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arXiv · 1704.03546

Brill-Noether theory for curves on generic abelian surfaces

Arend Bayer·Chunyi Li
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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.

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Arend Bayer, Chunyi Li. 2018-05-14. Brill-Noether theory for curves on generic abelian surfaces. https://arxiv.org/abs/1704.03546

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