Theta characteristics and the fixed locus of [-1] on some varieties of Kummer type
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866913751975854080 |
|---|---|
| author | Honigs, Katrina McDonald, Graham |
| author_facet | Honigs, Katrina McDonald, Graham |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_13129 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Theta characteristics and the fixed locus of [-1] on some varieties of Kummer type Honigs, Katrina McDonald, Graham Algebraic Geometry 14J50, 14J42, 14K25 (primary), 15A63 (secondary) 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. |
| title | Theta characteristics and the fixed locus of [-1] on some varieties of Kummer type |
| topic | Algebraic Geometry 14J50, 14J42, 14K25 (primary), 15A63 (secondary) |
| url | https://arxiv.org/abs/2307.13129 |