\documentclass[reqno]{amsart}
\pagestyle{plain}
\usepackage{amsmath}
%\usepackage{hyperref}
\usepackage{amscd}
\usepackage{graphics}

\usepackage{latexsym}
\addtolength{\textwidth}{25mm}
\addtolength{\textheight}{20mm}
\addtolength{\oddsidemargin}{-.5\oddsidemargin}
\addtolength{\evensidemargin}{-.5\evensidemargin}
\addtolength{\topmargin}{-.5\topmargin}
\addtolength{\headsep}{-.55\headsep}
%\addtolength{\footskip}{-.5\footskip}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{thm}[theorem]{Theorem}
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{lem}[theorem]{Lemma}
\newtheorem{prop}[theorem]{Proposition}
\newtheorem{claim}[theorem]{Claim}

\theoremstyle{definition}
\newtheorem{defn}[theorem]{Definition}
\newtheorem{notat}[theorem]{Notation}
\newtheorem{conv}[theorem]{Convention}
\newtheorem{hyp}[theorem]{Hypothesis}
\newtheorem{conj}[theorem]{Conjecture}
\newtheorem{prin}[theorem]{Principle}
\newtheorem{ques}[theorem]{Question}
\newtheorem{rmk}[theorem]{Remark}

\newtheorem{ex}{Example}
\newtheorem{corex}[ex]{Corollary and Example}

\theoremstyle{remark}
\newtheorem{Remark}{Remark}[section]

\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\AAA}{\mathbb{A}}
\newcommand{\PP}{\mathbb{P}}
\newcommand{\PPP}{\PP_{\CC}}
\newcommand{\SP}{\text{Spec }}
\newcommand{\HB}[3]{\text{Hilb}^{#1}_{{#2}/{#3}}}
\newcommand{\GR}[3]{\text{G}_{#1}(#2,#3)}
\newcommand{\mgn}[2]{\text{M}_{#1,#2}}
\newcommand{\Mgn}[2]{\mathcal{M}_{#1,#2}}
\newcommand{\mgnb}[2]{\overline{\text{M}}_{#1,#2}}
\newcommand{\Mgnb}[2]{\overline{\mathcal{M}}_{#1,#2}}
\newcommand{\kgn}[4]{\text{M}_{#1,#2}(#3,#4)}
\newcommand{\Kgn}[4]{\mathcal{M}_{#1,#2}(#3,#4)}
\newcommand{\kgnb}[4]{\overline{\text{M}}_{#1,#2}(#3,#4)}
\newcommand{\Kgnb}[4]{\overline{\mathcal{M}}_{#1,#2}(#3,#4)}
\newcommand{\kgnbo}[4]{\overline{\text{M}}_{#1,#2}(#3,#4)^o}
\newcommand{\Kgnbo}[4]{\overline{\mathcal{M}}_{#1,#2}(#3,#4)^o}
\newcommand{\Kbm}[2]{\overline{\mathcal{M}}_{#1}(#2)}
\newcommand{\kbm}[2]{\overline{M}_{#1}(#2)}
\newcommand{\kbmo}[2]{{M}_{#1}(#2)}
\newcommand{\Kbmo}[2]{{\mathcal{M}}_{#1}(#2)}
\newcommand{\kbmoo}[2]{{M}_{#1}(#2)^o}
\newcommand{\Kbmoo}[2]{{\mathcal{M}}_{#1}(#2)^o}
\newcommand{\Na}[2]{{\mathcal{N}}(#1,#2)}
\newcommand{\nNa}[2]{{\mathcal{N}}(#1,#2)_s}


\newcommand{\lt}{\left}
\newcommand{\rt}{\right}
\newcommand{\bc}[2]{\binom{#1}{#2}}

\newcommand{\mc}{\mathcal}
\newcommand{\cB}{\mathcal B}
\newcommand{\cC}{\mathcal C}
\newcommand{\cE}{{\mathcal E}}
\newcommand{\cF}{{\mathcal F}}
\newcommand{\cG}{{\mathcal G}}
\newcommand{\cX}{{\mathcal X}}
\newcommand{\cD}{{\mathcal D}}
\newcommand{\lci}{\mathcal {LCI}}
\newcommand{\fe}{\mathcal {FE}}

\newcommand{\g}{{\frak g}}
\newcommand{\gs}{{\frak g}^{\ast}}
\newcommand{\HH}{{\cal H}}
\newcommand{\OO}{\mathcal O}
\newcommand{\Ci}{C^{\infty}}
\newcommand{\B}{{\cal B}}
\newcommand{\Cs}{C^{\ast}}
\newcommand{\Ws}{W^{\ast}}
\newcommand{\Metc}{M^{d,w}}
\newcommand{\Hetc}{H^{d,w}}
\newcommand{\Betc}{(B,\Sigma,b_0)}

