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\title{Fano hypersurfaces in positive characteristic}
\author{\firstname{Yi} \lastname{Zhu}}
\address{United States}
\email{math.zhu@gmail.com}

\begin{abstract}
We prove that a general Fano hypersurface in a projective space over an algebraically closed field is separably rationally connected.
\end{abstract}

\dateposted{2024-02-02}
\begin{document}
\maketitle

\section{Introduction}

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

\begin{defi}[{\cite[IV.3]{Kollar}}]
Let $X$ be a smooth variety defined over $k$.

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

A variety $X$ is \emph{separably rationally connected} if there exists a proper rational curve $f:\bP^1\rightarrow 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 \emph{very free} curves.
\end{defi}

We refer to Koll\'ar's book~\cite{Kollar} or the work of Koll\'ar--Miyaoka--Mori~\cite{KMM} 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.

\begin{ques}[Koll\'ar] In arbitrary characteristic, is every smooth Fano variety separably rationally connected?
\end{ques}

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

\begin{theo}\label{mainn}
In arbitrary characteristic, a general Fano hypersurface of degree $n$ in $\bP^n_k$ contains a minimal very free rational curve of degree $n$, i.e., the pullback of the tangent bundle has the splitting type $\cO(2)\oplus\cO(1)^{\oplus(n-2)}$.
\end{theo}

\begin{theo}\label{main}
In arbitrary characteristic, a general Fano hypersurface in $\mathbb{P}^n_k$ is separably rationally connected.
\end{theo}

de Jong and Starr~\cite{dJS1} proved that every family of separably rationally connected varieties over a curve admits a rational section. Thus using Theorem~\ref{main}, we give another proof of Tsen's theorem.

\begin{coro}
Every family of Fano hypersurfaces in $\bP^n$ over a curve admits a rational section.
\end{coro}

\subsection*{Acknowledgment}

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

\section{Typical Curves and Deformation Theory}

\begin{nota}
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.
\[
\xymatrix{0\ar[r]& TC \ar[r]& TX|_C\ar[r]& \mathcal{N}_{C|X} \ar[r]& 0}
\]
By adjunction, the degree of $TX|_C$ is the degree of $\cO_{\bP^n}(1)|_C$. Thus the degree of the normal bundle $\mathcal{N}_{C|X}$ is $e-2$ and the rank is $n-2$.
\end{nota}

\begin{defi}\label{typicaldef}
Let $e$ be a positive integer $\le n$. A smooth rational curve $C$ of degree $e$ contained in the smooth locus of $X$ is \emph{typical}, if its normal bundle is the following:
\begin{equation*}
\mathcal{N}_{C|X}\cong
\begin{cases}
\cO_C^{\oplus(n-3)}\oplus\cO_C(-1),&\text{if }\ e= 1,\\
\cO_C^{\oplus(n-e)}\oplus\cO_C(1)^{\oplus(e-2)}, &\text{if }\ e\ge 2.
\end{cases}
\end{equation*}
The curve $C$ is a \emph{typical line}, resp., \emph{typical conic} if moreover the degree of $C$ is one, resp., two.
\end{defi}

\begin{rema}\leavevmode
\begin{enumerate}
\item For a typical line $L$ on $X$, there is a canonically defined \emph{trivial subbundle} $\cO_L^{\oplus(n-2)}$ in $\N_{L|X}$.
\item When $e=n$, typical rational curves of degree $n$ are very free.
\end{enumerate}
\end{rema}

\begin{lemm}\label{typicalcurve}
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:
\begin{enumerate}
\item \label{9.1} $h^1(C, \mathcal{N}_{C|X}(-1))=0$,
\item \label{9.2} $h^1(C,\mathcal{N}_{C|X}(-2))\le n-e$.
\end{enumerate}
\end{lemm}

\begin{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\dots\oplus\mathcal{O}_C(a_{n-2})$, where $a_1\ge\cdots\ge a_{n-2}$. Condition~\eqref{9.1} is equivalent to that $a_{n-2}\ge 0$. Condition~\eqref{9.2} implies that at most $n-e$ of the $a_i$'s are $0$. By degree count, $C$ is a typical rational curve of degree $e$.
\end{proof}

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

\begin{lemm}\label{typicalline}
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:
\begin{enumerate}
\item $h^1(C, \mathcal{N}_{L|X})=0$,
\item $h^1(C,\mathcal{N}_{L|X}(-1))\le1$.
\end{enumerate}
\end{lemm}

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

Let $R_{e,n}$ be the Hilbert scheme parameterizing flat projective families of one-dimensional subschemes in $\X$ with the Hilbert polynomial $P(d)=ed+1$. By~\cite[Theorem~1.4]{Kollar}, $R_{e,n}$ is projective over $H_n$.

