Curves in linear systems on abelian surfaces

Abstract: Let $A$ be an abelian surface. We investigate curves in a linear system on the dual abelian surface $\hat{A}$. There is an isomorphism of moduli spaces due to Yoshioka between the space $K_3(A)$ parametrizing 0-dimensional length-4 subschemes on $A$ that sum to the identity in the group law, and the space $K_{\hat{A}}(0,\hat{\ell},-1)$ parametrizing certain rank 1 torsion free sheaves supported on curves in a linear system on $\hat{A}$. Leveraging this isomorphism together with quadratic forms associated to symmetric line bundles on $A$, we develop a computational method that allows us to characterize the singularities of the curves that correspond to a finite distinguished subset of $K_3(A)$. In this talk we will describe these methods and compute an example of a curve with two nodal singularities.

This is joint work with Katrina Honigs and Peter McDonald.

algebraic geometrynumber theory

Audience: researchers in the discipline

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SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

We normally meet in-person in the indicated room. For online editions, we use Zoom and distribute the link through the mailing list. If you wish to be put on the mailing list, please subscribe to ntag-external using lists.sfu.ca

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