Seshadri constants on abelian surfaces

TL;DR

This paper presents a comprehensive method to compute Seshadri constants on any complex abelian surface, enabling detailed analysis of their structure and complexity, which varies significantly even in low Picard number cases.

Contribution

It introduces an effective algorithm for computing Seshadri constants and their curves on all abelian surfaces, extending understanding beyond previously known special cases.

Findings

Computed Seshadri constants for all nef line bundles on abelian surfaces.

Developed a method to analyze the complexity of Seshadri functions.

Found that Seshadri function complexity can be as high as the Cantor function in Picard number two cases.

Abstract

So far, Seshadri constants on abelian surfaces are completely understood only in the cases of Picard number one and on principally polarized abelian surfaces with real multiplication. Beyond that, there are partial results for products of elliptic curves. In this paper, we show how to compute the Seshadri constant of any nef line bundle on any abelian surface over the complex numbers. We develop an effective algorithm depending only on the basis of the N\'eron-Severi group to compute not only the Seshadri constants but also the numerical data of their Seshadri curves. Access to the Seshadri curves allows us to plot Seshadri functions and better understand their structure. We show that already in the case of Picard number two the complexity of Seshadri functions can vary to a great degree. Our results indicate that aside from finitely many cases the complexity of the Seshadri function is at least as high as in the Cantor function.