Derived equivalences of canonical covers of hyperelliptic and Enriques surfaces in positive characteristic

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 Fourier-Mukai set of canonical 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.

Derived isogenies and isogenies for abelian surfaces

“…For untwisted case, i.e. X = X and Y = Y , this is exactly [18,Proposition 3.1]. Here we extend it to the twisted case.…”

On the Brauer group of bielliptic surfaces (with an appendix by Jonas Bergström and Sofia Tirabassi)

“…We believe that many of these topics might be well known by experts, but we were not able to find a rigorous literature, thus we wrote this for the reader convenience. In the first part of this paragraph, we will follow closely the narrative of [HLT19]. Let A be an elliptic curve over C with identity element p 0 , then there is a lattice Λ such that A ≃ C/Λ.…”

Supersingular Irreducible Symplectic Varieties

“…As in the proof of Theorem 4.10, up to an algebraically closed base field extension κ k, we can take a projective lift (S, H, L, E) of the 4-tuple (S κ , H, L, E) over some discrete valuation ring W ′ of characteristic zero and H, L, E ∈ Pic(S W ′ ). By considering the relative moduli space of stable sheaves, these data gives rise to a lift (36) M H (S, v) → Spec W ′ of M H (S κ , v) over W ′ . Let K ′ be the fraction field of W ′ and S K ′ be the generic fiber of S → Spec W ′ .…”

“…When A is the product of elliptic curves, we know that A ′ has to be isomorphic to A (cf. [36]). In general, the supersingular abelian surface A ′ is expected to be isomorphic to either A or its dual A ∨ (cf.…”