Fano Hypersurfaces in Positive Characteristic

We prove that a general Fano hypersurface in a projective space over an algebraically closed field of arbitrary characteristic is separably rationally connected.


Introduction
In this paper, we work with varieties over an algebraically closed field k of arbitrary characteristic.Definition 1.1 ( [Kol96] IV.3).Let X be a variety defined over k.
A variety X is rationally connected if there is a family of irreducible proper rational curves g : U → Y and an evaluation morphism u : U → X such that the morphism u (2) A variety X is separably rationally connected if there exists a proper rational curve f : P 1 → X such that the image lies in the smooth locus of X and the pullback of the tangent sheaf f * T X is ample.Such rational curves are called very free curves.
We refer to Kollár's book [Kol96] or the work of Kollár-Miyaoka-Mori [KMM92] for the background.If X is separably rationally connected, then X is rationally connected.The converse is true when the ground field is of characteristic zero by using the generic smoothness for the dominant map u (2) .In positive characteristics, the converse statement is open.
In characteristic zero, a very important class of rationally connected varieties are Fano varieties, i.e., smooth varieties with ample anticanonical bundles.In positive characteristic, we only know that they are rationally chain connected.Question 1.2 (Kollár).In arbitrary characteristic, are Fano varieties separably rationally connected?
The question is open even for Fano hypersurfaces in projective spaces.In this paper, we prove the following theorem.
Theorem 1.3.In arbitrary characteristic, a general Fano hypersurface of degree n in P n k contains a minimal very free rational curve of degree n, i.e., the pullback of the tangent bundle has the splitting type O(2) ⊕ O(1) ⊕(n−2) .Theorem 1.4.In arbitrary characteristic, a general Fano hypersurface in P n k is separably rationally connected.
de Jong-Starr [dJS03] proved that every family of separably rationally connected varieties over a curve admits a rational section.Thus using Theorem 1.4, we give another proof of Tsen's theorem.

YI ZHU
Corollary 1.5.Every family of Fano hypersurfaces in P n over a curve admits a rational section.
Acknowledgment.The author would like to thank his advisor Professor Jason Starr for helpful discussions.

Typical Curves and Deformation Theory
Let n be an integer ≥ 3. Let X be a hypersurface of degree n in P n .Let C be a smoothly embedded rational curve of degree e in X.We have the normal bundle short exact sequence.
By adjunction, the degree of T X| C is the degree of O P n (1)| C .Thus the degree of the normal bundle N C|X is e − 2 and the rank is n − 2.
Definition 2.1.Let e be a positive integer ≤ n.A smoothly embedded rational curve C of degree e in X is typical, if the normal bundle is the following: The curve C is a typical line if the degree of C is one.
Note that when e = n, typical rational curves of degree n are very free.
Lemma 2.2.Let L be a smoothly embedded line in a hypersurface X of degree n.Then L is typical if and only if both of the following conditions hold: (1) Proof.We may assume that Together with condition (2), a n−2 is either 0 or −1.When a n−2 = 0, N L|X is semipositive, contradicting with the fact that the degree of Because of the degree of the normal bundle, L is typical.
Lemma 2.3.Let C be a smoothly embedded rational curve of degree e in a hypersurface X of degree n, where 2 ≤ e ≤ n.Then C is typical if and only if both of the following conditions hold: Proof.Recall that the rank of the normal bundle N C|X is n − 2 and the degree is e − 2. We may assume that Condition ( 1) is equivalent to that a n−2 ≥ 0. Condition (2) implies that at most n − e of a i 's are 0. By degree count, C is a typical rational curve of degree e.Typical rational curves in the hypersurface X are deformation open as very free curves in the following sense.
Let H n be the Hilbert scheme of hypersurfaces of degree n in P n .It is isomorphic to some projective space.Let X → H n be the universal hypersurface.The morphism X → H n is flat projective and there exists a relative very ample invertible sheaf O X (1) on X .
Let R e,n be the Hilbert scheme parameterizing flat projective families of onedimensional subschemes in X with the Hilbert polynomial P (d) = ed+1.By [Kol96] Theorem 1.4, R e,n is projective over H n .
Let C be the universal families over R e,n , denoted by π : C → R e,n .We have the following diagram, where i is a closed immersion.
Proposition 2.4.Let e be a positive integer ≤ n.There exists an open subset in R e,n parameterizing typical curves of degree e in hypersurfaces of degree n.
Proof.Every typical curve of degree e in a hypersurface of degree n gives a point in R e,n .Any small deformation of a smoothly embedded rational curve is still a smoothly embedded rational curve.Thus the proposition follows by Lemma 2.2, Lemma 2.3 and the upper semicontinuity theorem [Har77] III.12.8.

