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

2023/07/24 by Honigs, Katrina, McDonald, Graham
#14J42 #14J50 #14K25 (primary) #15A63 (secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2307.13129

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-1A≅ K_A(0,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 |L|, where L is the dual (1,d)-polarization on 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.

Related

• Kummer Surfaces, Isogenies and Theta Functions
• Theta groups and projective models of hyperkähler varieties
• The loci of abelian varieties with points of high multiplicity on the theta divisor
• Fixed loci of symplectic automorphisms of K3^[n] and n-Kummer type manifolds
• Symplectic involutions of K3^[n] type and Kummer n type manifolds