Nef cones of projective bundles over surfaces and Seshadri constants.

In this article, we give a description of the closed cone of curves of the projective bundle $\mathbb{P}(E)$ over a smooth projective surface $X$. Using this, we calculate the nef cone of $\mathbb{P}(E)$ over $X$ in some cases under some suitable assumptions on $X$ as well as on the vector bundle $E$. We give a necessary and sufficient condition for ampleness of semistable vector bundles with vanishing discriminant on a smooth projective variety $X$. We also calculate the Seshadri constants of ample vector bundles in the following cases : (1) for a completely decomposable ample vector bundles on $\mathbb{P}^2$ at a closed point in $\mathbb{P}^2$, (2) for a semistable ample vector bundles with vanishing discriminant on some special ruled surfaces at special points, and in all other cases, we give bounds on Seshadri constants of any ample vector bundles on these spaces.