The action of the Cremona group on rational curves of ${\mathbb{P}}^3$

A Cremona transformation is a birational self-map of the projective space ${\mathbb{P}}^n$. Cremona transformations of ${\mathbb{P}}^n$ form a group and this group has a rational action on subvarieties of ${\mathbb{P}}^n$ and hence on its Hilbert scheme. We study this action on the family of rational curves of ${\mathbb{P}}^3$ and we prove the rectifiability of any one dimensional family. This shows that any uniruled surface is Cremona equivalent to a scroll and it answers a question of Bogomolov-B"ohning related to the study of uniformly rational varieties. We provide examples of infinitely many scrolls in the same Cremona orbit and we show that a “general” scroll is not in the Cremona orbit of a “general” rational surface.