Remarks on ample vector bundles of curve genus two

Abstract.

Let $\mathcal{E}$ be an ample vector bundle of rank n – 1 on a smooth complex projective variety X of dimension n≥ 3 such that X is a $\mathbb {P}^{n-1}$-bundle over $\mathbb {P}^1$ and that $\mathcal{E}_{F}\, \cong\, \mathcal {O}_\mathbb {P}(2)\, \oplus\, \mathcal {O}_ \mathbb {P}(1)^{\oplus\, (n-2)}$ for any fiber F of the bundle projection $X\, \to \mathbb {P}^1$. The pairs $({\it X}, \mathcal{E})$ with $g({\it X}, \mathcal{E})$ = 2 are classified, where $g({\it X}, \mathcal{E})$ is the curve genus of $({\it X}, \mathcal{E})$. This allows us to improve some previous results.