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

Katrina Honigs, Graham McDonald

TL;DR

This work analyzes fixed loci of symplectic involutions on hyperkähler varieties of Kummer type by exploiting a Fourier--Mukai isomorphism $\Psi$ between $K_{d-1}(A)$ and $K_{{ ext{A}}}(0,\hat{l},-1)$. It develops a theta-characteristic framework, visualized via Hudson tables, to track how isolated fixed points map under $\Psi$ and to classify their supporting curves; for odd $d$, the quantity $q(\xi)$ encodes the eigenspace in which the image curve lies. The authors provide a complete description in the fourfold case ($d=3$) showing a partition of $S'$ into $35$ fixed points into a $15$–point subset mapping to a fixed curve $C$ and a $20$–point subset mapping to nodal curves $B_x$, together with the corresponding $R$-set structure and a precise $\mathrm{Sp}(A[2])$–orbit analysis. They also present a numerical table for the KMO fixed-loci formula, together with examples in the $(1,5)$–polarized setting that reveal richer orbit structures and singularity behavior in the supporting curves. Overall, the work provides a concrete, combinatorial bridge between fixed loci, theta-characteristics, and moduli-theoretic Fourier--Mukai transforms in Kummer-type hyperkähler geometry, with practical tables that aid further computations of fixed-locus components.

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.

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

TL;DR

This work analyzes fixed loci of symplectic involutions on hyperkähler varieties of Kummer type by exploiting a Fourier--Mukai isomorphism between and . It develops a theta-characteristic framework, visualized via Hudson tables, to track how isolated fixed points map under and to classify their supporting curves; for odd , the quantity encodes the eigenspace in which the image curve lies. The authors provide a complete description in the fourfold case () showing a partition of into fixed points into a –point subset mapping to a fixed curve and a –point subset mapping to nodal curves , together with the corresponding -set structure and a precise –orbit analysis. They also present a numerical table for the KMO fixed-loci formula, together with examples in the –polarized setting that reveal richer orbit structures and singularity behavior in the supporting curves. Overall, the work provides a concrete, combinatorial bridge between fixed loci, theta-characteristics, and moduli-theoretic Fourier--Mukai transforms in Kummer-type hyperkähler geometry, with practical tables that aid further computations of fixed-locus components.

Abstract

We study some combinatorial aspects of the fixed loci of symplectic involutions acting on hyperkähler varieties of Kummer type. Given an abelian surface with a -polarization , there is an isomorphism between a hyperkähler of Kummer type that parametrizes length- subschemes of and one that parametrizes degree line bundles supported on curves in , where is the dual -polarization on . We examine the bijection this isomorphism gives between isolated points in the fixed loci of when 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.
Paper Structure (19 sections, 14 theorems, 32 equations, 2 figures, 3 tables)

This paper contains 19 sections, 14 theorems, 32 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background and notation
  3. A formula for the number of fixed loci
  4. Theta characteristics on abelian surfaces with odd degree polarizations
  5. The principally polarized case
  6. $(1,d)$-polarized surfaces with $d$ odd
  7. $2$-torsion base points of a polarization
  8. Isomorphism of moduli spaces
  9. The isomorphism
  10. The support of $\Psi(\xi)$
  11. Criteria for $(\star)$ to not be an isomorphism
  12. The fourfolds case
  13. The set $S$
  14. The set $R$
  15. The bijection $\Psi$ and supporting curves
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $q$ be a theta characteristic associated with $L$. For any $\xi\in S'$, the value of $q(\xi):=\sum_{i=1}^{d}q(u_i)\in \mathbb F_2$ determines which eigenspace of $[-1_{\hat{A}}]$ acting on $|\hat{L}|$ contains the supporting curve of $\Psi(\xi)$.

Figures (2)

  • Figure 1: On the left is a singular Kummer surface in green with a plane in red. The $16$ singular points are marked. The point in green is dual to the plane. The six points intersecting the plane are red. On the right we show a simplified diagram of this surface with the singular points labelled so the reader can see them more clearly.
  • Figure 2: The Hudson table

Theorems & Definitions (32)

  • Theorem 1.1: Prop. \ref{['determinant']}
  • Theorem 1.2: Prop. \ref{['twoorbits']}
  • Theorem 1.3: Proposition \ref{['Bx']}
  • Theorem 3.1: KMO Kamenova, Mongardi, Oblomkov
  • Theorem 3.2: Song OEIS
  • Remark 3.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 22 more