# Comptes Rendus
Mathématique

Yi Zhu

Fano hypersurfaces in positive characteristic

Volume 362 (2024), p. 107-115

Online since: 2 February 2024

https://doi.org/10.5802/crmath.438

<figure><img src="image_1.png" /></figure>

This article is licensed under the
CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
http://creativecommons.org/licenses/by/4.0/
Research article / *Article de recherche*
Algebraic geometry / *Géométrie algébrique*

# Fano hypersurfaces in positive characteristic

Yi Zhu<sup>a</sup>

<sup>a</sup> United States

E-mail: math.zhu@gmail.com

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

Manuscript received 22 February 2022, revised 15 November 2022, accepted 16 October 2022.

## 1. Introduction

In this paper, we work with varieties over an algebraically closed field $k$ of arbitrary characteristic.

**Definition 1 ([5, IV.3]).** Let $X$ be a smooth variety defined over $k$.

A variety $X$ is rationally connected if there is a family of irreducible proper rational curves $g: U \to Y$ and a morphism $u: U \to X$ such that the morphism $u^{(2)}: U \times_Y U \to X \times_X X$ is dominant.

A variety $X$ is separably rationally connected if there exists a proper rational curve $f: \mathbb{P}^1 \to X$ such that the image lies in the smooth locus of $X$ and the pullback of the tangent sheaf $f^*TX$ is ample. Such rational curves are called very free curves.

We refer to Kollár's book [5] or the work of Kollár-Miyaoka-Mori [6] 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 generic smoothness. In positive characteristic, 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 2 (Kollár).** In arbitrary characteristic, is every smooth Fano variety separably rationally connected?

The question is open even for Fano hypersurfaces in projective spaces. In this paper, we prove the following theorem.

**Theorem 3.** In arbitrary characteristic, a general Fano hypersurface of degree $n$ in $\mathbb{P}_k^n$ contains a minimal very free rational curve of degree $n$, i.e., the pullback of the tangent bundle has the splitting type $\mathcal{O}(2) \oplus \mathcal{O}(1)^{\oplus(n-2)}$.

**Theorem 4.** In arbitrary characteristic, a general Fano hypersurface in $\mathbb{P}_k^n$ is separably rationally connected.
de Jong and Starr [4] proved that every family of separably rationally connected varieties over a curve admits a rational section. Thus using Theorem 4, we give another proof of Tsen's theorem.

**Corollary 5.** Every family of Fano hypersurfaces in $\mathbb{P}^n$ over a curve admits a rational section.

### Acknowledgment

The author would like to thank his advisor Professor Jason Starr for helpful discussions.

## 2. Typical Curves and Deformation Theory

**Notation 6.** Let $n$ be an integer $\ge 3$. Let $X$ be a hypersurface of degree $n$ in $\mathbb{P}^n$. Let $C$ be a smooth rational curve of degree $e$ contained in the smooth locus of $X$. Consider the normal bundle exact sequence.

$$0 \longrightarrow TC \longrightarrow TX|_C \longrightarrow \mathcal{N}_{C|X} \longrightarrow 0$$

By adjunction, the degree of $TX|_C$ is the degree of $\mathcal{O}_{\mathbb{P}^n}(1)|_C$. Thus the degree of the normal bundle $\mathcal{N}_{C|X}$ is $e-2$ and the rank is $n-2$.

**Definition 7.** Let $e$ be a positive integer $\le n$. A smooth rational curve $C$ of degree $e$ contained in the smooth locus of $X$ is typical, if its normal bundle is the following:

$$\mathcal{N}_{C|X} \cong \begin{cases} \mathcal{O}_C^{\oplus(n-3)} \oplus \mathcal{O}_C(-1), & \text{if } e=1, \\ \mathcal{O}_C^{\oplus(n-e)} \oplus \mathcal{O}_C(1)^{\oplus(e-2)}, & \text{if } e \ge 2. \end{cases}$$

The curve $C$ is a typical line, resp., typical conic if moreover the degree of $C$ is one, resp., two.

