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

ABSTRACT

We 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.

KEYWORDS:

• Algebraic surface
• Brauer group
• class group

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16K50
• Secondary: 14F22
• 14J26
• 14C20

Acknowledgments

The author is grateful to the referee for helpful suggestions that led to significant improvements.