Blow ups and base changes of symmetric powers and Chow groups

TL;DR

This paper investigates conditions under which certain morphisms between symmetric powers of smooth projective curves and their blow-ups induce injective maps on Chow groups, revealing new insights into their algebraic cycle structures.

Contribution

It establishes criteria for injectivity of pushforward maps on Chow groups for blow-ups and base changes of symmetric powers of curves.

Findings

Injectivity of pushforward homomorphisms for blow-ups along non-singular subvarieties.

Injectivity results for base changes of symmetric power embeddings.

Conditions under which Chow groups are preserved under these morphisms.

Abstract

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-singular subvariety. In this note we are going to discuss when the pushforward homomorphism 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 to prove that base changes of embeddings of one symmetric power into another, with respect to closed embeddings induces an injection at the level of Chow groups.