**Remark 8.**

(1) For a typical line $L$ on $X$, there is a canonically defined trivial subbundle $\mathcal{O}_L^{\oplus(n-2)}$ in $\mathcal{N}_{L|X}$.

(2) When $e=n$, typical rational curves of degree $n$ are very free.

**Lemma 9.** Let $C$ be a smooth rational curve of degree $e$ on the smooth locus of a hypersurface $X$ of degree $n$, where $2 \le e \le n$. Then $C$ is typical if and only if both of the following conditions hold:

(1) $h^1(C, \mathcal{N}_{C|X}(-1)) = 0,$

(2) $h^1(C, \mathcal{N}_{C|X}(-2)) \le n-e.$

**Proof.** Recall that the rank of the normal bundle $\mathcal{N}_{C|X}$ is $n-2$ and the degree is $e-2$. Assume that $\mathcal{N}_{C|X}$ has the splitting type $\mathcal{O}_C(a_1) \oplus \cdots \oplus \mathcal{O}_C(a_{n-2})$, where $a_1 \ge \cdots \ge a_{n-2}$. Condition (1) is equivalent to that $a_{n-2} \ge 0$. Condition (2) implies that at most $n-e$ of the $a_i$'s are 0. By degree count, $C$ is a typical rational curve of degree $e$. $\square$

<figure><img src="image_2.png" /></figure>

Similarly, we have the following cohomological criterion for typical lines.

**Lemma 10.** Let $L$ be a smooth line on the smooth locus of $X$. Then $L$ is typical if and only if both of the following conditions hold:

(1) $h^1(C, \mathcal{N}_{L|X}) = 0,$

(2) $h^1(C, \mathcal{N}_{L|X}(-1)) \le 1.$

Let $H_n$ be the Hilbert scheme of hypersurfaces of degree $n$ in $\mathbb{P}^n$. It is isomorphic to a projective space. Let $\mathcal{X} \to H_n$ be the universal hypersurface. The morphism $\mathcal{X} \to H_n$ is flat projective and there exists a relative very ample invertible sheaf $\mathcal{O}_{\mathcal{X}}(1)$ on $\mathcal{X}$.

Let $R_{e,n}$ be the Hilbert scheme parameterizing flat projective families of one-dimensional subschemes in $\mathcal{X}$ with the Hilbert polynomial $P(d) = ed+1$. By [5, Theorem 1.4], $R_{e,n}$ is projective over $H_n$.
Let $\mathcal{C}$ be the universal family over $R_{e,n}$, denoted by $\pi : \mathcal{C} \to R_{e,n}$. We have the following diagram,

<figure><img src="image_3.png" /></figure>

where $i$ is a closed immersion.

Typical rational curves on hypersurfaces are deformation open in the following sense.

**Proposition 11.** Let $e$ be a positive integer $\le n$. There exists an open subset in $R_{e,n}$ parameterizing typical curves of degree $e$ in hypersurfaces of degree $n$.

**Proof.** By Lemma 10 and Lemma 9, every typical curve $C$ on a hypersurface $X$ gives an unobstructed point in $R_{e,n}$. By the upper semicontinuity theorem [3, III.12.8], a small deformation of $(X_t, C_t)$ is still typical and $C_t$ is contained in the smooth locus of $X_t$. $\square$

**Proposition 12.** Let $X$ be a hypersurface of degree $n$ in $\mathbb{P}^n$. Let $L$ and $M$ be two typical lines on $X$ intersecting transversally at one point $p$. Assume that the following conditions hold:

(1) the direction $T_pL$ lies outside the trivial subbundle of $\mathcal{N}_{M|X}$;

(2) the direction $T_pM$ lies outside the trivial subbundle of $\mathcal{N}_{L|X}$.

Then the pair $(X, D = L \cup M) \in R_{2,n}$ can be smoothed to a pair $(X', C)$ where $C$ is a typical conic in $X'$. Furthermore, there exists an open neighborhood of $(X, D = L \cup M)$ in which any smoothing of $(X, D = L \cup M)$ is a typical conic.

