# Nef cones of projective bundles over surfaces and Seshadri constants

## TL;DR

This paper characterizes the nef cone of divisors on projective bundles over surfaces, computes Seshadri constants for certain vector bundles, and advances understanding of the geometry of these bundles and their positivity properties.

## Contribution

It provides explicit descriptions of nef cones for projective bundles over specific surfaces and calculates Seshadri constants for semistable ample bundles with vanishing discriminant.

## Key findings

- Explicit nef cone descriptions for certain projective bundles
- Calculated Seshadri constants at special points
- Enhanced understanding of positivity in vector bundles

## Abstract

In this article, we give a description of the closed cone of curves of the projective bundle $\mathbb{P}(E)$ over a smooth projective variety $X$. Using duality, we then calculate the nef cone of divisors in $\mathbb{P}(E)$ over some special surfaces $X$ and for some special bundles on $X$. As an application, we also calculate the Seshadri constants of semistable ample vector bundles with vanishing discriminant on some special ruled surfaces at special points.