Unramified Brauer classes on cyclic covers of the projective plane

Colin Ingalls; Andrew Obus; Ekin Ozman; Bianca Viray

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arXiv · 1310.8005

Unramified Brauer classes on cyclic covers of the projective plane

Colin Ingalls·Andrew Obus·Ekin Ozman·Bianca Viray

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Abstract

Let X --> P^2 be a p-cyclic cover branched over a smooth, connected curve C of degree divisible by p, defined over a separably closed field of prime-to-p characteristic. We show that all (unramified) p-torsion Brauer classes on X that are fixed by Aut(X/P^2) arise as pullbacks of certain Brauer classes on k(P^2) that are unramified away from C and a fixed line L. We completely characterize these Brauer classes on k(P^2) and relate the kernel of the pullback map to the Picard group of X. If p = 2, we give a second construction, which works over any base field of characteristic not 2, that uses Clifford algebras arising from symmetric resolutions of line bundles on C to yield Azumaya representatives for the 2-torision Brauer classes on X. We show that, when p=2 and sqrt{-1} is in our base field, both constructions give the same result.

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Colin Ingalls, Andrew Obus, Ekin Ozman, Bianca Viray. 2013-10-30. Unramified Brauer classes on cyclic covers of the projective plane. https://arxiv.org/abs/1310.8005

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