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 set of Fourier–Mukai partners of a canonical cover of a hyperelliptic or Enriques surface over an algebraically closed field of characteristic greater than three is trivial. These results extend earlier results of Bridgeland–Maciocia and Sosna to positive characteristic.
Highlights
The main motivation of this paper is the recent series of results in the study of equivalences of derived categories of sheaves of smooth projective varieties over fields other than the field of complex numbers
Over finite fields, the first named author proves that the Hasse–Weil zeta function of an abelian variety, as well as of smooth varieties of dimension at most three, is unaltered under equivalences of derived categories [9,11]
Ward in his thesis [35] produces examples of genus one curves over Q admitting an arbitrary number of distinct Fourier–Mukai partners, revealing in this way consistent differences with the case of elliptic curves over C
Summary
The main motivation of this paper is the recent series of results in the study of equivalences of derived categories of sheaves of smooth projective varieties over fields other than the field of complex numbers. Lieblich and Olsson in [17] extend to positive characteristic seminal works of Mukai and Orlov concerning derived equivalences of K 3 surfaces They prove that any Fourier–Mukai partner of a K 3 surface X over an algebraically closed field of characteristic p = 2 is a moduli space of Gieseker-stable. 6 we observe that one can push the techniques of [17] a little further in order to prove that K 3 surfaces that are canonical covers of Enriques surfaces in characteristic p > 3 do not admit any non-trivial Fourier–Mukai partner This in particular extends the second part of the result of Sosna [34, Theorem 1.1] to positive characteristic.
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