Higher syzygies on abelian surfaces

TL;DR

This paper extends the understanding of higher syzygies on abelian surfaces using infinitesimal Newton-Okounkov bodies and shows their dependence on Seshadri constants, improving bounds for polarization.

Contribution

It introduces a new approach linking higher syzygies to Seshadri constants and extends previous results with novel bounds for polarized abelian surfaces.

Findings

Higher syzygies are determined by Seshadri constants for large L^2.

Extension of Lazarsfeld-Pareschi-Popa results to abelian surfaces.

Improved lower bounds for L^2 in higher syzygies of polarized abelian surfaces.

Abstract

Based on the theory of an infinitesimal Newton-Okounkov body, we extend the results of Lazarsfeld-Pareschi-Popa on abelian surfaces. Moreover, we show that the higher syzygies of $(X,L)$ are completely determined by its Seshadri constant when $L^{2}$ is large. As an application, we improve the existing lower bound of $(L^{2})$ for higher syzygies of a polarized abelian surface $(X,L)$.