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Pure and Applied Mathematics Quarterly
Volume 13 (2017)
Special Issue in Honor of Yuri Manin: Part 1 of 2
Guest Editors: Lizhen Ji, Kefeng Liu, Yuri Tschinkel, and Shing-Tung Yau
Brill–Noether theory for curves on generic abelian surfaces
Pages: 49 – 76
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 $V^r_d (\lvert H \rvert)$ parametrizing curves $C$ in the primitive linear system $(\lvert H \rvert)$ 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.
Both authors are supported by ERC starting grant no. 337039“WallXBirGeom”.
Received 10 July 2017
Published 14 September 2018