\def\mapright#1{-{#1}\to}
\def\Ker{{\rm Ker}\,}
\def\dd{{\rm d}}
\def\Proj{{\rm Proj}\,}
\def\Spec{{\rm Spec}\,}
\def\Hom{{\rm Hom}\,}
\def\mC{{\mathbb C}}
\def\mP{{\mathbb P}}
\def\PP{{\mathbb P}}
\def\cO{{\mathcal O}}
\def\cZ{{\mathcal Z}}
\def\cM{{\mathcal M}}
\def\cC{{\mathcal C}}

\begin{document}

\title[Appendix]
{A fact about linear spaces on hypersurfaces}


\author[Starr]{Jason Michael Starr}
\address{Department of Mathematics \\
  Massachusetts Institute of Technology \\ Cambridge MA 02139}
\email{jstarr@math.mit.edu} 
\date{\today}


\begin{abstract}
A smooth, nondegenerate
hypersurface in projective space contains no linear subvarieties of
greater than half its dimension.  It can contain
linear subvarieties of half its dimension.  This note proves that a
smooth hypersurface of degree $d\geq 3$
contains at most finitely many such subvarieties.
\end{abstract}

\maketitle

\medskip\noindent 
Let $k$ be a field.  Let $X\subset \PP^n_k$ be a hypersurface of
degree $d > 1$.  For each integer $m > 0$, denote by $F_m(X)$ the
\emph{Fano scheme of $m$-planes in} $X$, cf. ~\cite{AKFano}.  There
are a number of natural questions about $F_m(X)$: is this scheme
nonempty, is this scheme connected, is this scheme irreducible, is
this scheme reduced, is this scheme smooth, what is the dimension of
this scheme, 
what is the degree of this
scheme, etc.?  For each of these questions, the answer is uniform for
a \emph{generic hypersurface}.  More precisely, 
there is a non-empty open subset $U$ of the parameter space %X
of hypersurfaces, such that for
every point in $U$ the answer to the question is the same.  For a generic
hypersurface, the answer is often easy to find:  The total space of the
relative Fano scheme of the universal hypersurface is itself a
projective bundle over the Grassmannian $\mathbb{G}(m,n)$, so if %X
the question for a generic hypersurface can be reformulated as a
question about the total space of the relative Fano scheme, it is easy
to answer the question.
However, much less
is known if $X$ is assumed to be smooth, but \emph{not} generic.

\medskip\noindent
There are a few easy results, such as the following.
\begin{prop} \label{prop-1}
\marginpar{prop-1}
Let $X \subset \PP^n$ be a hypersurface of degree $d > 1$ and let $m$
be an integer such that $2m \geq n$.
Every $m$-plane $\Lambda \subset X$ intersects the singular locus of $X$.
In particular, if $X$ is smooth then $F_m(X)$ is empty.
\end{prop}

\medskip\noindent
\begin{proof}
Choose a system of homogeneous coordinates $x_0,\dots,x_n$ on $\PP^n$
such that $\Lambda$ is given by $x_{m+1}=\cdots =x_n=0$.  %X
Let $F$ be a defining equation for $X$.  Because $\Lambda \subset X$,
$F(x_0,\dots,x_m,0,\dots,0) = 0$.  Also,
$$
\frac{\partial F}{\partial x_i}(x_0,\dots,x_m,0,\dots,0) = 0,
$$
for $i=0,\dots,m$.
Because $d>1$, for $i=1,\dots,n-m$, the homogeneous polynomial on $\Lambda$,
$$
\frac{\partial F}{\partial x_{m+i}}(x_0,\dots,x_m,0,\dots,0),
$$
is nonconstant.  Since $n-m \leq m$, these $n-m$ nonconstant
homogeneous polynomials 
have a common zero in $\Lambda$. 
By the Jacobian criterion, this is a singular point of $X$.
\end{proof}

\medskip\noindent
\begin{rmk}  This also follows easily from the Lefschetz hyperplane
  theorem.
\end{rmk}

\medskip\noindent
What happens if $n=2m+1$?  If $m>1$ or if $m=1$ and $d>3$, then a
\emph{generic} hypersurface $X\subset \PP^n$ contains no $m$-plane.
However there do exist smooth hypersurfaces 
containing an $m$-plane.  For instance, 
if $\text{char}(k)$ does not divide
$d$ then the Fermat hypersurface $x_0^d + \dots + x_n^d=0$ %X
is smooth and contains many $m$-planes, e.g.,
$x_0+x_1=x_2+x_3=\cdots= x_{n-1}+x_n=0$ when $d$ is odd.  %X
However, if $d \geq 3$,
a smooth
hypersurface cannot contain a positive-dimensional family of
$m$-planes.  This was proved independently by Olivier Debarre, using a
different argument.  

