The brauer group and ramified double covers of surfaces

Abstract

Let Z be a nonsingular plane curve of even degree and U = P ² — Z. Let : X —> P ² be tbe double cover ramified over Z and V = ^—1(U). It is shown that the kernel of the restriction map on Brauer groups^" : B(U) —> B(V), is isomorphic to Z/2(²) where ρ— 2 < r <ρ— 1, p being the Picard number of X and if ^" : Pic P² —> Pic X is an isomorphism, exactly half of the algebra classes of order 2 in B(X) are restrictions of algebra classes of order 2 in B(U) whose ramification has been split by V. The kernel of the corestriction map 2 B(V) — ₂B(U) is shown to be the subgroup consisting of elements fixed by the Galois group.