An explicit derived equivalence of Azumaya algebras on K3 surfaces via Koszul duality

Abstract

We consider moduli spaces of Azumaya algebras on K3 surfaces and construct an example. In some cases we show a derived equivalence which corresponds to a derived equivalence between twisted sheaves. We prove if A and A are Morita equivalent Azumaya algebras of degree r then 2r divides c 2 (A) − c 2 (A ). In particular this implies that if A is an Azumaya algebra on a K3 surface and c 2 (A) is within 2r of its minimal bound then the moduli stack of Azumaya algebras with the same underlying gerbe, if non-empty, is a proper algebraic space.

Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems

“…The aim of this section is provide a proof for quadric fibrations with simple degeneration. For quadric surface bundles, this fact has been noted in [69,Lemma 4.2] and [51,Thm. 4.5].…”

“…In particular, projecting T x Q ∩ Q away from x, we find Q ′ as a quadric hypersurface in the target P n−3 , which we call the quadric reduction by hyperbolic splitting associated to Q and x. Recently, this construction was considered [51] in the study of nets of quadrics over P 2 and associated K3 surfaces. Conversely, consider a quadric Q ′ ⊂ P n−3 and an embedding of P n−3 as a hyperplane of P n−2 .…”

“…It would be interesting to give a direct explicit isomorphism between the Severi-Brauer scheme of B 0 and the connected part of the Stein factorization of ΛG(Q). See [51,Thm. 4.5] for an approach in the case of surface bundles.…”

Contractions of hyper-Kähler fourfolds and the Brauer group

“…The rulings of the quadric surface defined by M x are the fibers of the conic bundle E over the points in π −1 (x) for x ∈ P 2 . Following [IK,, [Ku1,§3], the (sheaf of) even Clifford algebra(s) of M is the sheaf…”

“…An even theta characteristic on C defines a quadratic form q : O 6 → O(1) on P 2 ([IK, Theorem 4.2, Proposition 3.3]. There is an Azumaya algebra A of rank r = 2 4 = 16 on the K3 double plane K with Cl 0 (q) = π * A where Cl 0 (q) is the even Clifford algebra of q ( [IK,, [Ku1,§3]). By construction, Cl 0 (q) has a filtration with quotients O,…”