Morphisms between Cremona groups, and characterization of rational varieties

Abstract

AbstractWe classify all (abstract) homomorphisms from the group$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}{\sf PGL}_{r+1}(\mathbf{C})$to the group${\sf Bir}(M)$of birational transformations of a complex projective variety$M$, provided that$r\geq \dim _\mathbf{C}(M)$. As a byproduct, we show that: (i)${\sf Bir}(\mathbb{P}^n_\mathbf{C})$is isomorphic, as an abstract group, to${\sf Bir}(\mathbb{P}^m_\mathbf{C})$if and only if$n=m$; and (ii)$M$is rational if and only if${\sf PGL}_{\dim (M)+1}(\mathbf{C})$embeds as a subgroup of${\sf Bir}(M)$.