**Proof.** Let $D$ be the union of the lines $L$ and $M$. Since $D$ is a local complete intersection in the smooth locus of $X$, the normal bundle $\mathcal{N}_{D|X}$ is locally free. We have the following short exact sequence.

$$0 \longrightarrow \mathcal{N}_{L|X} \longrightarrow \mathcal{N}_{D|X|L} \longrightarrow T_pM \longrightarrow 0$$

By [2, Lemma 2.6], the locally free sheaf $\mathcal{N}_{D|X|L}$ is the sheaf of rational sections of $\mathcal{N}_{L|X}$ which has at most one pole at the direction of $T_pM$. Since $\mathcal{N}_{L|X} \cong \mathcal{O}_L^{\oplus(n-3)} \oplus \mathcal{O}_L(-1)$, condition (2) implies that $\mathcal{N}_{D|X|L}$ is isomorphic to $\mathcal{O}_L^{\oplus(n-2)}$.

By the same argument, condition (1) implies that the sheaf $\mathcal{N}_{D|X|M}$ is isomorphic to $\mathcal{O}_M^{\oplus(n-2)}$. Now we have the following short exact sequence.

$$0 \longrightarrow \mathcal{N}_{D|X|M}(-p) \longrightarrow \mathcal{N}_{D|X} \longrightarrow \mathcal{N}_{D|X|L} \longrightarrow 0$$ $$\mathcal{O}_M(-1)^{\oplus(n-2)} \qquad \mathcal{O}_L^{\oplus(n-2)}$$

First we claim that there is a smoothing of $D$. Since $h^1(D, \mathcal{N}_{D|X}) = 0$, the pair $(X, D)$ is unobstructed in $R_{2,n}$, cf. [5, I.2]. By [7, Lemma 3.17], it suffices to show that the map

$$H^0(D, \mathcal{N}_{D|X}) \to H^0(L, \mathcal{N}_{D|X|L}) \to T_pM$$

is surjective. Since $H^1(M, \mathcal{N}_{D|X|M}(-p)) = 0$, the first map is surjective. Since $H^1(L, \mathcal{N}_{D|X|L}) = 0$, the second map is also surjective.

Let $q, r$ be two distinct points on $L - \{p\}$. We have

$$h^1(D, \mathcal{N}_{D|X}(-q)) = 0 \quad \text{and} \quad h^1(D, \mathcal{N}_{D|X}(-q-r)) = n-2.$$

Now for any smoothing $(X_t, D_t)$ of $(X, D)$ over a curve $T$, we can specify two distinct points $q_t$ and $r_t$ on $D_t$ which specialize to $q$ and $r$ on $D$. After shrinking $T$, we may assume that the conic
$D_t$ is contained in the smooth locus of $X_t$. By the upper semicontinuity theorem and Lemma 9, $D_t$ is a typical conic on $X_t$. $\square$

**Definition 13.** Let $X$ be a hypersurface of degree $n$ in $\mathbb{P}^n$. A typical comb with $m$ teeth on $X$ is a reduced curve in $X$ with $m + 1$ irreducible components $C, L_1, \dots, L_m$ satisfying the following conditions:

(1) $C$ is a typical conic on $X$;

(2) $L_1, \dots, L_m$ are disjoint typical lines on $X$ 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 14.** Let $X$ be a hypersurface of degree $n$ in $\mathbb{P}^n$. Let $D = C \cup L_1 \cup \dots \cup L_{n-2}$ be a typical comb with $n-2$ teeth on $X$. Let $p_i$ be the intersection point $L_i \cap C$. Assume that the following conditions hold:

(1) the direction $T_{p_i}C$ lies outside the trivial subbundle of $\mathcal{N}_{L_i|X}$;

(2) the directions $T_{p_i}L_i$ are general in $\mathcal{N}_{C|X}$ so that the sheaf $\mathcal{N}_{D|X|C}$ is isomorphic to $\mathcal{O}_C(1)^{\oplus(n-2)}$.

