Some Remarks on Brauer Groups of K3 Surfaces

We discuss the geometry of the genus one fibrations associated to an elliptic fibration on a K3 surface. We show that the two-torsion subgroup of the Brauer group of a general elliptic fibration is naturally isomorphic to the two-torsion of the Jacobian of a curve associated to the fibration. We remark that this is related to Recillas’ trigonal construction. Finally we discuss the two-torsion in the Brauer group of a general K3 surface with a polarisation of degree two. Elements of the Brauer group of a projective variety have various geometric incarnations, like Azumaya algebras and Severi-Brauer varieties. Moreover, an element α in the Brauer group of a projective K3 surface X determines a polarized Hodge substructure T<α> of the transcendental lattice TX of X. We are interested in finding geometric realizations of the Hodge structure T<α> and in relating such a realization to the other incarnations. In this paper we collect some examples and add a few new results. Recently, there has been much interest in Brauer groups of K3 surfaces ([C2], [DP], [Wi]), but our aim is much more modest. For certain elements α ∈ Br(X) the Hodge structure T<α> is the transcendental lattice of another K3 surface Y . The surface Y is not necessarily unique. Two K3 surfaces with isomorphic transcendental lattice are said to be Fourier-Mukai partners. Results from Mukai and Orlov show that if Z is a K3 surface then TY and TZ are Hodge isometric if and only if the bounded derived categories of coherent sheaves on Y and Z are isomorphic. The number of Fourier-Mukai partners of a given K3 surface is determined in [HLOY], see also [St]. In sections 1 and 2 we briefly recall some basic definitions. In section 3 we consider the case when the K3 surface X has an elliptic fibration (with a section). In that case T<α> is always the transcendental lattice of a K3 surface Y , but Y is far from unique in general. These surfaces are studied in section 4. They have a fibration in genus one curves (without a section). In fact, any pair of elements ±α ∈ Br(X) determines a unique fibration in genus one curves on such a surface. Somewhat surprisingly, Y often has more than one genus one fibration. We use wellknown results on K3 surfaces to determine the number of these surfaces, their automorphisms and the genus one fibrations. In Example 4.9 we work out the numerology for certain cyclic subgroups of Br(X). In section 5 we consider projective models for some genus one fibrations. In section 6 we prove that the two-torsion in the Brauer group of a general elliptic fibration X → P is naturally isomorphic to the two-torsion in the Jacobian of a certain curve in X. We observe that the pairs X, Y , where Y is the surface associated to an element α of order two in Br(X), are closely related to Recillas’ trigonal construction. This construction is a bijection between trigonal curves with a point of order two and tetragonal curves. In section 7 we remark that classical formulas allow one to give the conic bundle, and hence the Azumaya algebra, on X associated to α.