The Brauer group of an affine rational surface with a non-rational singularity

Abstract

The object of study is the family of normal affine algebraic surfaces defined by equations of the form zn=(y−a1x)⋯(y−anx)(x−1). Each surface X in this family is rational and contains a non-rational singularity. Using an explicit resolution of the singularity, many computations involving Weil divisors and Azumaya algebras on X are completely carried out. The Picard group and Brauer group are shown to depend in subtle ways on the values a1,…,an. For an odd prime n, and for a general choice of X, the Picard group and the Brauer group are computed.