Let $\C$ be the universal family over $R_{e,n}$, denoted by $\pi : \C\rightarrow R_{e,n}$. We have the following diagram,
\[
\xymatrix{
\C \ar[d]_\pi \ar[r]^<<<<<<i & \X \times_{H_n} R_{e,n}\ar[ld] \\
R_{e,n}&}
\]
where $i$ is a closed immersion.

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

\begin{prop}\label{typicaldefopen}
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 $n$.
\end{prop}

\begin{proof}
By Lemma~\ref{typicalline} and Lemma~\ref{typicalcurve}, every typical curve $C$ on a hypersurface $X$ gives an unobstructed point in $R_{e,n}$. By the upper semicontinuity theorem~\cite[III.12.8]{Hartshorne}, a small deformation of $(X_t,C_t)$ is still typical and $C_t$ is contained in the smooth locus of $X_t$.
\end{proof}

\begin{prop}\label{typicalconic}
Let $X$ be a hypersurface of degree $n$ in $\bP^n$. Let $L$ and $M$ be two typical lines on $X$ intersecting transversally at one point $p$. Assume that the following conditions hold:
\begin{enumerate}
\item \label{12.1} the direction $T_p L$ lies outside the trivial subbundle of $\mathcal{N}_{M|X}$;
\item \label{12.2} the direction $T_p M$ lies outside the trivial subbundle of $\N_{L|X}$.
\end{enumerate}
Then the pair $(X,D=L\cup M)\in R_{2,n}$ can be smoothed to a pair $(X^\prime, C)$ where $C$ is a typical conic in $X^\prime$. 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.
\end{prop}

\begin{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.
\[
\xymatrix{0 \ar[r]& \mathcal{N}_{L|X} \ar[r]& \mathcal{N}_{D|X}|_L \ar[r]& T_p M \ar[r]& 0}
\]
By~\cite[Lemma~2.6]{GHS}, 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 $T_p M$. Since $\mathcal{N}_{L|X}\cong\cO_L^{\oplus(n-3)}\oplus\cO_L(-1)$, condition \eqref{12.2} implies that $\N_{D|X}|_L$ is isomorphic to $\cO_L^{\oplus(n-2)}$.

By the same argument, condition~\eqref{12.1} implies that the sheaf $\mathcal{N}_{D|X}|_M$ is isomorphic to $\cO_M^{\oplus(n-2)}$. Now we have the following short exact sequence.
\[
\xymatrix{0 \ar[r]& \mathcal{N}_{D|X}|_M(-p) \ar[r]\ar@{=}[d]& \mathcal{N}_{D|X} \ar[r]& \mathcal{N}_{D|X}|_L \ar[r]\ar@{=}[d]& 0\\
& \cO_M(-1)^{\oplus(n-2)} && \cO_L^{\oplus(n-2)}}
\]

First we claim that there is a smoothing of $D$. Since $h^1(D,\N_{D|X})=0$, the pair $(X,D)$ is unobstructed in $R_{2,n}$, cf.~\cite[I.2]{Kollar}. By~\cite[Lemma~3.17]{Starr0}, it suffices to show that the map
\[
H^0(D,\mathcal{N}_{D|X})\rightarrow H^0(L,\N_{D|X}|L)\rightarrow T_p M
\]
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~\ref{typicalcurve}, $D_t$ is a typical conic on $X_t$.
\end{proof}

\begin{defi}\label{tc}
Let $X$ be a hypersurface of degree $n$ in $\bP^n$. A \emph{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:
\begin{enumerate}
\item $C$ is a typical conic on $X$;
\item $L_1,\dots,L_m$ are disjoint typical lines on $X$ and each $L_i$ intersects $C$ transversally at $p_i$.
\end{enumerate}
The conic $C$ is called the \emph{handle} of the comb and $L_i$'s are called the \emph{teeth}.
\end{defi}

