The unramified Brauer groups of normic bundles

TL;DR

This paper develops a systematic method to compute the unramified Brauer groups of certain algebraic varieties defined by normic bundles, aiding in understanding their arithmetic properties.

Contribution

It introduces a partial compactification approach that aligns the Brauer group with the unramified Brauer group for these varieties.

Findings

Provides a systematic method for computing unramified Brauer groups

Establishes a partial compactification that preserves the Brauer group

Enables analysis of arithmetic properties of normic bundle varieties

Abstract

We produce a partial compactification of the variety given by P(t)=N_{K/k}(\mathbf z) whose Brauer group coincides with the unramified Brauer group, where K is an \'etale k-algebra and P(t)\in k[t] is a nonconstant polynomial. Then we obtain a systematic method to compute the unramified Brauer group for all such varieties.