\medskip\noindent
The setup is as follows.  
Let $n=2m+1$.  Let $X \subset \PP^n$ be a hypersurface of
degree $d$.  Let $\Lambda_1,\Lambda_2 \subset X$ be $m$-planes.
Denote by $Z$ the intersection $\Lambda_1 \cap \Lambda_2$.  This is
either
empty or else an
$r$-plane for some integer $r$.  If $Z$ is empty, define $r$ to be
$-1$.

\medskip\noindent 
Denote by $X_\text{sm} \subset X$ the smooth locus of $X$, i.e., the
maximal open subscheme that is smooth.  Denote $\Lambda_{i,\text{sm}}
= \Lambda_i \cap X_\text{sm}$ for $i=1,2$.
There are Chow classes,
$$
[\Lambda_{i,\text{sm}}] \in A_m(X_\text{sm}), \
i=1,2.
$$  
Because
$X_\text{sm}$ is smooth, the intersection product
$[\Lambda_{1,\text{sm}}] \cdot 
[\Lambda_{2,\text{sm}}] \in A_0(X_\text{sm})$ is defined.  

\medskip\noindent
\begin{lem} \label{lem-2}
\marginpar{lem-2}
If $Z$ is contained in $X_\text{sm}$, then the degree of
$[\Lambda_{1,\text{sm}}] \cdot
[\Lambda_{2,\text{sm}}]$ is $( 1 - (1-d)^{r+1})/d$. %X
\end{lem}

\begin{proof}
If $r=-1$, i.e., if $Z$ is empty, this is obvious.  Therefore suppose
that $Z$ is an $r$-plane for some $r \geq 0$.  By the excess
intersection formula, the class $[\Lambda_{1,\text{sm}}]\cdot
[\Lambda_{2,\text{sm}}]$ is the
pushforward from $Z$ of the refined intersection product,
$(\Lambda_{1,\text{sm}}\cdot \Lambda_{2,\text{sm}})^Z$.  And by
~\cite[Prop. 9.1.1]{F}, the refined intersection product is,
$$
(\Lambda_{1,\text{sm}}\cdot \Lambda_{2,\text{sm}})^Z = \lt\{
c(N_{\Lambda_{1,\text{sm}}/X_{\text{sm}}})/c(N_{Z/\Lambda_{2,\text{sm}}})\cap
[Z] \rt\}_r.
$$
Denote $H = c_1(\OO_Z(1))$.  The normal bundle of $Z$ in
$\Lambda_{1,\text{sm}}$ is $\OO_Z(1)^{m-r}$.  The restriction to $Z$
of the normal bundle of $\Lambda_{2,\text{sm}}$ in $\PP^n$ is
$\OO_Z(1)^{n-m}= \OO_Z(1)^{m+1}$.  
And the restriction to $Z$ of the normal bundle of
$X$ in $\PP^n$ is $\OO_Z(d)$.  Therefore,
$$
c(N_{\Lambda_{1,\text{sm}}/X_{\text{sm}}})/c(N_{Z/\Lambda_{2,\text{sm}}})
= \frac{(1+H)^{m+1}}{(1+H)^{m-r}(1+dH)} = \frac{(1+H)^{r+1}}{1+dH}.
$$
Expanding this out gives,
$$
\left(\sum_{i=0}^{r+1} \binom{r+1}{i} H^i\right) %X
\left(\sum_{j=0}^{\infty} (-1)^j %X
  d^j H^j\right). %X
$$
In particular, the coefficient of $H^r$ is,
\begin{align*} %X
\sum_{i=0}^r \binom{r+1}{i} (-1)^{r-i} d^{r-i} &= %X
\frac{-1}{d} \sum_{i=0}^r \binom{r+1}{i} (-1)^{r+1-i} d^{r+1-i}\\ %X
&= ( 1 - (1-d)^{r+1})/d.%X
\end{align*}%X
\end{proof}

\begin{prop} \label{prop-3}
\marginpar{prop-3}
If $d\geq 3$, if $\Lambda_1$ and $\Lambda_2$ are distinct, and if at
least one of $\Lambda_1$, $\Lambda_2$ is contained in $X_{\text{sm}}$,
then $[\Lambda_{1,\text{sm}}]$ is not numerically equivalent to
$[\Lambda_{2,\text{sm}}]$.  
\end{prop}

