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Classifying elements of the Brauer group of a variety X over a p-adic field according to the p-adic accuracy needed to evaluate them gives a filtration on Br X. We show that, on the p-torsion, this filtration coincides with a modified version of that defined by Katos Swan conductor, and that the refined Swan conductor controls how the evaluation maps vary on p-adic discs, giving a geometric characterisation of the refined Swan conductor. We give applications to the study of rational points on varieties over number fields.
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Let $k$ be a field and $X/k$ be a smooth quasiprojective orbifold. Let $Xto underline{X}$ be its coarse moduli space. In this paper we study the Brauer group of $X$ and compare it with the Brauer group of the smooth locus of $underline{X}$.
Let $k$ be a field finitely generated over the finite field $mathbb F_p$ of odd characteristic $p$. For any K3 surface $X$ over $k$ we prove that the prime to $p$ component of the cokernel of the natural map $Br(k)to Br(X)$ is finite.
Transcendental Brauer elements are notoriously difficult to compute. Work of Wittenberg, and later, Ieronymou, gives a method for computing 2-torsion transcendental classes on surfaces that have a genus 1 fibration with rational 2-torsion in the Jacobian fibration. We use ideas from a descent paper of Poonen and Schaefer to remove this assumption on the rational 2-torsion.
We show that the Brauer group of any moduli space of stable pairs with fixed determinant over a curve is zero.
We prove that a smooth proper universally CH_0-trivial variety X over a field k has universally trivial Brauer group. This fills a gap in the literature concerning the p-torsion of the Brauer group when k has characteristic p.
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