A Reider-type theorem for higher syzygies on abelian surfaces

TL;DR

This paper extends Reider-type theorems to higher syzygies on abelian surfaces using infinitesimal Newton--Okounkov bodies, confirming a conjecture for large enough polarization degree.

Contribution

It introduces a new Reider-type theorem for higher syzygies on abelian surfaces and verifies a conjecture for primitive polarization of type (1,d) with d ≥ 23.

Findings

Reider-type theorem for higher syzygies established

Conjecture of Gross and Popescu confirmed for d ≥ 23

Application of infinitesimal Newton--Okounkov bodies in syzygy theory

Abstract

Building on the theory of infinitesimal Newton--Okounkov bodies and previous work of Lazarsfeld--Pareschi--Popa, we present a Reider-type theorem for higher syzygies of ample line bundles on abelian surfaces. As an application of our methods we confirm a conjecture of Gross and Popescu on abelian surfaces with a very ample primitive polarization of type $(1,d)$, whenever $d\geq 23$.