A birational Torelli theorem with a Brauer class

TL;DR

This paper establishes a Torelli-type theorem linking the birational geometry of moduli spaces of vector bundles with the isomorphism class of the underlying genus 2 curves, using Brauer classes and Jacobian properties.

Contribution

It proves that birational maps preserving Brauer classes between certain moduli spaces imply the isomorphism of the underlying curves, extending Torelli theorems to these moduli spaces.

Findings

Birational maps preserving Brauer classes imply Jacobian isomorphism.

Curves without real multiplication are isomorphic if their moduli spaces are birationally equivalent.

Results extend to Kummer surfaces and quadratic line complexes.

Abstract

Let $\text{M}_C( 2, \mathcal{O}_C) \cong \mathbb{P}^3$ denote the coarse moduli space of semistable vector bundles of rank $2$ with trivial determinant over a smooth projective curve $C$ of genus $2$ over $\mathbb{C}$. Let $\beta_C$ denote the natural Brauer class over the stable locus. We prove that if $f^*( \beta_{C'}) = \beta_C$ for some birational map $f$ from $\text{M}_C( 2, \mathcal{O}_C)$ to $\text{M}_{C'}( 2, \mathcal{O}_{C'})$, then the Jacobians of $C$ and of $C'$ are isomorphic as abelian varieties. If moreover these Jacobians do not admit real multiplication, then the curves $C$ and $C'$ are isomorphic. Similar statements hold for Kummer surfaces in $\mathbb{P}^3$ and for quadratic line complexes.