The basepoint-freeness threshold of a very general abelian surface

TL;DR

This paper computes the basepoint-freeness threshold for very general abelian surfaces of Picard rank 1 using cohomological rank functions and Bridgeland stability, revealing new insights into their syzygies.

Contribution

It introduces explicit calculations of cohomological rank functions for these surfaces and links them to Bridgeland stability to advance understanding of their syzygies.

Findings

Explicit basepoint-freeness thresholds computed.

Connections established between cohomological functions and stability.

New information on syzygies of polarized abelian surfaces.

Abstract

For abelian surfaces of Picard rank 1, we perform explicit computations of the cohomological rank functions of the ideal sheaf of one point, and in particular of the basepoint-freeness threshold. Our main tool is the relation between cohomological rank functions and Bridgeland stability. In virtue of recent results of Caucci and Ito, these computations provide new information on the syzygies of polarized abelian surfaces.