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

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.