Then the pair $(X, D) \in \mathbb{R}_{n,n}$ can be smoothed to a pair $(X', C')$ where $C'$ is a very free rational curve on $X'$.

**Proof.** The proof is very similar to the proof of Proposition 12. Here we only sketch the proof. Condition (1) implies that the sheaf $\mathcal{N}_{D|X|L_i}$ is isomorphic to $\mathcal{O}_{L_i}^{\oplus(n-2)}$ for each $i$. We have the following short exact sequence.

$$ \begin{aligned} 0 & \longrightarrow \bigoplus_i \mathcal{N}_{D|X|L_i}(-p) & \longrightarrow \mathcal{N}_{D|X} & \longrightarrow \mathcal{N}_{D|X|C} & \longrightarrow 0 \\ & \Vert & & \Vert & \\ \mathcal{O}_{L_i}(-1)^{\oplus(n-2)} & & \mathcal{O}_C(1)^{\oplus(n-2)} & & \end{aligned} $$

Since $H^1(D, \mathcal{N}_{D|X}) = 0$, $D$ is unobstructed. By diagram chasing, the map $H^0(D, \mathcal{N}_{D|X}) \to \bigoplus_i T_{p_i}L_i$ is surjective. Thus we can smooth the typical comb $D$.

Now we may choose a smoothing $(X_t, D_t)$ and specify two distinct points $(q_t, r_t)$ which specialize to two distinct points $(q, r)$ on $C - \{p_1, \dots, p_{n-2}\}$. By the long exact sequence in cohomology associated to the above exact sequence, we know that $h^1(D, \mathcal{N}_{D|X}(-q-r)) = 0$. By the upper semicontinuity theorem, a general smoothing of the pair $(X, D)$ gives a very free curve in a general hypersurface. $\square$

### 3. An Example

In this section, we construct a hypersurface of degree $n$ in $\mathbb{P}^n$, which contains a special configuration of typical lines. Later we will use this example to produce a very free curve in a general hypersurface.
**Notation 15.** Let $n$ be an integer $\ge 4$. Let $[x_0 : \cdots : x_n]$ be homogeneous coordinates for $\mathbb{P}^n$. Let $X$ be a hypersurface of degree $n$ in the projective space $\mathbb{P}^n$ defined by the following equation $F(x_0, \dots, x_n)$. Let $e_i$ be the point in $\mathbb{P}^n$ represented by the $i$-th unit vector in $k^{n+1}$.

$$
\begin{align*}
x_0^{n-1} x_n &+ x_1^{n-3} x_n^2 x_0 \\
&+ x_1^{n-4} x_n^3 x_3 \\
&\vdots \\
&+ x_1 x_n^{n-2} x_{n-2} \\
&+ x_n^{n-1} x_{n-1}
\end{align*}
\begin{align*}
&+(x_1^{n-1} + x_0 x_1^{n-2} + \cdots + x_0^{n-3} x_1^2) x_2 \\
&+(x_0 x_1^{n-2} + \cdots + x_0^{n-3} x_1^2) x_3 \\
&\vdots \\
&+(x_0^{n-4} x_2^3 + x_0^{n-3} x_2^2) x_{n-2} \\
&+ x_0^{n-3} x_1^2 x_{n-1}
\end{align*}
\begin{align*}
&+(x_2^{n-1} + x_0 x_2^{n-2} + \cdots + x_0^{n-3} x_2^2) x_3 \\
&+(x_0 x_2^{n-2} + \cdots + x_0^{n-3} x_2^2) x_4 \\
&\vdots \\
&+(x_0^{n-4} x_2^3 + x_0^{n-3} x_2^2) x_{n-1} \\
&+ x_0^{n-3} x_2^2 x_1
\end{align*}
\begin{align*}
&+ \cdots \\
&+ \cdots \\
&\vdots \\
&+ \cdots \\
&+ \cdots
\end{align*}
\begin{align*}
&+(x_{n-1}^{n-1} + x_0 x_{n-1}^{n-2} + \cdots + x_0^{n-3} x_{n-1}^2) x_1 \\
&+(x_0 x_{n-1}^{n-2} + \cdots + x_0^{n-3} x_{n-1}^2) x_2 \\
&\vdots \\
&+(x_0^{n-4} x_{n-1}^3 + x_0^{n-3} x_{n-1}^2) x_{n-3} \\
&+ x_0^{n-3} x_{n-1}^2 x_{n-2}
\end{align*}
$$

