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Register a new userFourier-Mukai partners of Enriques and bielliptic surfaces in positive characteristic
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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.
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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.
Let $V$ be a $6$-dimensional complex vector space with an involution $sigma$ of trace $0$, and let $W subseteq operatorname{Sym}^2 V^vee$ be a generic $3$-dimensional subspace of $sigma$-invariant quadratic forms. To these data we can associate an Enriques surface as the $sigma$-quotient of the complete intersection of the quadratic forms in $W$. We exhibit noncommutative Deligne-Mumford stacks together with Brauer classes whose derived categories are equivalent to those of the Enriques surfaces.
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.
Over complex numbers, the Fourier-Mukai partners of abelian varieties are well-understood. A celebrated result is Orlovs derived Torelli theorem. In this note, we study the FM-partners of abelian varieties in positive characteristic. We notice that, in odd characteristics, two abelian varieties of odd dimension are derived equivalent if their associated Kummer stacks are derived equivalent, which is Krug and Sosnas result over complex numbers. For abelian surfaces in odd characteristic, we show that two abelian surfaces are derived equivalent if and only if their associated Kummer surfaces are isomorphic. This extends the result [doi:10.1215/s0012-7094-03-12036-0] to odd characteristic fields, which solved a classical problem originally from Shioda. Furthermore, we establish the derived Torelli theorem for supersingular abelian varieties and apply it to characterize the quasi-liftable birational models of supersingular generalized Kummer varieties.
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 give a counting formula in case we have an ordinary tame untwisted K3 surface. We also show that every tame twisted Fourier Mukai partner of a K3 surface of finite height is a moduli space of twisted sheaves over it.
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