Brauer groups of certain affine cubic surfaces

We study the Brauer groups of affine surfaces that are complements of singular hyperplane sections of smooth cubic surfaces over a field $k$ of characteristic $0$. We determine the Brauer group over the algebraic closure as a Galois module for all the possible singular hyperplane sections. For the case when the hyperplane section is geometrically the union of three lines, we give explicit examples where transcendental elements of order $2$ and $3$ exist over $\mathbb{Q}$. We end with an application on the integral Brauer-Manin obstruction to the integral Hasse principle.

Publication Details

Journal
ArXiv.org
Published
2025-09-19
Primary Topic
Geochemistry and Geologic Mapping
Type
article
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article

Brauer groups of certain affine cubic surfaces

ArXiv.org
Geochemistry and Geologic Mapping
article

Brauer groups of certain affine cubic surfaces

article en

Abstract

We study the Brauer groups of affine surfaces that are complements of singular hyperplane sections of smooth cubic surfaces over a field $k$ of characteristic $0$. We determine the Brauer group over the algebraic closure as a Galois module for all the possible singular hyperplane sections. For the case when the hyperplane section is geometrically the union of three lines, we give explicit examples where transcendental elements of order $2$ and $3$ exist over $\mathbb{Q}$. We end with an application on the integral Brauer-Manin obstruction to the integral Hasse principle.

ArXiv.org
Openalex Percentile: Top 29%
Geochemistry and Geologic Mapping
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Brauer groups of certain affine cubic surfaces · ArXiv.org (2025) | TGRS Research Map | TGRS