The Brauer Group of an Affine Double Plane Associated to a Hyperelliptic Curve

TL;DR

This paper investigates the divisor class group and Brauer group of affine double planes defined by z^2 = f, analyzing cases where f factors into linear forms or defines a hyperelliptic curve, with explicit computations of related algebraic groups.

Contribution

It provides explicit calculations of divisor class groups and Brauer groups for affine double planes associated with hyperelliptic curves and ruled surfaces, extending previous understanding.

Findings

Computed divisor class groups and Brauer groups for specific affine double planes.

Identified the structure of the relative Brauer group in these cases.

Analyzed the cases where the surface is birational to a ruled surface or hyperelliptic curve.

Abstract

For an affine double plane defined by an equation of the form z^2 = f, we study the divisor class group and the Brauer group. Two cases are considered. In the first case, f is a product of n linear forms in k[x,y] and X is birational to a ruled surface P^1 x C, where C is rational if n is odd and hyperelliptic if n is even. In the second case, f is the equation of an affine hyperelliptic curve. On the open set where the cover is unramified, we compute the groups of divisor classes, the Brauer groups, the relative Brauer group, as well as all of the terms in he exact sequence of Chase, Harrison and Rosenberg.