In this article, we prove that a tame twisted K3 surface over an algebraically closed field of positive characteristic has only finitely many tame twisted Fourier-Mukai partners and we also give a counting formula for the case of an ordinary tame untwisted K3 surface.

Kawamata has shown that the quasi-Albanese map of a quasi-projective variety with log-irregularity equal to the dimension and log-Kodaira dimension 0 is birational. In this note, we show that under these hypotheses the quasi-Albanese map is proper in codimension 1 as conjectured by Iitaka. 

Let V be a smooth quasi-projective complex surface such that the first three logarithmic plurigenera 𝑃₁ (𝑉), 𝑃₂ (𝑉)and 𝑃₃ (𝑉) are equal to 1 and the logarithmic irregularity 𝑞(𝑉) is equal to 2. We prove that the quasi-Albanesemorphism 𝑎_𝑉 : 𝑉 → 𝐴(𝑉) is birational and there exists a finite set S such that 𝑎𝑉 is proper over 𝐴(𝑉) \ 𝑆, thusgiving a sharp effective version of a classical result of Iitaka [12].

We give a notion of ordinary Enriques surfaces and their canonical lifts in any positive characteristic, and we prove Torelli-type results for this class of Enriques surfaces.

We provide explicit generators of the torsion of the second cohomology of bielliptic surfaces, and we use this to study the pullback map between the Brauer group of a bielliptic surface and that of its canonical cover.

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 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.

We show that an inequality, proven by Küronya-Pintye, which governs the behavior of the log-canonical threshold of an ideal over Pn and that of its Castelnuovo-Mumford regularity, can be applied to the setting of principally polarized abelian varieties by substituting the Castelnuovo-Mumford regularity with Θ-regularity of Pareschi-Popa.