The action of the Cremona group on rational curves of ℙ³

2015/05/04 by Angelini, Elena, Mella, Massimiliano
#Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1505.00563

Abstract

A Cremona transformation is a birational self-map of the projective space ℙⁿ . Cremona transformations of ℙⁿ form a group and this group has a rational action on subvarieties of ℙⁿ and hence on its Hilbert scheme. We study this action on the family of rational curves of ℙ³ 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-Böhning 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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