Nef cone and Seshadri constants on products of projective bundles over curves

Abstract

Let $X=\mathbb{P}(E_1){\times }_C\mathbb{P}(E_2)$ where $C$ is a smooth curve and let $E_1$, $E_2$ be vector bundles over $C$. In this paper, we extend the results in \cite{K-M-R} by computing the nef cone of $X$ without restriction on the rank or semistability of $E_1$ and $E_2$. We also study the Seshadri constants of ample line bundles on $X$. We calculate the Seshadri constants in some cases and give bounds in some of the remaining cases.

Final version. To appear in Journal of Ramanujan Mathematical Society