Morphisms from a very general hypersurface

Abstract

Let $X$ be a very general hypersurface of degree $d$ in the projective $(n+1)$-space with $n\ge 3$, and $f:X\to Y$ a non-birational surjective morphism to a normal projective variety $Y$. We first prove that $Y$ is a klt Fano variety if $\mathrm{deg}f\ge C$ for some constant $C=C(n,d)$ depending only on $n$ and $d$. Next we prove an optimal upper bound $\mathrm{deg}f\le \mathrm{deg}X$ provided that $Y$ is factorial, $\mathrm{deg}f$ is prime and $\mathrm{deg}f\ge E(n)$ for some constant $E(n)$ (with $E(n)=n(n+1)$ when $Y$ is smooth). As a corollary, we show that $Y\cong {\mathbf{P}}^n$ under some conditions on $Y$ and $\mathrm{deg}f$.

minor changes; slightly re-ordered the sections; Pure and Applied Mathematics Quarterly (to appear, James McKernan's issue)