Harvard-MIT Algebraic Geometry Seminar
		       May 7, 2002 at 3:00 p.m.
			    MIT Room 4-163



		      Morphisms of hypersurfaces



			 David Sheppard (MIT)



Abstract:

Given a nonconstant morphism f:X_d --> Y_e of smooth hypersurfaces of the
indicated degrees in P^n, with n\geq 4, the Grothendieck-Lefshcetz
Theorem says that f^*O_Y(1)=O_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 P^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 P^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
--> Y_3 is a morphism of hypersurfaces in P^4 and d\leq 5, then f is
either constant or d=3 and f is an isomorphism.