Triple solids and scrolls

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