Some remarks about the Zariski topology of the Cremona group

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