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