The algebraic Brauer group of a pinched variety

TL;DR

This paper investigates the algebraic Brauer group of pinched varieties, providing explicit descriptions of the kernel and cokernel of related Brauer group maps, and applies these results to compute Brauer groups of singular curves over various fields.

Contribution

It offers a detailed description of the Brauer group maps for pinched varieties and extends classical theorems to singular curves over local fields.

Findings

Computed the Brauer group of singular rational curves over any field.

Described kernels and cokernels of Brauer group pullback maps for pinched varieties.

Extended classical theorems to broader classes of singular curves.

Abstract

Added lemma provided by Michel Brion. Other (minor) changes. Submitted version. Let k be any field, let X' be a projective and geometrically integral k-scheme and let Y' be a finite closed subscheme of X'. If f: Y'-> Y is a schematically dominant morphism between finite k-schemes and X is obtained by pinching X' along Y' via f, we describe the kernel (and, in certain cases, the cokernel) of the induced pullback map Br_{1}X -> Br_{1}X' between the corresponding algebraic (cohomological) Brauer groups of X and X'_solely_ in terms of the Brauer groups of the residue fields of Y and Y' and the Amitsur subgroups of X and X' in Br k. As an application, we compute the Brauer group of a projective and geometrically integral k-rational curve (with arbitrary singularities) over any field k. We also extend a well-known theorem of Roquette and Lichtenbaum to a class of singular curves over any local field.