Non-Fine Moduli Spaces of Sheaves on K3 Surfaces

Details Non-Fine Moduli Spaces of Sheaves on K3 Surfaces Non-Fine Moduli Spaces of Sheaves on K3 Surfaces

2001-08-27

arxiv.org/abs/math/0108180v1

Mathematics

Algebraic Geometry

22 pages, LaTeX 2e

Scientific paper

In general, if M is a moduli space of stable sheaves on X, there is a unique alpha in the Brauer group of M such that a pi_M^* alpha^{-1}-twisted universal sheaf exists on X times M. In this paper we study the situation when X and M are K3 surfaces, and we identify alpha in terms of Mukai's map between the cohomology of X and of M (defined by means of a quasi-universal sheaf). We prove that the derived category of sheaves on X and the derived category of alpha-twisted sheaves on M are equivalent. This suggests a conjecture which describes, in terms of Hodge isometries of lattices, when derived categories of twisted sheaves on two K3 surfaces are equivalent. If proven true, this conjecture would generalize a theorem of Orlov and a recent result of Donagi-Pantev.

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