Let $p$ be the point $[1:0:\cdots:0]$ and $q$ be the point $[0:1:0:\cdots:0]$. Let $L_i$ be the line spanned by $\{e_0, e_i\}$ for $i=1, \dots, n-1$ 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 $L_1, \dots, L_n$. The following picture shows the configuration of the points and the lines in the projective space.

<figure><img src="image_4.png" /></figure>

**Lemma 16.**

(1) Both $p$ and $q$ lie in the smooth locus of $X$.

(2) The tangent space $T_pX$ is the hyperplane $\{x_n = 0\}$, which is spanned by the lines $L_1, \dots, L_{n-1}$.

(3) The tangent space of $T_qX$ is the hyperplane $\{x_2 = 0\}$.

**Proof.** By taking the partial derivatives of $F$, we have $\frac{\partial F}{\partial x_i}(p) = 0$ for $i = 0, \dots, n-1$ and $\frac{\partial F}{\partial x_n}(p) = 1$. Similarly, we have $\frac{\partial F}{\partial x_i}(q) = 0$ for $i \ne 2$ and $\frac{\partial F}{\partial x_2}(q) = 1$. $\square$

**Lemma 17.** The lines $L_1, \dots, L_{n-1}$ are in the smooth locus of $X$.
**Proof.** We will prove the case for line $L_1$. The remaining cases can be computed directly by the same method. Denote $L_1 = \{[x_0 : x_1 : 0 : \cdots : 0] \in \mathbb{P}^n\}$. By restricting the partial derivatives of the defining equation of the hypersurface X on $L_1$, we get the following.

$$
\begin{align*}
\left. \frac{\partial F}{\partial x_2} \right|_{L_1} &= x_1^{n-1} + x_0 x_1^{n-2} + \cdots + x_0^{n-3} x_1^2 \\
\left. \frac{\partial F}{\partial x_3} \right|_{L_1} &= x_0 x_1^{n-2} + \cdots + x_0^{n-3} x_1^2 \\
&\vdots \\
\left. \frac{\partial F}{\partial x_{n-2}} \right|_{L_1} &= x_0^{n-4} x_1^3 + x_0^{n-3} x_1^2 \\
\left. \frac{\partial F}{\partial x_{n-1}} \right|_{L_1} &= x_0^{n-3} x_1^2 \\
\left. \frac{\partial F}{\partial x_n} \right|_{L_1} &= x_0^{n-1}
\end{align*}
\tag{1}
$$

For points on $L_1$ with $x_0 \neq 0$, we have $\frac{\partial F}{\partial x_n}|_{L_1} \neq 0$. At the point $q$, $\frac{\partial F}{\partial x_2}|_{L_1} \neq 0$. Hence every point on the line $L_1$ is a smooth point of X. $\square$

**Lemma 18.** 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.

$$
\begin{align*}
\left. \frac{\partial F}{\partial x_0} \right|_{L_n} &= x_1^{n-3} x_n^2 \\
\left. \frac{\partial F}{\partial x_3} \right|_{L_n} &= x_1^{n-4} x_n^3 \\
&\vdots \\
\left. \frac{\partial F}{\partial x_{n-2}} \right|_{L_n} &= x_1 x_n^{n-2} \\
\left. \frac{\partial F}{\partial x_{n-1}} \right|_{L_n} &= x_n^{n-1} \\
\left. \frac{\partial F}{\partial x_2} \right|_{L_n} &= x_1^{n-1}
\end{align*}
\tag{2}
$$

For points on $L_n$ with $x_1 \neq 0$, we have $\frac{\partial F}{\partial x_2}|_{L_n} \neq 0$. For points on $L_n$ with $x_n \neq 0$, we have $\frac{\partial F}{\partial x_{n-1}}|_{L_n} \neq 0$. Hence every point on the line $L_n$ is a smooth point of X. $\square$

**Proposition 19.** With the setup as in Notation 15, X satisfies the following properties.

(1) The lines $L_1, \dots, L_n$ are typical in X.

