Criteria for the ampleness of certain vector bundles

TL;DR

This paper establishes criteria for the ampleness of vector bundles over surfaces, extending previous results to higher ranks and providing counterexamples to demonstrate the sharpness of these criteria.

Contribution

It generalizes a known theorem for rank two bundles to higher ranks and offers counterexamples to show the criteria are optimal.

Findings

Vector bundles are ample under certain restrictions and numerical conditions.

The criteria are proven to be sharp through counterexamples.

The work extends existing theorems to higher-rank vector bundles.

Abstract

We prove that certain vector bundles over surfaces are ample if they are so when restricted to divisors, certain numerical criteria hold, and they are semistable (with respect to $\det(E)$). This result is a higher-rank version of a theorem of Schneider and Tancredi for vector bundles of rank two over surfaces. We also provide counterexamples indicating that our theorem is sharp.