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Let $\Sym^m X$ denote the $m$-th symmetric power of a smooth projective curve$X$. Let $\wt{\Sym^m X}$ be the blow up of $\Sym^m X$ along some non-singularsubvariety. In this note we are going to discuss when the pushforwardhomomorphism induced by the natural morphism from $\wt{\Sym^m X}$ to $\Sym^n X$is injective at the level of Chow groups for $m\leq n$. Also we are going toprove that base changes of embeddings of one symmetric power into another, withrespect to closed embeddings induces an injection at the level of Chow groups.

Blow ups and base changes of symmetric powers and Chow groups