Theta characteristics and the fixed locus of [-1] on some varieties of Kummer type

Abstract

We study some combinatorial aspects of the fixed loci of symplectic involutions acting on hyperkähler varieties of Kummer type. Given an abelian surface $A$ with a $(1,d)$-polarization $L$, there is an isomorphism $K_{d-1}A\cong K_{\hat{A}}(0,\hat{l},-1)$ between a hyperkähler of Kummer type that parametrizes length-$d$ subschemes of $A$ and one that parametrizes degree $d-1$ line bundles supported on curves in $|\hat{L}|$, where $\hat{L}$ is the dual $(1,d)$-polarization on $\hat{A}$. We examine the bijection this isomorphism gives between isolated points in the fixed loci of $[-1_A]$ when $d$ is odd, which has a combinatorics related to theta characteristics. Along the way, we give a table of numerical values for a formula of Kamenova, Mongardi, and Oblomkov counting the number of components of a symplectic involution acting on a Kummer-type variety.

To appear in Kyoto J. Math. 22 pages, 2 figures