Unramified Brauer Classes on Cyclic Covers of the Projective Plane

Abstract

Let ${X} \rightarrow \mathbb{P}^{2}$ be a p-cyclic cover branched over a smooth, connected curve C of degree divisible by p, defined over a separably closed field of characteristic diffierent from p. We show that all (unramified) p-torsion Brauer classes on X that are fixed by Aut$({X}/\mathbb{P}^{2})$ arise as pull-backs of certain Brauer classes on ${\rm{k}}(\mathbb{P}^{2})$ that are unramified away from C and a fixed line L. We completely characterize these Brauer classes on ${\rm{k}}(\mathbb{P}^{2})$ 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 Clifiord algebras arising from symmetric resolutions of line bundles on C to yield Azumaya represen- tatives for the 2-torsion Brauer classes on X. We show that when $\sqrt{-1}$ is in our base field, both constructions give the same result.

Mathematics Subject Classiffication (2010). Primary: 14F22; Secondary: 12G05, 14J28, 14J50, 15A66, 16K50.

With an Appendix by Hugh Thomas