Lemma 2.5.
There exists an open subset in R e,n such that for every closed point (C, X) in the open subset, C lies in the smooth locus of X.
Proof.Let S ⊂ X be the relative singular locus in the universal hypersurface.S is a closed subset of X .Since π is proper, the locus π(i −1 (R e,n × Hn S)) is a closed subset of R e,n parametrizing the point (C , X ) such that C intersects the singular locus of X .Thus the complement U is open in R e,n and satisfies the desired property.
Let L be a typical line in a hypersurface X of degree n in P n .By definition, N L|X ∼ = O ⊕(n−3) ⊕O(−1).We have a canonically defined trivial subbundle O ⊕(n−2) of N L|X .
Proposition 2.6.Let X be a hypersurface of degree n in P n .Let L and M be two typical lines in X intersecting transversally at only one point p.Assume that the following conditions hold: (1) the direction T p L is not in the trivial subbundle of N M |X ; (2) the direction T p M is not in the trivial subbundle of N L|X .
Then the pair (L ∪ M, X) ∈ R 2,n can be smoothed to a pair (C, X ) where C is a typical conic in X .Furthermore, there exists an open neighborhood of (L ∪ M, X) in which any smoothing of (L ∪ M, X) is a typical conic.
Proof.Let D be the union of the lines L and M .Since D is a local complete intersection and lies in the smooth locus of X, the normal bundle N D|X is locally free.We have the following short exact sequence.
By [GHS03] Lemma 2.6, the locally free sheaf N D|X | L is the sheaf of rational sections of N L|X which has at most one pole at the direction of By the same argument, condition (1) implies that the sheaf N D|X | M is isomorphic to O ⊕(n−2) .Now we have the following short exact sequence.
First we claim that D can be smoothed.Since h 1 (D, N D|X ) = 0, the pair (D, X) is unobstructed in R 2,n , cf. [Kol96] I.2.By [Sta09] Lemma 3.17, it suffices to show that the map Let q, r be two distinct points on L − {p}.By the long exact sequence associated to the above short exact sequence at h 1 , we get h 1 (D, N D|X (−q)) = 0 and Now for any smoothing (D t , X t ) of (D, X) over T , we can specify two distinct points p t and q t on D t which specialize to q and r on D. By Lemma 2.5, after shrinking T , the conic D t lies in the smooth locus of X t .Thus D t is smoothly embedded.By the upper semicontinuity theorem and Lemma 2.3, D t is a typical conic in X t .Definition 2.7.Let X be a hypersurface of degree n in P n .A typical comb with m teeth in X is a reduced curve in X with m+1 irreducible components C, L 1 , • • • , L m satisfying the following conditions: (1) C is a typical conic in X; (2) L 1 , • • • , L m are disjoint typical lines in X and each L i intersects C transversally at p i .
The conic C is called the handle of the comb and L i 's are called the teeth.
Proposition 2.8.Let X be a hypersurface of degree n in Assume that the following conditions hold: (1) the direction T pi C is not in the trivial subbundle of N Li|X ; (2) the directions Then the pair (D, X) ∈ R n,n can be smoothed to a pair (C , X ) where C is a very free curve in X .
Proof.The proof is very similar to the proof of Proposition 2.6.Here we only sketch the proof.Condition (1) implies that the sheaf for each i.We have the following short exact sequence.
Since H 1 (D, N D|X ) = 0, D is unobstructed.By diagram chasing, the map H 0 (D, N D|X ) → i T pi C ⊗ T pi L i is surjective.Thus we can smooth the typical comb D. Now we may choose a smoothing (D t , X t ) and specify two distinct points (q t , r t ) which specialize to two distinct points (q, r) on C − {p 1 , • • • , p n−2 }.By the long exact sequence, we know that h 1 (D, N D|X (−q − r)) = 0.By Lemma 2.5 and the upper semicontinuity theorem, a general smoothing of the pair (D, X) gives a very free curve in a general hypersurface.

