Brill-Noether theory for curves on generic abelian surfaces

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)(vertical bar H vertical bar) parametrizing curves C in the primitive linear system vertical bar H vertical bar 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.