Morphisms between Cremona groups and characterization of rational varieties (Q2874605)

The main result of this paper is the following. Let \(M\) be a connected smooth complex projective variety of dimension \(n\). Suppose that there exists an injective homomorphism of groups \(f : \mathrm{PGL}(r+1, \mathbb{C}) \rightarrow \mathrm{Bir}(M)\), where \(\mathrm{Bir}(M)\) is the group of birational automorphism of M. Then \(r \leq n\). Moreover, if \(r=n\) then \(M\) is birational to \(\mathbb{P}^n_{\mathbb{C}}\).NEWLINENEWLINEThis result has the following two interesting consequences.NEWLINENEWLINEThe first consequence is that the \(n\)-Cremona group \(\mathrm{Bir}(\mathbb{P}^n_{\mathbb{C}})\) embeds into the \(m\)-Cremona group \(\mathrm{Bir}(\mathbb{P}^m_{\mathbb{C}})\) if and only if \(n\leq m\). In particular, \(\mathrm{Bir}(\mathbb{P}^n_{\mathbb{C}}) \cong \mathrm{Bir}(\mathbb{P}^m_{\mathbb{C}})\) if and only if \(n=m\).NEWLINENEWLINEThe second result is that a connected smooth complex projective variety \(M\) of dimension \(n\) is rational if and only if \(\mathrm{Bir}(M)\) is isomorphic as an abstract group to \(\mathrm{Bir}(\mathbb{P}_{\mathbb{C}}^n)=\mathrm{PGL}(n+1,\mathbb{C})\).