Pure and Applied Mathematics Quarterly

Volume 13 (2017)

Number 1

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

DOI: http://dx.doi.org/10.4310/PAMQ.2017.v13.n1.a2

Authors

Arend Bayer (School of Mathematics and Maxwell Institute, University of Edinburgh, Scotland, United Kingdom)

Chunyi Li (School of Mathematics and Maxwell Institute, University of Edinburgh, Scotland, United Kingdom)

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 $V^r_d (\lvert H \rvert)$ 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