  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.
