# A note on Mumford-Roitman argument on Chow schemes

## TL;DR

This paper investigates the fibers of natural maps related to Chow schemes and algebraic groups, focusing on the Mumford-Roitman argument and the structure of Chow groups in algebraic geometry.

## Contribution

It provides new insights into the fibers of maps from algebraic groups to their quotients and from Hilbert schemes to Chow groups, extending Mumford-Roitman techniques.

## Key findings

- Characterization of fibers of the map from a projective algebraic group to its quotient.
- Definition of a natural map from Hilbert schemes to Chow groups of relative zero cycles.
- Analysis of the fibers of these maps in algebraic geometry contexts.

## Abstract

In this note we are going to understand two questions. One is the fiber of the natural map from a projective algebraic group $G$ to $G/\Gamma$, where $\Gamma$ denotes the $\Gamma$-equivalence on $G$. The other one is to define a natural map from Hilbert scheme of the generic fiber of a fibration $X\to S$ to the Chow group of relative zero cycles on $X\to S$ and to understand the fibers of this map.