Triple solids and scrolls

Abstract

Let $Y$ be a smooth projective variety of dimension $n\ge 2$ endowed with a finite morphism $\varphi :Y\to {\mathbb{P}}^n$ of degree $3$, and suppose that $Y$, polarized by some ample line bundle, is a scroll over a smooth variety $X$ of dimension $m$. Then $n\le 3$ and either $m=1$ or $2$. When $m=1$, a complete description of the few varieties $Y$ satisfying these conditions is provided. When $m=2$, various restrictions are discussed showing that in several instances the possibilities for such a $Y$ reduce to the single case of the Segre product ${\mathbb{P}}^2\times {\mathbb{P}}^1$. This happens, in particular, if $Y$ is a Fano threefold as well as if the base surface $X$ is ${\mathbb{P}}^2$.