We prove that a twisted Enriques (respectively, untwisted bielliptic) surface over an algebraically closed field of positive characteristic at least 3 (respectively, at least 5) has no non-trivial Fourier-Mukai partners.
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 canonic
al 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.