An Example
In this section, we construct a hypersurface of degree n in P n , which contains a special configuration of lines.Later we will use this example to produce a very free curve in a general hypersurface.
Let n be an integer ≥ 4. Let [x 0 : • • • : x n ] be the homogeneous coordinates for P n .Let X be a hypersurface of degree n in the projective space P n defined by the following equation.Lemma 3.2.
(1) Both p and q lie in the smooth locus of X.
(2) The tangent space T p X is the hyperplane {x n = 0}, which is spanned by the lines L 1 , . . ., L n−1 .
(3) The tangent space of T q X is the hyperplane {x 2 = 0}.
Proof.We will prove the case for line L 1 .The remaining cases can be computed directly by the same method.Denote L 1 = {[x 0 : x 1 : 0 : • • • : 0] ∈ P n }.By restricting the partial derivatives of the defining equation of the hypersurface X on L 1 , we get the following. (3.1) For points on L 1 with x 0 = 0, we have ∂F ∂xn | L1 = 0.At the point q, ∂F ∂x2 | L1 = 0. Hence every point on the line L 1 is a smooth point of X.
Lemma 3.4.The line L n is in the smooth locus of X.
Proof.By restricting the partial derivatives of the defining equation of X on L n , we get the following. (3.2) For points on L n with x 1 = 0, we have ∂F ∂x2 | Ln = 0.For points on L n with x n = 0, we have ∂F ∂xn−1 | Ln = 0. Hence every point on the line L n is a smooth point of X. Proposition 3.5.With the notations as above, X satisfies the following properties.
Proof.Let L be a line in X.We have the following short exact sequences.0 degree n containing D t in P n t form a trivial projective bundle over (T, 0).Thus the family π : D → (T, 0) admits a lifting to a flat family of pairs π : (D, X T ) → (T, 0) in R n,n such that the special fiber (D 0 , X 0 ) is (C, X) which is constructed in Section 3.All the following steps of the proof requires to shrink T if necessary.By Proposition 2.6 and Corollary 3.6, we may assume that the handle D t is a typical conic in X t for every t ∈ T − {0}.By Proposition 2.4 and Corollary 3.7 (1), all the teeth of the comb D t are typical.Thus for every t ∈ T − {0}, we get a typical comb D t as in Definition 2.7.Now the theorem follows if we verify the two conditions in Proposition 2.3.Since they are open conditions, it suffices to check on the special fiber (C, X), which is proved in Corollary 3.7.

Notation 3. 1 .
Let p be the point [1 : 0 : • • • : 0] and q be the point [0 : 1 : 0 : • • • : 0].Let L i be the line spanned by {e 0 , e i } for i = 1, . . ., n−1 and L n be the line spanned by {e 1 , e n }.It is easy to check that they all lie in the hypersurface X.Let C be the union of L 1 , • • • , L n .The following picture shows the configuration of the points and the lines in the projective space.
m m m m m m m m m m m m m m (T, 0) Proof of Theorem 1.4.By[Kol96] IV.3.11 and Lemma 2.5, it suffices to produce one very free curve in a hypersurface of degree d.Let Y be a general smooth Fano hypersurface of degree d in P n .When d = n, this is proved in Theorem 1.3.When d < n, we may choose a general linear subspace L of dimension d such that Y ∩ L is smooth and contains a very free curve f : P 1 → Y ∩ L by Theorem 1.3.By the normal bundle exact sequence,0 − −−− → T (Y ∩ L) − −−− → T Y − −−− → N Y ∩L|Y − −−− → 0 the sheaf f * T (Y ∩ L) is positive and the sheaf N Y ∩L|Y is isomorphic to N L|P n , which is O(1) ⊕(n−d) .Therefore the pullback bundle f * T Y is positive.Thus f : P 1 → Y ∩ L → Y is a very free curve in Y .