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The Brauer-Manin obstruction of Symmetric products

Yongqi Liang, Xingyu Liu, Hui Zhang

Abstract

This article focuses on smooth, projective, and geometrically integral varieties $X$ over a field $k$, whose geometric Picard group $Pic(X_{\overline{k}})$ is torsion-free. We establish an isomorphism$Br(X)/Br(k) \simeq Br_{nr}\bigl(Sym_{X/k}^{n}\bigr)/Br(k)$, where $Sym_{X/k}^{n}$ denotes the $n$-th symmetric product. Using this isomorphism, we investigate the relationship between the Brauer--Manin obstruction to the Hasse principle and weak approximation for rational points on the smooth proejctive model $Sym_{X/k}^{n,sm}$, and the corresponding obstruction for $0$-cycles of degree $n$ on $X$.

The Brauer-Manin obstruction of Symmetric products

Abstract

This article focuses on smooth, projective, and geometrically integral varieties over a field , whose geometric Picard group is torsion-free. We establish an isomorphism, where denotes the -th symmetric product. Using this isomorphism, we investigate the relationship between the Brauer--Manin obstruction to the Hasse principle and weak approximation for rational points on the smooth proejctive model , and the corresponding obstruction for -cycles of degree on .
Paper Structure (11 sections, 16 theorems, 61 equations)

This paper contains 11 sections, 16 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminary
  3. The varieties with torsion free geometric Picard groups
  4. Preliminary
  5. The Weil restriction
  6. Weil restriction as a geometric quotient
  7. The Symmetric product
  8. The homomorphisms for Brauer groups
  9. The generic fiber of $f_{n}$
  10. The Brauer group for the Symmetric product
  11. The Hasse principle and weak approximation

Key Result

Lemma 2.3

For a regular extension $K/k$, we have the following exact sequence:

Theorems & Definitions (34)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 24 more