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arXiv · 1505.00563

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

Elena Angelini·Massimiliano Mella
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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.

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Elena Angelini, Massimiliano Mella. 2015-05-04. The action of the Cremona group on rational curves of $ \mathbb{P}^{3} $. https://arxiv.org/abs/1505.00563

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