Triple solids and scrolls

Let Y be a smooth complex projective variety of dimension n ≥ 2 endowed with a finite morphism φ : Y → 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 ≤ 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 P^2 × P^1. This happens, in particular, if Y is a Fano threefold as well as if the base surface X is P^2. .