Rank Two Sheaves on K3 Surfaces: A Special Construction

TL;DR

This paper explores the relationship between certain K3 surfaces and moduli spaces of rank two vector bundles, providing explicit constructions, conditions for their properties, and calculations of Fourier-Mukai transforms.

Contribution

It introduces a special construction linking K3 surfaces of degree 8 with associated surfaces and moduli spaces, extending previous work with explicit examples and calculations.

Findings

M' can be fine, compact, and non-empty under certain conditions.

M' is often birational or isomorphic to the associated K3 surface M.

Explicit Fourier-Mukai transform calculations are provided for specific cases.

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.