Fourier-Mukai partners of canonical covers of bielliptic and Enriques surfaces

TL;DR

This paper investigates the derived categories of canonical covers of bielliptic and Enriques surfaces, establishing uniqueness and limitations of Fourier-Mukai partners, with implications for related geometric structures.

Contribution

It proves the non-existence of non-trivial Fourier-Mukai partners for Enriques surfaces' canonical covers and limits the number for bielliptic surfaces, extending to related geometric objects.

Findings

Canonical cover of Enriques surface has no non-trivial Fourier-Mukai partners.

Canonical cover of bielliptic surface has at most one Fourier-Mukai partner.

No exceptional or spherical objects in the derived category of a bielliptic surface.

Abstract

We prove that the canonical cover of an Enriques surface does not admit non-trivial Fourier-Mukai partners. We also show that the canonical cover of a bielliptic surface has at most one non-isomorphic Fourier-Mukai partner. The first result is then applied to birational Hilbert schemes of points and the second to birational generalised Kummer varieties. An appendix establishes that there are no exceptional or spherical objects in the derived category of a bielliptic surface.