The Brauer-Manin obstruction of Symmetric products

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$$Br(X)/Br(k) \simeq Br_{nr}\bigl(Sym_{X/k}<sup>{n}\bigr)/Br(k)$$, where $Sym_{X/k}<sup>{n}$ 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 $Sym_{X/k}<sup>{n,sm}$, and the corresponding obstruction for $0$-cycles of degree $n$ on $X$.