On homomorphisms between Cremona groups

2016/03/10 by Urech, Christian
#14E07 (Primary) #14L30 #32M05 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1603.03294

Abstract

We look at algebraic embeddings of the Cremona group in n variables Cr_n(C) to the group of birational transformations Bir(M) of an algebraic variety M. First we study geometrical properties of an example of an embedding of Cr₂(C) into Cr₅(C) that is due to Gizatullin. In a second part, we give a full classification of all algebraic embeddings of Cr₂(C) into Bir(M), where dim(M)=3, and generalize this result partially to algebraic embeddings of Cr_n(C) into Bir(M), where dim(M)=n+1, for arbitrary n≥ 2. In particular, this yields a classification of all algebraic PGL_n+1(C)-actions on smooth projective varieties of dimension n+1 that can be extended to rational actions of Cr_n(C).

Related

• Algebraic growth of the Cremona group
• The action of the Cremona group on rational curves of ℙ³
• The essential and the Cremona dimensions of a group
• Some remarks about the Zariski topology of the Cremona group
• Quelques propriétés des transformations birationnelles du plan projectif complexe