Projective manifolds with hyperplane sections being five-sheeted covers of projective space

Abstract

Let $L$ be a very ample line bundle on a smooth complex projective variety $X$ of dimension $\ge 7$. We classify the polarized manifolds $(X,L)$ such that there exists a smooth member $A$ of $\left|L\right|$ endowed with a branched covering of degree five $\pi :A\to {\mathbf{P}}^n$. The cases of $\mathrm{deg}\pi =2$ and $3$ are already studied by Lanteri-Palleschi-Sommese.