Dominant rational maps from a very general hypersurface in the projective space

Yongnam Lee, De-Qi Zhang

Published 2019-08-19Version 1

In this paper we study dominant rational maps from a very general hypersurface $X$ of degree at least $n+3$ in the projective $(n+1)$-space ${\mathbb P}^{n+1}$ to smooth projective $n$-folds $Y$. Based on Lefschetz theory, Hodge theory, and Cayley-Bacharach property, we prove that there is no dominant rational map from $X$ to $Y$ unless $Y$ is uniruled if the degree of the map is a prime number. Furthermore, we prove that $Y$ is rationally connected when $n=3$ and the degree of the map is a prime number.