Three embeddings of the Klein simple group into the Cremona group of rank three (Q692787)

A classical and fundamental problem of birational geometry is the description of the Cremona group \(\mathrm{Cr}_n(\mathbb{C})\) of birational automorphisms of \(\mathbb{P}^n\). The cases \(n=1,2\) are well understood. For \(n=1\) it is \(\mathrm{PGL}(2,\mathbb{C})\) and for \(n=2\) it is generated by a single quadratic transformation and automorphisms. Moreover, the finite subgroups of \(\mathrm{Cr}_2(\mathbb{C})\) have been classified [\textit{I. V. Dolgachev} and \textit{V. A. Isovskikh}, Progress in Mathematics 269, 443--548 (2009; Zbl 1219.14015)]. However, at the time of this writing, very little is known about the structure of the Cremona groups \(\mathrm{Cr}_n(\mathbb{C})\), for \(n \geq 3\). One possible way to analyze their structure is to study their finite subgroups. For \(n=3\) it is known that the only finite simple non-abelian subgroups of \(\mathrm{Cr}_3(\mathbb{C})\) are \(A_4, \mathrm{PSL}_2(\mathbb{F}_7), A_6, A_7, \mathrm{PSL}_2(\mathbb{F}_8)\) and \(\mathrm{PSU}(\mathbb{F}_2)\) [\textit{Y. Prokhorov}, J. Algebr. Geom. 21, No3, 563--600 (2012; Zbl 1257.14011)]. Moreover, the authors have shown that up to conjugation there is exactly one subgroup of \(\mathrm{Cr}_3(\mathbb{C})\) isomorphic to \(\mathrm{PSL}_2(\mathbb{F}_8)\), exactly one subgroup isomorphic to \(A_7\), two subgroups isomorphic to \(\mathrm{PSU}(\mathbb{F}_2)\) and at least five isomorphic to \(A_6\). In this paper it is shown that the Cremona group \(\mathrm{Cr}_3(\mathbb{C})\) contains at least 3 non-conjugate subgroups isomorphic to \(\mathrm{PSL}_2(\mathbb{F}_7)\). In order to prove this the authors construct a smooth rational threefold \(X\) such that \(\mathrm{Aut}(X) \cong \mathrm{PSL}_2(\mathbb{F}_7)\). Moreover they show that \(X\) is birationally \(G\)-superrigid, where \(G=\mathrm{PSL}_2(\mathbb{F}_7)\). Consequently \(\mathrm{Bir}^G(X)=\mathrm{Aut}(X)=\mathrm{PSL}_2(\mathbb{F}_7)\), where \(\mathrm{Bir}^G(X)\) is the group of \(G\)-invariant birational automorphisms of \(X\). From this it follows that the normalizer of \(\mathrm{PSL}_2(\mathbb{F}_7)\) in \(\mathrm{Bir}(X)\) is isomorphic to \(\mathrm{PSL}_2(\mathbb{F}_7)\). Then considering that \(\mathrm{Bir}(X)\cong \mathrm{Cr}_3(\mathbb{C})\), \(\mathrm{PSL}_2(\mathbb{F}_7)\) is embedded in \(\mathrm{Cr}_3(\mathbb{C})\) with normalizer isomorphic to \(\mathrm{PSL}_2(\mathbb{F}_7)\). Another embedding of \(\mathrm{PSL}_2(\mathbb{F}_7)\) in \(\mathrm{Cr}_3(\mathbb{C})\) is obtained by considering a faithful irreducible four dimensional representation of \(\mathrm{SL}_2(\mathbb{F}_7)\). Then the authors shown that this embedding of \(\mathrm{PSL}_2(\mathbb{F}_7)\) has normalizer \(\mathrm{PSL}_2(\mathbb{F}_7) \times \mathbb{Z}_2\). From this it follows that there are at least three non-conjugate subgroups of \(\mathrm{Cr}_3(\mathbb{C})\) isomorphic to \(\mathrm{PSL}_2(\mathbb{F}_7)\).