree. Duke Math. J., 95(1):125–160, 1998.

[3] J. Kollár, Y. Miyaoka, and S. Mori. Rational connectedness and boundedness of Fano manifolds. J. Differential Geom., 36(3):765–779, 1992.

[4] J. Starr. Hypersurfaces of low degree are rationally simply-connected. preprint, 2004.

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

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