Triple solids and scrolls

Abstract: Let $Y$ be a smooth projective variety of dimension $n \geq 2$ endowed with a finite morphism $\phi:Y \to \mathbb P^n$ 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 $n \leq 3$ 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 $\mathbb P^2$.