Unramified Brauer groups of symmetric products and the Brauer-Manin obstructions
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908986385629184 |
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| author | Liang, Yongqi Liu, Xingyu Zhang, Hui |
| author_facet | Liang, Yongqi Liu, Xingyu Zhang, Hui |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_06872 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Unramified Brauer groups of symmetric products and the Brauer-Manin obstructions Liang, Yongqi Liu, Xingyu Zhang, Hui Algebraic Geometry Number Theory 14F22, 14G12 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. |
| title | Unramified Brauer groups of symmetric products and the Brauer-Manin obstructions |
| topic | Algebraic Geometry Number Theory 14F22, 14G12 |
| url | https://arxiv.org/abs/2601.06872 |