The Brauer group of an affine double plane associated to a hyperelliptic curve. Issue 4 (3rd April 2017)

• Record Type: Journal Article
• Authors: Ford, Timothy J.
• Is Part Of: Communications in algebra. Volume 45:Issue 4(2017)
• Page Start: 1416
• Page End: 1442
• Publication Date: 2017-04-03
• Subjects: Algebraic surface -- Brauer group -- class group; 16K50 -- Secondary: 14F22 -- 14J26 -- 14C20; Algebra -- Periodicals; 512.005
• DOI: 10.1080/00927872.2016.1175608
• Languages: English
• ISSNs: 0092-7872

Abstract

ABSTRACT: We study the Brauer group of an affine double plane π : X → 𝔸 2 defined by an equation of the form z 2 = 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 ℙ 1 × C, where C is rational if n is odd and hyperelliptic if n is even. In the second case, f = y 2 − 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.