Finite collineation groups and birational rigidity

Abstract

We classify finite groups G in ${\mathrm {PGL}}_{4}({\mathbb {C}})$ such that ${\mathbb {P}}^3$ is G-birationally rigid.

Appendix A: Finite primitive groups

In diagrams (A.1), (A.2), and (A.3) below, we present all the finite primitive subgroups of ${\mathrm {PGL}}_4({\mathbb {C}})$ together with some inclusions between them, including those that are necessary for the proof of Theorem 1.1. We emphasize that we will not need the whole classification for the proof, but only the information about the few smallest groups.

By $\varvec{\mu }_n$ we denote the cyclic group of order n. The inclusions are depicted by arrows going from a smaller group to a larger one. We point out that there are indeed several additional inclusions (apart from those that are obtained merely by composing those inclusions that we list). For instance, the group ${\mathfrak {A}}_4\times {\mathfrak {A}}_4$ from diagram (A.1) and the group ${\mathfrak {A}}_6$ from diagram (A.3) are both subgroups of $\varvec{\mu }_2^4\rtimes {\mathfrak {A}}_6$ from diagram (A.2).

In diagram (A.1), we list all finite primitive subgroups of ${\mathrm {PGL}}_4({\mathbb {C}})$ that leave invariant a quadric surface. The larger “connected component” of the diagram contains the groups $1^\circ$–$12^\circ$ described in [4, §§121, 122]; the smaller “connected component” contains the group (B) described in [4, §102] and the group (H) described in [4, §119]. We denote by $\widehat{G\times G}$ the group generated by $G\times G$ and an involution that exchanges the factors of $G\times G$. This group is isomorphic to $(G\times G)\rtimes \varvec{\mu }_2$. However, we reserve the notation $({\mathfrak {A}}_4\times {\mathfrak {A}}_4)\rtimes \varvec{\mu }_2$ not for a group of the latter type, but for a different subgroup of ${\mathrm {PGL}}_4({\mathbb {C}})$ generated by ${\mathfrak {A}}_4$ and an element of order 2 that does not exchange the factors; this element can be thought of as a product of two elementary transpositions in the factors of ${\mathfrak {S}}_4\times {\mathfrak {S}}_4\supset {\mathfrak {A}}_4\times {\mathfrak {A}}_4$. We refer the reader to [4, §121] for details. The above group is contained as a subgroup of index two in two other primitive subgroups of ${\mathrm {PGL}}_4({\mathbb {C}})$ preserving a quadric surface so that in both cases an additional element of order 2 exchanges the factors of ${\mathfrak {A}}_4\times {\mathfrak {A}}_4\subset ({\mathfrak {A}}_4\times {\mathfrak {A}}_4)\rtimes \varvec{\mu }_2$, see [4, §122] for details. We denote these two groups by $\widehat{({\mathfrak {A}}_4\times {\mathfrak {A}}_4)\rtimes \varvec{\mu }_2^{(1)}}$ and $\widehat{({\mathfrak {A}}_4\times {\mathfrak {A}}_4)\rtimes \varvec{\mu }_2^{(2)}}$.

(A.1)

In diagram (A.2), we list all finite primitive subgroups of ${\mathrm {PGL}}_4({\mathbb {C}})$ that contain a subgroup isomorphic to $\varvec{\mu }_2^4$ except for those that leave invariant a quadric surface. These are the groups $13^\circ$–$21^\circ$ described in [4, §124]. We denote by $\mathrm {D}_{10}$ the dihedral group of order 10. By F.G we mean a non-split extension of a group G by a group F.

(A.2)

Finally, in diagram (A.3), we list the remaining finite primitive subgroups of ${\mathrm {PGL}}_4({\mathbb {C}})$. These are the groups (A), (C), (D), (E), and (F) described in [4, §102], and the groups (G) and (K) described in [4, §119].

(A.3)