Derived Partners of Enriques Surfaces

TL;DR

This paper constructs noncommutative stacks and sheaves of Azumaya algebras to study derived categories of Enriques surfaces, providing new geometric insights and realizations of Brauer classes.

Contribution

It introduces a novel noncommutative geometric framework for Enriques surfaces using Azumaya algebras and stacks, extending previous theoretical results.

Findings

Equivalence of derived categories between Enriques surfaces and noncommutative stacks

Construction of geometric realizations of Brauer classes

New insights into the structure of Enriques surfaces via noncommutative geometry

Abstract

Let $V$ be a $6$-dimensional complex vector space with an involution $\sigma$ of trace $0$, and let $W \subset \Sym^2 V^\vee$ be a generic $3$-dimensional subspace of $\sigma$-invariant quadratic forms. To these data we can associate an Enriques surface as the $\sigma$-quotient of the complete intersection of the quadratic forms in $W$. We exhibit noncommutative Deligne-Mumford stacks together with sheaves of Azumaya algebras on them whose derived categories are equivalent to those of the Enriques surfaces. This provides a more accessible treatment of of Theorem 6.16 in https://www.ams.org/journals/jams/2021-34-02/S0894-0347-2021-00963-3/ .. We also construct geometric realizations of the Brauer classes coming from these sheaves of Azumaya algebras which may be of independent interest.