Rank 2 Sheaves on K3 Surfaces: A Special Construction

Abstract

Abstract. Let X be a K3 surface of degree 8 in P 5 with hyperplane section H. We associate to it another K3 surface M which is a double cover of P 2 ramified on a sextic curve C. In the generic case when X is smooth and a complete intersection of three quadrics, there is a natural correspondence between M and the moduli space M of rank two vector bundles on X with Chern classes c 1 = H and c 2 = 4. We build on previous work of Mukai and others, giving conditions and examples where M is fine, compact, non-empty; and birational or isomorphic to M . We also present an explicit calculation of the Fourier-Mukai transform when X contains a line and has Picard number two.

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

“…However it is not a complete intersection so the methods in this paper do not apply. We describe this case in [12].…”

“…In particular if ρ(M ) = 1 then I does not admit a section therefore the period of B x is 2. if X contains a line then X M and I lifts to a vector bundle on X × M . In this case the moduli problem is symmetric [12].…”

“…It has period 1 if there exists a class u ∈ Pic(M) which has odd intersection with the hyperplane class. This is related to a classical moduli problem as well [22], [15].…”

Brauer Groups on K3 Surfaces and Arithmetic Applications

“…We therefore obtain a degree two K3 surface S as the double cover of P 2 branched over the sextic C. We say that X and S are projectively dual varieties. Proof This is Example 0.9 of Mukai [34], and Example 2.2 in [36]; a very detailed discussion is given by Ingalls and Khalid [22]. The basic idea is as follows.…”

“…so E * can also be regarded as a quotient bundle on Gr (2,4). Restricting E * and F to X via the embedding X ⊂ Z p ∼ = Gr(2, 4) yields two stable vector bundles on X with Mukai vectors v = (2, h, 2) (see [22] for details, particularly page 450 and Corollary 3.5). Note that we had to choose an identification Z p ∼ = Gr(2, 4), but a different identification yields the same pair of bundles E * | X and F | X , up to interchanging them (the automorphism group of Gr(2, 4) has two connected components, and as homogeneous bundles, E * and F are invariant under pullbacks by automorphisms in the connected component of the identity, and interchanged by pullbacks by automorphisms in the other component).…”