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We completely describe the Brill-Noether theory for curves in the primitivelinear system on generic abelian surfaces, in the following sense: givenintegers $d$ 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 sufficientcondition for this variety to be non-empty, and show that it is either adisjoint union of Grassmannians, or irreducible. Moreover, we show that, whennon-empty, it is of expected dimension.  This completes prior results by Knutsen, Lelli-Chiesa and Mongardi.

Brill-Noether theory for curves on generic abelian surfaces