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

TL;DR

This paper explores the combinatorial structure of fixed loci of symplectic involutions on Kummer-type hyperk"ahler varieties, revealing connections to theta characteristics and providing a numerical table for component counts.

Contribution

It establishes a bijection between fixed points of involutions on Kummer varieties and theta characteristics, and offers new numerical data on involution components.

Findings

Bijection between fixed points and theta characteristics for odd d

Numerical table for involution component counts

Connection between involution fixed loci and combinatorics of theta characteristics

Abstract

We study some combinatorial aspects of the fixed loci of symplectic involutions acting on hyperk\"ahler 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 hyperk\"ahler 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.