\begin{prop}\label{typicalcomb}
Let $X$ be a hypersurface of degree $n$ in $\bP^n$. Let $D=C\cup L_1\cup\cdots\cup L_{n-2}$ 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:
\begin{enumerate}
\item \label{14.1} the direction $T_{p_i} C$ lies outside the trivial subbundle of $\mathcal{N}_{L_i|X}$;
\item \label{14.2} the directions $T_{p_i} L_i$ are general in $\N_{C|X}$ so that the sheaf $\N_{D|X}|_C$ is isomorphic to $\cO_C(1)^{\oplus(n-2)}$.
\end{enumerate}
Then the pair $(X, D)\in R_{n,n}$ can be smoothed to a pair $(X^\prime,C')$ where $C'$ is a very free rational curve on $X^\prime$.
\end{prop}

\begin{proof}
The proof is very similar to the proof of Proposition~\ref{typicalconic}. Here we only sketch the proof. Condition~\eqref{14.1} implies that the sheaf $\N_{D|X}|_{L_i}$ is isomorphic to $\cO_{L_i}^{\oplus(n-2)}$ for each $i$. We have the following short exact sequence.
\[
\xymatrix{0 \ar[r]& \bigoplus_i\mathcal{N}_{D|X}|_{L_i}(-p) \ar[r]\ar@{=}[d]& \mathcal{N}_{D|X} \ar[r]& \mathcal{N}_{D|X}|_C \ar[r]\ar@{=}[d]& 0\\
& \cO_{L_i}(-1)^{\oplus(n-2)} && \cO_C(1)^{\oplus(n-2)}}
\]
Since $H^1(D,\N_{D|X})=0$, $D$ is unobstructed. By diagram chasing, the map $H^0(D,\N_{D|X})\rightarrow \bigoplus_i T_{p_i} L_i$ 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 $h^1(D, \N_{D|X}(-q-r))=0$. By the upper semicontinuity theorem, a general smoothing of the pair $(X,D)$ gives a very free curve in a general hypersurface.
\end{proof}


\section{An Example}\label{sec3}

In this section, we construct a hypersurface of degree $n$ in $\bP^n$, which contains a special configuration of typical lines. Later we will use this example to produce a very free curve in a general hypersurface.

\begin{nota}\label{spiky}
Let $n$ be an integer $\ge 4$. Let $\lbrack x_0:\dots:x_n\rbrack$ be homogeneous coordinates for $\mathbb{P}^n$. Let $X$ be a hypersurface of degree $n$ in the projective space $\bP^n$ 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 $k^{n+1}$.
\[
\begin{array}{llll}
x_0^{n-1}x_n &+x_1^{n-3}x_n^2x_0 & +(x_1^{n-1}+x_0x_1^{n-2}+\dots+x_0^{n-3}x_1^2)x_2 &+(x_2^{n-1}+x_0x_2^{n-2}+\dots+x_0^{n-3}x_2^2)x_3
\\
& +x_1^{n-4}x_n^3x_3 &+(x_0x_1^{n-2}+\dots+x_0^{n-3}x_1^2)x_3 &+(x_0x_2^{n-2}+\dots+x_0^{n-3}x_2^2)x_4\\[-0.25em]
&\;\vdots&\;\vdots&\;\vdots \\
&+x_1x_n^{n-2}x_{n-2} &+(x_0^{n-4}x_1^{3}+x_0^{n-3}x_1^2)x_{n-2} &+(x_0^{n-4}x_2^{3}+x_0^{n-3}x_2^2)x_{n-1}\\
&+x_n^{n-1}x_{n-1}&+x_0^{n-3}x_1^2x_{n-1} &+x_0^{n-3}x_2^2x_{1}\\%\}
\\
&+\;\cdots& +(x_{n-1}^{n-1}+x_0x_{n-1}^{n-2}+\dots+x_0^{n-3}x_{n-1}^2)x_1&\\
&+\;\cdots&+(x_0x_{n-1}^{n-2}+\dots+x_0^{n-3}x_{n-1}^2)x_2&\\[-0.25em]
&\;\vdots&\;\vdots\\
&+\;\cdots&+(x_0^{n-4}x_{n-1}^3+x_0^{n-3}x_{n-1}^2)x_{n-3}&\\
&+\;\cdots& +x_0^{n-3}x_{n-1}^2x_{n-2}&
\end{array}
\]

