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Morphisms between Cremona groups, and characterization of rational varieties
- Part of
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- Journal:
- Compositio Mathematica / Volume 150 / Issue 7 / July 2014
- Published online by Cambridge University Press:
- 25 June 2014, pp. 1107-1124
- Print publication:
- July 2014
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- Article
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We 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 
${\sf Bir}(M)$.
