On homomorphisms between Cremona groups

Abstract

We look at algebraic embeddings of the Cremona group in n variables Crn(C) to the groups of birational transformations Bir(M ) of an algebraic variety M . First we study geometrical properties of an example of an embedding of Cr 2 (C) into Cr 5 (C) that is due to Gizatullin. In a second part, we give a full classification of all algebraic embeddings of Cr 2 (C) into Bir(M ), where dim(M ) = 3 and generalize this result partially to algebraic embeddings of Crn(C) into Bir(M ), where dim(M ) = n + 1, for arbitrary n. 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 Crn(C).Example 1.1. Assume that a variety M is birationally equivalent to P n ×N for some variety N . The standard action on the first factor yields an injective homomorphism of Cr n into Bir(P n ×N ) and therefore also into Bir(M ). We call embeddings of this type standard embeddings. In particular, we obtain in that way for all nonnegative integers m an injective homomorphism Cr n → Bir(P n × P m ).Example 1.2. A variety M is called stably rational if there exists a n such that M × P n is rational. There exist varieties of dimension larger than or equal to 3 that are stably rational but not rational (see [BCTSSD85]). We will see that two standard embeddings f 1 : Cr n → Bir(P n ×N ) and f 2 : Cr n → Bir(P n ×M ) are conjugate if and only if N and M are birationally equivalent (Lemma 3.3). So every class of birationally equivalent stably rational varieties of dimension k defines a different conjugacy class of injective homomorphisms Cr n → Bir(P m ) for m = n + k.