\medskip\noindent
\begin{proof}
Let $\Lambda_1$ be contained in $X_{\text{sm}}$.  By
Lemma~\ref{lem-2},
$$
\text{deg}([\Lambda_1]\cdot [\Lambda_1]) = ( 1 - (1-d)^{m+1})/d. %X
$$
Also, $Z = \Lambda_1\cap \Lambda_2$ is contained in $X_\text{sm}$.  So
by Lemma~\ref{lem-2},
$$
\text{deg}([\Lambda_1]\cdot[\Lambda_{2,\text{sm}}]) =
( 1 - (1-d)^{r+1})/d. %X
$$
Because $d-1 \geq 2$ and $r < m$, we have  %X
$(d-1)^{r+1} < (d-1)^{m+1}$. Therefore $( 1 - (1-d)^{r+1})/d\neq  ( 1 - (1-d)^{m+1})/d$, %X
and so $[\Lambda_1]$ is not numerically equivalent to %X
$[\Lambda_{2,\text{sm}}]$.
\end{proof}

\begin{cor}[Debarre] \label{cor-4}
\marginpar{cor-4}
There are only finitely many $m$-planes contained in $X_{\text{sm}}$.
\end{cor}

\begin{proof}
By Proposition ~\ref{prop-3}, distinct $m$-planes contained in
$X_{\text{sm}}$ are not algebraically equivalent.  Therefore every
irreducible component of $F_m(X)$ that contains a point parametrizing
an $m$-plane in $X_{\text{sm}}$ is just a point.  Because $F_m(X)$ is
quasi-compact, the number of these irreducible components is finite.  
\end{proof}

\begin{rmk} \label{rmk-5}
\marginpar{rmk-5}
Debarre's proof shows more than Corollary ~\ref{cor-4}: 
for any $m$-plane $\Lambda$ contained in
$X_{\text{sm}}$, $h^0(\Lambda,N_{\Lambda/X}) = 0$.  It follows that
each such point is a connected component of $F_m(X)$, and that
$F_m(X)$ is reduced at this point.  
\end{rmk}

\medskip\noindent
A natural question is, what is the maximal number of $m$-planes
contained in a smooth hypersurface of degree $d$ in $\PP^{2m+1}$?  There is
a naive upper bound that grows as $d^{(m+1)^2}$, but this is
too large.  The Fermat hypersurface contains $C_m d^{m+1}$
distinct $m$-planes, where $C_m = (2m+1)(2m-1)\cdot \dots \cdot 3
\cdot 1$.  Joe Harris points out that for $m=1$, the degree of the
\emph{flecnodal curve} gives an upper bound of $11d^2-24d$.  

\medskip\noindent
More
generally, define the \emph{parabolic locus} $P(X) \subset X$ to be the set of
points $p\in X$ such that there is a line $L \subset \PP^{2m+1}$ that
has contact of order $3m+1$ with $X$ at $p$.  This is the pushforward
in $X$ of a subscheme in $\PP(T_X)$ that is the zero locus of a
section of a locally free sheaf.  If $X$ is generic, this section is a
regular section.  Then a Chern class computation gives that the degree
of $P(X)$ is a polynomial $p_m(d)$ of degree $m+1$ in $d$ whose leading term
is,
$$
\lt( \frac{(3m+1)!}{2} - 1 \rt) d^{m+1}.
$$
Therefore, for arbitrary $X$, $\text{deg}(P_m(X)) \leq p_m(d)$, where
$P_m(X)$ is the $m$-cycle of all $m$-dimensional irreducible
components of $P(X)$ (weighted by multiplicity).  

\medskip\noindent
Of course every
$m$-plane $\Lambda$ is contained in $P(X)$.  It is not clear that
every $m$-plane is contained in $P_m(X)$, i.e., that $\Lambda$ is an 
irreducible component of $P(X)$.  And, indeed, this
fails if $d \leq 3m$.  For $d \geq 3m$, it may be true that every
$m$-plane is an irreducible component of $P(X)$.  

\medskip\noindent
In the special case that $m=1$, 
$P(X)$ is a curve, the \emph{parabolic curve}, for all $d\geq 3$.
Therefore, the number of lines in a smooth surface of
degree $d\geq 3$ in $\PP^3$ is at most
$11d^2-24d$.  Note this gives the correct answer for $d=3$.  For $d=4$
this gives the wrong answer;
Segre proved the maximal number of lines on a quartic surface is $64$,
cf. ~\cite{SegreLines}.  In fact, by a more involved analysis, Segre
proved that the number of lines on a smooth surface of degree $d\geq 3$
is at most $11d^2 -28d+12$.
The Fermat surface
contains $3d^2$ lines.  The true maximum is probably strictly %X
between $3d^2$ and $11d^2-28d+12$.





\bibliography{my}
\bibliographystyle{abbrv}

\end{document}