(2) For $i = 1, \dots, n-1$, the trivial subbundle of the normal bundle $\mathcal{N}_{L_i|X}$ at $p$ is generated by $\partial_{\overline{i+1}} - \partial_{\overline{i+2}}, \dots, \partial_{\overline{i+n-3}} - \partial_{\overline{i+n-2}}$, where the notation $\overline{j}$ is $j$ if $j$ is less than $n$ and $j - (n-1)$ if otherwise.

(3) The trivial subbundle of the normal bundle $\mathcal{N}_{L_1|X}$ at $q$ is generated by $\partial_3, \dots, \partial_{n-1}$.

(4) The trivial subbundle of the normal bundle $\mathcal{N}_{L_n|X}$ at $q$ is generated by $\partial_3, \dots, \partial_{n-1}$.
**Proof.** Let $L$ be a line in $X$. We have the following short exact sequences.

<figure><img src="image_5.png" /></figure>

The associated long exact sequence is the following.

$$H^0(L, \mathcal{N}_{L|X}(-1)) \to k^n \xrightarrow{\alpha} H^0(L, \mathcal{O}_L(n-1)) \to H^1(L, \mathcal{N}_{L|X}(-1)) \to 0$$

where the map $\alpha$ sends the natural basis of $k^n$ to the derivatives of $F$ with respect to the normal directions of $L$ in $\mathbb{P}^n$. By Lemma 10, $L$ is typical if and only if the image of $\alpha$ is of codimension one in $H^0(L, \mathcal{O}_L(n-1))$.

When $L = L_1$, by (1), $\frac{\partial F}{\partial x_2}|_{L_1}, \dots, \frac{\partial F}{\partial x_n}|_{L_1}$ form a codimensional-one subspace of $H^0(L_1, \mathcal{O}_{L_1}(n-1))$. Thus we get that $H^1(L_1, \mathcal{N}_{L_1|X}(-1))$ is one dimensional, i.e., $L_1$ is typical in $X$.

By the short exact sequence above, $N_{L_1|X}(-1)$ is a subbundle of the trivial bundle $\mathcal{O}_{L_1}^{\oplus(n-1)}$ which maps to 0 in $\mathcal{O}_{L_1}(n-1)$. Let $\partial_2, \dots, \partial_n$ be the generators of $\mathcal{O}_{L_1}^{\oplus(n-1)}$. We get $N_{L_1|X}(-1)$ is generated by $x_0(\partial_2 - \partial_3) - x_1(\partial_3 - \partial_4), \dots, x_0(\partial_{n-2} - \partial_{n-1}) - x_1\partial_{n-1}, x_0^2\partial_{n-1} - x_1^2\partial_n$ as an $\mathcal{O}_{L_1}$-module. If we restrict the bundle at $p$ and $q$, we get properties (2) and (3) for $L_1$.

When $L = L_2, \dots, L_{n-1}$, we can prove properties (2) and (3) in a similar way. When $L = L_n$, (4) follows from the same computation as above by applying (2). $\square$

With the description of the trivial subbundles of the normal bundles of lines in $X$ as above, we get the following corollaries.

**Corollary 20.** With the setup as in Notation 15, we have the following statements.

(1) The lines $L_1$ and $L_n$ are typical in $X$.

(2) The direction $T_q L_1$ lies outside the trivial subbundle of $\mathcal{N}_{L_n|X}$.

(3) The direction $T_q L_n$ lies outside in the trivial subbundle of $\mathcal{N}_{L_1|X}$.

**Corollary 21.** With the setup as in Notation 15, we have the following statements.

(1) The lines $L_2, \dots, L_{n-2}$ are typical in $X$.

