Derived equivalences of gerbey curves

TL;DR

This paper investigates derived equivalences between certain algebraic stacks over genus 1 curves, developing a new theory of integral transforms to determine when two stacky genus 1 curves are derived equivalent.

Contribution

It introduces a novel framework of integral transforms for algebraic stacks and applies it to classify derived equivalences of stacky genus 1 curves.

Findings

Characterization of derived equivalences via integral transforms

Conditions under which Picard stacks are derived equivalent

Answers to when Picard stacks are inverse or iterated equivalents

Abstract

We study derived equivalences of certain stacks over genus $1$ curves, which arise as connected components of the Picard stack of a genus $1$ curve. To this end, we develop a theory of integral transforms for these algebraic stacks. We use this theory to answer the question of when two stacky genus $1$ curves are derived equivalent. We use integral transforms and intersection theory on stacks to answer the following questions: if $C'=Pic^d(C)$, is $C=Pic^f(C')$ for some integer $f$? If $C'=Pic^d(C)$ and $C''=Pic^f(C')$, then is $C''=Pic^g(C)$ for some integer $g$?