2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72074We 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.This replacement has a more general theorem (thm 6.2) on Brauer groups of double covers of rational surfaces. Corrected some typos, updated references. To appear in Advances in MathematicsAlgebraic Geometrytext