Let $p$ be the point $\lbrack 1:0:\dots:0\rbrack$ and $q$ be the point $\lbrack 0:1:0:\dots:0\rbrack$. 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.
\end{nota}

\begin{center}
\includegraphics[scale=0.5]{spiky.jpg}
\end{center}

\begin{lemm}\leavevmode
\begin{enumerate}
\item Both $p$ and $q$ lie in the smooth locus of $X$.
\item The tangent space $T_p X$ is the hyperplane $\{x_n=0\}$, which is spanned by the lines $L_1,\dots,L_{n-1}$.
\item The tangent space of $T_q X$ is the hyperplane $\{x_2=0\}$.
\end{enumerate}
\end{lemm}

\begin{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 $i\neq 2$ and $\frac{\partial F}{\partial x_2}(q)=1$.
\end{proof}

\begin{lemm}
The lines $L_1,\dots, L_{n-1}$ are in the smooth locus of $X$.
\end{lemm}

\begin{proof}
We will prove the case for line $L_1$. The remaining cases can be computed directly by the same method. Denote $L_1=\{[x_0:x_1:0:\dots:0]\in\mathbb{P}^n\}$. By restricting the partial derivatives of the defining equation of the hypersurface $X$ on $L_1$, we get the following.
\begin{equation}
\begin{aligned}
\frac{\partial F}{\partial x_2}\Biggr|_{L_1}&=x_1^{n-1}+x_0x_1^{n-2}+\dots+x_0^{n-3}x_1^2\\
\frac{\partial F}{\partial x_3}\Biggr|_{L_1}&=x_0x_1^{n-2}+\dots+x_0^{n-3}x_1^2\\
&\;\;\vdots\\
\frac{\partial F}{\partial x_{n-2}}\Biggr|_{L_1}&=x_0^{n-4}x_1^{3}+x_0^{n-3}x_1^2\\
\frac{\partial F}{\partial x_{n-1}}\Biggr|_{L_1}&=x_0^{n-3}x_1^2\\
\frac{\partial F}{\partial x_n}\Biggr|_{L_1}&=x_0^{n-1}
\end{aligned}\label{L1}
\end{equation}

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 $X$.
\end{proof}

\begin{lemm}
The line $L_n$ is in the smooth locus of $X$.
\end{lemm}

\begin{proof}
By restricting the partial derivatives of the defining equation of $X$ on $L_n$, we get the following.
\begin{equation}
\begin{aligned}
\frac{\partial F}{\partial x_0}\Biggr|_{L_n}&=x_1^{n-3}x_n^2\\
\frac{\partial F}{\partial x_3}\Biggr|_{L_n}&=x_1^{n-4}x_n^3\\
&\;\;\vdots\\
\frac{\partial F}{\partial x_{n-2}}\Biggr|_{L_n}&=x_1x_n^{n-2}\\
\frac{\partial F}{\partial x_{n-1}}\Biggr|_{L_n}&=x_n^{n-1}\\
\frac{\partial F}{\partial x_2}\Biggr|_{L_n}&=x_1^{n-1}
\end{aligned}\label{Ln}
\end{equation}

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 $X$.
\end{proof}

\begin{prop}\label{x0p1}
With the setup as in Notation~\ref{spiky}, $X$ satisfies the following properties.
\begin{enumerate}%\alphenumi
\item \label{19.1} The lines $L_1,\dots,L_n$ are typical in $X$.
\item \label{19.2} For $i=1,\dots, n-1$, the trivial subbundle of the normal bundle $\mathcal{N}_{L_i|X}$ at $p$ is generated by $\partial_{\bar{i+1}}-\partial_{\bar{i+2}},\dots,\partial_{\bar{i+n-3}}-\partial_{\bar{i+n-2}}$, where the notation $\bar{j}$ is $j$ if $j$ is less than $n$ and $j-(n-1)$ if otherwise.
\item \label{19.3} The trivial subbundle of the normal bundle $\mathcal{N}_{L_1|X}$ at $q$ is generated by $\partial_3,\dots, \partial_{n-1}$
\item \label{19.4} The trivial subbundle of the normal bundle $\mathcal{N}_{L_n|X}$ at $q$ is generated by $\partial_3,\dots,\partial_{n-1}$.
\end{enumerate}
\end{prop}

