Derived categories of torsors for abelian schemes

TL;DR

This paper investigates derived equivalences among genus 1 curves and their moduli spaces, revealing non-isomorphic examples over non-closed fields and establishing criteria for derived equivalence.

Contribution

It provides necessary and sufficient conditions for derived equivalence of genus 1 curves and explores derived categories of torsors for abelian schemes, extending understanding of twisted derived equivalences.

Findings

Existence of non-isomorphic derived equivalent genus 1 curves over non-closed fields

Criteria for derived equivalence of genus 1 curves and their moduli spaces

Partial answers to twisted derived equivalences on elliptic fibrations

Abstract

In the first part of our paper, we show that there exist non-isomorphic derived equivalent genus $1$ curves, and correspondingly there exist non-isomorphic moduli spaces of stable vector bundles on genus $1$ curves in general. Neither occurs over an algebraically closed field. We give necessary and sufficient conditions for two genus $1$ curves to be derived equivalent, and we go on to study when two principal homogeneous spaces for an abelian variety have equivalent derived categories. We apply our results to study twisted derived equivalences of the form $D^b(J,\alpha)\simeq D^b(J,\beta)$, when $J$ is an elliptic fibration, giving a partial answer to a question of C\u{a}ld\u{a}raru.