(2) The direction $T_p L_1$ lies outside the trivial subbundle of $\mathcal{N}_{L_i|X}$ for $2 \le i \le n-1$.

(3) The directions $T_p L_2, \dots, T_p L_{n-1}$ span the normal bundle $\mathcal{N}_{L_1|X}$ at $p$.

# 4. Proof of the Main Theorem

**Lemma 22.** Let $C$ be the union of $n$ lines $L_1, \dots, L_n$ in $\mathbb{P}^n$ as in Notation 15. The following properties hold for $C$ for every positive integer $d$:

(1) $h^0(C, \mathcal{O}_C(d)) = nd + 1$ and $h^1(C, \mathcal{O}_C(d)) = 0$.

(2) $h^1(C, \mathcal{I}_C(d)) = 0$.

(3) $h^0(C, \mathcal{I}_C(d)) = h^0(\mathbb{P}^n, \mathcal{O}(d)) - nd - 1$.

**Remark 23.** The curve $C$ is an example of curves with rational $n$-fold point, cf. [1, 3.7]. The following lemma is an analogue of [1, Lemma 3.8].

**Proof.** This can be computed directly. For any $d > 0$, when $i = 1, \dots, n-1$, the homogeneous polynomials of degree $d$ that do not vanish on $L_i$ are generated by $\{x_0^d, x_0^{d-1}x_i, \dots, x_i^d\}$. The homogeneous polynomials of degree $d$ that do not vanish on $L_n$ are generated by $\{x_1^d, x_1^{d-1}x_i, \dots, x_n^d\}$. Since every global section of $\mathcal{O}_C(d)$ is obtained by gluing global sections on each component, which imposes exactly $n-1$ linear conditions, we have

$$h^0(\mathcal{O}_C(d)) = n(d+1) - (n-1) = nd + 1$$
and $h^0(\mathcal{O}_{\mathbb{P}^n}(d)) \to h^0(\mathcal{O}_C(d))$ is surjective for any $d$. In particular, the arithmetic genus of $C$ is zero. Condition (1) is proved. The rest of the lemma follows by considering the long exact sequence in cohomology

$$0 \longrightarrow h^0(C, \mathcal{I}_C(d)) \longrightarrow h^0(\mathcal{O}_{\mathbb{P}^n}(d)) \longrightarrow h^0(\mathcal{O}_C(d)) \longrightarrow h^1(C, \mathcal{I}_C(d)). \quad \square$$

**Construction 24.** Let $C$ be the union of $n$ lines $L_1, \dots, L_n$ in $\mathbb{P}^n$ as in Notation 15. If we consider $L_1 \cup L_n$ as a conic in $\mathbb{P}^n$, there exists a smooth affine pointed curve $(T, 0)$ and a smoothing $D' \to (T, 0)$ satisfying the following conditions:

(1) The special fiber $D'_0$ is $L_1 \cup L_n$;

(2) For any $t \in T - \{0\}$, $D'_t$ is a smooth conic contained in the plane spanned by $L_1$ and $L_n$.

We may assume that there exists $n-2$ sections $s_i : (T, 0) \to D'$ for $i = 1, \dots, n-2$ such that $s_i(0) = p$ for all $i$'s and for $t \in T - \{0\}$, $s_i(t)$'s are all distinct on $D'_t$.

For any $s_i(t)$, there exists a unique line $L_{i+1}(t)$ through $s_i(t)$ parallel to $L_{i+1}$. After gluing the families of lines $L_{i+1}(t)$ on $D'_t$ at $s_i(t)$ for all $i$'s, we get a family of reducible curves $\pi : D \to (T, 0)$ satisfying the following conditions:

(1) The special fiber $D_0$ is $C$ constructed as in Notation 15.

(2) For any $t \in T - \{0\}$, $D_t$ is a comb with the handle $D'_t$ and with the teeth lines.

We have the following diagram.

<figure><img src="image_6.png" /></figure>

**Lemma 25.** The family $\pi : D \to (T, 0)$ is flat. Furthermore, $\pi_*\mathcal{I}_D(d)$ is locally free on $T$ for any $d > 0$, where $\mathcal{I}_D$ is the ideal sheaf of $D$ in $\mathbb{P}_T^n$.

