Brill-Noether theory for curves on generic abelian surfaces

TL;DR

This paper provides a complete description of the Brill-Noether theory for curves on generic abelian surfaces, detailing conditions for non-empty parameter spaces and their geometric structure, extending previous research in the field.

Contribution

It offers necessary and sufficient conditions for the non-emptiness and geometric structure of Brill-Noether varieties on curves in primitive linear systems on generic abelian surfaces.

Findings

Characterization of non-empty Brill-Noether varieties

Description of their geometric structure as Grassmannians or irreducible

Verification that these varieties have the expected dimension

Abstract

We completely describe the Brill-Noether theory for curves in the primitive linear system on generic abelian surfaces, in the following sense: given integers $d$ and $r$, consider the variety $V^r_d(|H|)$ parametrizing curves $C$ in the primitive linear system $|H|$ together with a torsion-free sheaf on $C$ of degree $d$ and $r+1$ global sections. We give a necessary and sufficient condition for this variety to be non-empty, and show that it is either a disjoint union of Grassmannians, or irreducible. Moreover, we show that, when non-empty, it is of expected dimension. This completes prior results by Knutsen, Lelli-Chiesa and Mongardi.