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{\bigboldfont
Harvard-M.I.T. Algebraic Geometry Seminar}

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{\large \bf
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MORPHISMS OF HYPERSURFACES
}

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{\bf 
DAVID SHEPPARD
}

Massachusetts Institute of Technology

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{\bf \textsc{Abstract:} }

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Given a nonconstant morphism $f:X_d \to Y_e$ of smooth hypersurfaces
of the indicated degrees in $\bP^n$, with $n\geq 4$, the
Grothendieck-Lefshcetz Theorem says that $f^*\cO_Y(1)=\cO_X(m)$ for
some positive integer $m$.  What can we say about $m$ in terms of $d$
and $e$?

\

One possibility is that $e$ divides $d$ and $m=d/e$.  This simply
means that $f$ extends to a morphism on all of $\bP^n$, and $X$ is the
preimage of $Y$ under this extended morphism.  To my knowledge, this
is the only possibility.  We will enumerate some cases where we can
show that $m=d/e$ is indeed the only possibility for $m$.

\

First we will show that $m$ is bounded in terms of $d$ and $e$.  Then
we will extend $f$ to a rational map on $\bP^n$ and consider the
preimage of $Y$, which is $X$ plus some other hypersurface $H$.  In
case $m\neq d/e$, {i.e.\ }$H$ is not empty, we will analyze the rational
map from $H$ to $Y$ and derive a contradiction in some cases.  In
particular, we will show that if $f:X_d \to Y_3$ is a morphism of
hypersurfaces in $\bP^4$ and $d\leq 5$, then $f$ is either constant or
$d=3$ and $f$ is an isomorphism.


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%%% Date goes here, and location (Harvard Rm 507 or MIT Room 4--163)
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{\large
May 7, 2002

3:00 p.m.

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MIT Room 4-163
}
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