**Proof.** The same computational argument as in the proof of Lemma 22 proves that $h^0(\mathbb{P}_T^n, I_{D_t}(d))$ and $h^1(\mathbb{P}_T^n, I_{D_t}(d))$ are constant for any $t \in T - \{0\}$. Thus the Hilbert polynomial is constant. Hence the family is flat over $T$. The remaining part of the lemma follows from the cohomology and base change theorem [3, III.12.9]. $\square$

**Proof of Theorem 3.** The theorem is known for $n=2,3$. We can assume that $n \ge 4$. By [5, IV.3.11], it suffices to produce one very free curve on a hypersurface of degree $n$. By Lemma 25, after shrinking $T$, hypersurfaces of degree $n$ containing $D_t$ in $\mathbb{P}_T^n$ form a trivial projective bundle over $(T, 0)$. Thus the family $\pi : D \to (T, 0)$ admits a lifting to a flat family of pairs $\pi : (\mathcal{X}_T, D) \to (T, 0)$ in $R_{n,n}$ such that the special fiber $(\mathcal{X}_0, D_0)$ is $(X, C)$ which is constructed in Section 3.

<figure><img src="image_7.png" /></figure>

All the following steps of the proof requires to shrink $T$ if necessary. By Proposition 12 and Corollary 20, we may assume that the handle $D'_t$ is a typical conic in $\mathcal{X}_t$ for every $t \in T - \{0\}$. By Proposition 11 and Corollary 21 (1), all the teeth of the comb $D_t$ are typical. Thus for every $t \in T - \{0\}$, we get a typical comb $D_t$ as in Definition 13. Now the theorem follows if we verify the two conditions in Proposition 9. Since they are open conditions, it suffices to check on the special fiber $(X, C)$, which is proved in Corollary 21. $\square$
**Proof of Theorem 4.** For a general Fano hypersurfaces of degree $d$ in $\mathbb{P}^n$, when $d = n$, this is proved in Theorem 3. When $d < n$, we may choose a general Fano hypersurface $Y$ of degree $d$ in $\mathbb{P}^d$ admitting a very free curve $f: \mathbb{P}^1 \to Y$. Construct the cone $X$ of $Y$ in $\mathbb{P}^n$. Note that $Y$ is the intersection of a projective subspace $L$ of dimension $d$ and $X$. By the normal bundle exact sequence,

$$0 \longrightarrow TY \longrightarrow TX \longrightarrow \mathcal{N}_{Y|X} \longrightarrow 0$$

the sheaf $f^*TY$ is positive and the sheaf $\mathcal{N}_{Y|X}$ is isomorphic to $\mathcal{N}_{L|\mathbb{P}^n}$, which is positive too. Therefore the pullback bundle $f^*TX$ is positive. $\square$

## References

[1] D. Chen, I. Coskun, "Towards Mori's program for the moduli space of stable maps", Am. J. Math. 133 (2011), no. 5, p. 1389-1419.

[2] T. Graber, J. Harris, J. M. Starr, "Families of rationally connected varieties", J. Am. Math. Soc. 16 (2003), no. 1, p. 57-67.

[3] R. Hartshorne, *Algebraic geometry*, Graduate Texts in Mathematics, vol. 52, Springer, 1977.

[4] A. J. de Jong, J. M. Starr, "Every rationally connected variety over the function field of a curve has a rational point", Am. J. Math. 125 (2003), no. 3, p. 567-580.

[5] J. Kollár, *Rational curves on algebraic varieties*, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 32, Springer, 1996.

[6] J. Kollár, Y. Miyaoka, S. Mori, "Rationally connected varieties", J. Algebra. Geom. 1 (1992), no. 3, p. 429-448.

[7] J. M. Starr, "Arithmetic over function fields", in *Arithmetic geometry*, Clay Mathematics Proceedings, vol. 8, American Mathematical Society, 2009, p. 375-418.