We study the Brauer group of an affine double plane pi : 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 left perpendicularx,yright perpendicular 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 = y(2) - p(x) is the equation of an affine hyperelliptic curve. For pi as well as the unramified part of pi, 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.
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