Derived equivalences of canonical covers of hyperelliptic and Enriques surfaces in positive characteristic

TL;DR

This paper proves that Fourier--Mukai partners of certain algebraic surfaces in positive characteristic are trivial, extending known results and showing that their canonical covers have no non-trivial derived equivalences.

Contribution

It establishes the triviality of Fourier--Mukai sets for canonical covers of hyperelliptic and Enriques surfaces in positive characteristic, generalizing previous characteristic-zero results.

Findings

Fourier--Mukai partners of abelian surfaces are moduli spaces of stable sheaves.

Fourier--Mukai sets of canonical covers of hyperelliptic and Enriques surfaces are trivial in characteristic > 3.

Results extend earlier work from characteristic zero to positive characteristic.

Abstract

We prove that any Fourier--Mukai partner of an abelian surface over an algebraically closed field of positive characteristic is isomorphic to a moduli space of Gieseker-stable sheaves. We apply this fact to show that the Fourier--Mukai set of canonical covers of hyperelliptic and Enriques surfaces over an algebraically closed field of characteristic greater than three is trivial. These results extend to positive characteristic earlier results of Bridgeland--Maciocia and Sosna.