\begin{proof}
Let $L$ be a line in $X$. We have the following short exact sequences.
\[
\xymatrix{0 \ar[r]& \mathcal{N}_{L|X}(-1) \ar[r]\ar@{=}[d]& \mathcal{N}_{L|\mathbb{P}^n}(-1) \ar[r]\ar@{=}[d]& \mathcal{N}_{X|\mathbb{P}^n}|_{L}(-1) \ar[r]\ar@{=}[d]& 0\phantom{.}\\
0 \ar[r]& \mathcal{N}_{L|X}(-1) \ar[r]& \cO_{L}^{\oplus(n-1)} \ar[r]& \cO_{L}(n-1) \ar[r]& 0.}
\]
The associated long exact sequence is the following.
\[
\xymatrix{H^0(L, \mathcal{N}_{L|X}(-1)) \rightarrow k^n \ar[r]^-{\alpha}& H^0(L, \cO_L(n-1))\ar[r]& H^1(L, \mathcal{N}_{L|X}(-1))\ar[r]& 0}
\]
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 $\bP^n$. By Lemma~\ref{typicalline}, $L$ is typical if and only if the image of $\alpha$ is of codimension one in $H^0(L,\cO_L(n-1))$.

When $L=L_1$, by~\eqref{L1}, $\frac{\partial F}{\partial x_2}|_{L_1},\dots, \frac{\partial F}{\partial x_n}|_{L_1}$ form a codimensional-one subspace of $H^0(L_1,\cO_{L_1}(n-1))$. 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 $\cO_{L_1}^{\oplus(n-1)}$ which maps to $0$ in $\cO_{L_1}(n-1)$. Let $\partial_2,\dots,\partial_n$ be the generators of $\cO_{L_1}^{\oplus(n-1)}$. 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 $\cO_{L_1}$-module. If we restrict the bundle at $p$ and $q$, we get properties \eqref{19.2} and \eqref{19.3} for $L_1$.

When $L=L_2,\dots, L_{n-1}$, we can prove properties \eqref{19.2} and \eqref{19.3} in a similar way. When $L=L_n$, \eqref{19.4} follows from the same computation as above by applying~\eqref{Ln}.
\end{proof}

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

\begin{coro}\label{cond1}
With the setup as in Notation~\ref{spiky}, we have the following statements.
\begin{enumerate}
\item The lines $L_1$ and $L_n$ are typical in $X$.
\item The direction $T_q L_1$ lies outside the trivial subbundle of $\N_{L_n|X}$.
\item The direction $T_q L_n$ lies outside in the trivial subbundle of $\N_{L_1|X}$.
\end{enumerate}
\end{coro}

\begin{coro}\label{cond2}
With the setup as in Notation~\ref{spiky}, we have the following statements.
\begin{enumerate}
\item The lines $L_2,\dots,L_{n-2}$ are typical in $X$.
\item The direction $T_p L_1$ lies outside the trivial subbundle of $\N_{L_i|X}$ for $2\le i\le n-1$.
\item The directions $T_p L_2,\dots,T_p L_{n-1}$ span the normal bundle $\N_{L_1|X}$ at $p$.
\end{enumerate}
\end{coro}


\section{Proof of the Main Theorem}

\begin{lemm}\label{h0C}
Let $C$ be the union of $n$ lines $L_1,\dots,L_n$ in $\bP^n$ as in Notation~\ref{spiky}. The following properties hold for $C$ for every positive integer $d$:
\begin{enumerate}
\item $h^0(C, \cO_C(d))=nd+1$ and $h^1(C,\cO_C(d))=0$.
\item $h^1(C,\I_C(d))=0$.
\item $h^0(C,\I_C(d))=h^0(\bP^n,\cO(d))-nd-1$.
\end{enumerate}
\end{lemm}

\begin{rema}
The curve $C$ is an example of curves with rational $n$-fold point, cf.~\cite[3.7]{ChenCoskun}. The following lemma is an analogue of~\cite[Lemma~3.8]{ChenCoskun}.
\end{rema}

\begin{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 $\cO_C(d)$ is obtained by gluing global sections on each component, which imposes exactly $n-1$ linear conditions, we have
\[
h^0(\cO_C(d))=n(d+1)-(n-1)=nd+1
\]
and $h^0(\cO_{\bP^n}(d))\to h^0(\cO_C(d))$ 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
\[
\xymatrix{0 \ar[r]& h^0(C,\I_C(d)) \ar[r]& h^0(\cO_{\bP^n}(d)) \ar[r]& h^0(\cO_C(d)) \ar[r]& h^1(C,\I_C(d)).}\qedhere
\]
\end{proof}

