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

Abstract

Let $L$ be a very ample line bundle on a smooth complex projective variety $X$ of dimension $\geq 6$. We classify the polarized manifolds $(X, L)$ such that there exists a smooth member $A$ of $|L|$ endowed with a branched covering of degree four $\pi \colon A \rightarrow \mathbf{P}^{n}$. The cases of $\deg \pi =2$ and $3$ are already studied by Lanteri-Palleschi-Sommese. Recently the case of $\deg \pi =5$ is studied by Amitani.