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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]^<<<<<*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*