\begin{enonce}{Construction}
Let $C$ be the union of $n$ lines $L_1,\dots,L_n$ in $\bP^n$ as in Notation~\ref{spiky}. If we consider $L_1\cup L_n$ as a conic in $\bP^n$, there exists a smooth affine pointed curve $(T,0)$ and a smoothing $D'\rightarrow (T,0)$ satisfying the following conditions:
\begin{enumerate}
\item The special fiber $D'_0$ is $L_1\cup L_n$;
\item For any $t\in T-\{0\}$, $D'_t$ is a smooth conic contained in the plane spanned by $L_1$ and $L_n$.
\end{enumerate}
We may assume that there exists $n-2$ sections $s_i:(T,0)\rightarrow D'$ 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 $\pi:D\rightarrow (T,0)$ satisfying the following conditions:
\begin{enumerate}
\item The special fiber $D_0$ is $C$ constructed as in Notation~\ref{spiky}.
\item For any $t\in T-\{0\}$, $D_t$ is a comb with the handle $D'_t$ and with the teeth lines.
\end{enumerate}

We have the following diagram.
\[
\xymatrix{D_0=C \ar[r] \ar[d] & D \ar[d]_\pi \ar[r]^i &\bP^n_T \ar@{->}[ld]^\pi \\0 \ar[r] &(T,0)}
\]
\end{enonce}

\begin{lemm}\label{flatlift}
The family $\pi:D\rightarrow (T,0)$ is flat. Furthermore, $\pi_*\I_D(d)$ is locally free on $T$ for any $d>0$, where $\I_D$ is the ideal sheaf of $D$ in $\bP^n_T$.
\end{lemm}

\begin{proof}
The same computational argument as in the proof of Lemma~\ref{h0C} proves that $h^0(\bP^n_t, I_{D_t}(d))$ and $h^1(\bP^n_t, I_{D_t}(d))$ 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~\cite[III.12.9]{Hartshorne}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{mainn}]
The theorem is known for $n=2,3$. We can assume that $n\ge 4$. By~\cite[IV.3.11]{Kollar}, it suffices to produce one very free curve on a hypersurface of degree $n$. By Lemma~\ref{flatlift}, after shrinking $T$, hypersurfaces of degree $n$ containing $D_t$ in $\bP^n_t$ form a trivial projective bundle over $(T,0)$. Thus the family $\pi:D\rightarrow (T,0)$ admits a lifting to a flat family of pairs $\pi:(\X_T,D)\rightarrow (T,0)$ in $R_{n,n}$ such that the special fiber $(\X_0,D_0)$ is $(X,C)$ which is constructed in Section~\ref{sec3}.
\[
\xymatrix{ D \ar[d]_\pi \ar[r]^i &\X_T \ar[r] \ar[ld] &\bP^n_T \ar[lld]^\pi \\(T,0)}
\]
All the following steps of the proof requires to shrink $T$ if necessary. By Proposition~\ref{typicalconic} and Corollary~\ref{cond1}, we may assume that the handle $D'_t$ is a typical conic in $\X_t$ for every $t\in T-\{0\}$. By Proposition~\ref{typicaldefopen} and Corollary~\ref{cond2} (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~\ref{tc}. Now the theorem follows if we verify the two conditions in Proposition~\ref{typicalcurve}. Since they are open conditions, it suffices to check on the special fiber $(X,C)$, which is proved in Corollary~\ref{cond2}.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{main}]
For a general Fano hypersurfaces of degree $d$ in $\bP^n$, when $d=n$, this is proved in Theorem~\ref{mainn}. When $d<n$, we may choose a general Fano hypersurface $Y$ of degree $d$ in $\bP^d$ admitting a very free curve $f:\bP^1\rightarrow Y$. Construct the cone $X$ of $Y$ in $\bP^n$. Note that $Y$ is the intersection of a projective subspace $L$ of dimension $d$ and $X$. By the normal bundle exact sequence,
\[
\xymatrix{0\ar[r]& TY \ar[r]& TX \ar[r]& \N_{Y| X}\ar[r]& 0}
\]
the sheaf $f^*TY$ is positive and the sheaf $\N_{Y| X}$ is isomorphic to $\N_{L|\bP^n}$, which is positive too. Therefore the pullback bundle $f^*TX$ is positive.
\end{proof}

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