Unramified Brauer groups of symmetric products and the Brauer-Manin obstructions

TL;DR

This paper investigates the Brauer groups of certain algebraic varieties and their symmetric products, establishing isomorphisms that relate obstructions to rational points and 0-cycles over number fields.

Contribution

It proves an isomorphism between the Brauer groups of varieties and their symmetric products, linking obstructions for rational points and 0-cycles.

Findings

Established an isomorphism between Brauer groups of $X$ and its symmetric products.

Connected Brauer--Manin obstructions for 0-cycles and rational points.

Applied results to relate obstructions on varieties and their symmetric products.

Abstract

This article focuses on smooth, projective, and geometrically integral varieties $X$ defined over a number field $k$ with torsion-free geometric Picard groups. We establish an isomorphism between the Brauer groups of $X$ and its symmetric products. As applications, we deduce the relationship between the Brauer--Manin obstruction to the Hasse principle and to weak approximation for $0$-cycles of degree $n$ on $X$ and the corresponding obstruction for rational points on smooth projective models of its $n$-fold symmetric product.