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

Source abstract

We study some combinatorial aspects of the fixed loci of symplectic involutions acting on hyper-Kähler varieties of Kummer type. Given an Abelian surface A with a (1,d)-polarization L, there is an isomorphism Kd−1A≅KAˆ(0,lˆ,−1) between a hyper-Kähler of Kummer type that parameterizes length-d subschemes of A and one that parameterizes degree d−1 line bundles supported on curves in |Lˆ|, where Lˆ is the dual (1,d)-polarization on Aˆ. We examine the bijection this isomorphism gives between isolated points in the fixed loci of [−1A] 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.