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 ${\rm deg} \, f \ge C$ for some constant $C = C(n, d)$ depending only on $n$ and $d$. Next we prove an optimal upper bound ${\rm deg} \, f \le {\rm deg} \, X$ provided that $Y$ is factorial, ${\rm deg} \, f$ is prime and ${\rm deg} \, f \ge E(n)$ for some constant $E(n)$ (with $E(n) = n(n+1)$ when $Y$ is smooth). As a corollary, we show that $Y\cong {\bf P}^n$ under some conditions on $Y$ and ${\rm deg} \, f$.