Some remarks about the Zariski topology of the Cremona group

TL;DR

This paper investigates the Zariski topology on the Cremona group of a rational variety, exploring how algebraic morphisms relate to the group's algebraic structure.

Contribution

It provides new insights into the relationship between the Zariski topology and the algebraic structure of the Cremona group of rational varieties.

Findings

Analysis of algebraic morphisms into the Cremona group

Relationship between Zariski topology and group structure

Implications for rational varieties

Abstract

For an algebraic variety $X$ we study the behavior of algebraic morphisms from an algebraic variety to the group $\bir(X)$ of birational maps of $X$ and obtain, as application, some insight about the relationship between the so-called Zariski topology of $\bir(X)$ and the algebraic structure of this group, where $X$ is a rational variety.