A Fourier-Mukai Approach to the Enumerative Geometry of Principally Polarized Abelian Surfaces

TL;DR

This paper employs Fourier-Mukai techniques to classify jumping schemes of twisted ideal sheaves on abelian surfaces, with applications to enumerative geometry and constraints on genus 5 curves.

Contribution

It introduces a novel Fourier-Mukai approach to classify loci of twisted ideal sheaves on abelian surfaces and applies this to geometric problems.

Findings

Classification of jumping schemes for twisted ideal sheaves

No smooth genus 5 curve on such a surface contains a g^1_3

Explicit description of singular divisors in |2l|

Abstract

We study twisted ideal sheaves of small length on an irreducible principally polarized abelian surface (T,l). Using Fourier-Mukai techniques we associate certain jumping schemes to such sheaves and completely classify such loci. We give examples of applications to the enumerative geometry of T and show that no smooth genus 5 curve on such a surface can contain a g^1_3. We also describe explicitly the singular divisors in the linear system |2l|.