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

TL;DR

This paper investigates the Cremona group's action on rational curves in projective 3-space, proving rectifiability of certain families and exploring the equivalence of uniruled surfaces to scrolls.

Contribution

It demonstrates the rectifiability of one-dimensional families of rational curves in P^3 and characterizes Cremona equivalence of uniruled surfaces to scrolls.

Findings

Any one-dimensional family of rational curves in P^3 is rectifiable.

Any uniruled surface is Cremona equivalent to a scroll.

There exist infinitely many scrolls within the same Cremona orbit.

Abstract

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.