Brill-Noether theory for curves on generic abelian surfaces

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 $V_d^r(|H|)$ 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.

22 pages. v2: minor changes addressing referee comments

References in corpus (1)

• Brill-Noether varieties of k-gonal curves