The Brauer group of an affine double plane associated to a hyperelliptic curve

Abstract

ABSTRACTWe study the Brauer group of an affine double plane π:X→𝔸2 defined by an equation of the form z2 = f in two separate cases. In the first case, f is a product of n linear forms in k[x,y] and X is birational to a ruled surface ℙ1×C, where C is rational if n is odd and hyperelliptic if n is even. In the second case, f = y2−p(x) is the equation of an affine hyperelliptic curve. For π as well as the unramified part of π, we compute the groups of divisor classes, the Brauer groups, the relative Brauer groups, and all of the terms in the sequences of Galois cohomology.