‐subgroups in the space Cremona group

Abstract

Abstract. We prove that if X is a rationally connected threefold and G is a p-subgroup in the group of birational selfmaps of X, then G is an abelian group generated by at most 3 elements provided that p 17. We also prove a similar result for p 11 under an assumption that G acts on a (Gorenstein) G-Fano threefold, and show that the same holds for p 5 under an assumption that G acts on a G-Mori fiber space.

Finite 3-subgroups in Cremona group of rank 3

“…In our case we can not repeat the argument used by Prokhorov and Shramov [PS18] to estimate the number of generators of p-subgroups. In order to do this they were looking for a point on the threefold fixed by the action of G and study its stabilizer.…”

“…We describe an example of the action of a 3-group on a surface which does not fix any point. This is a reason why we can not use the same strategy as in [PS18].…”

“…We consider the case of not necessarily abelian p-subgroups, though most part of them happen to be abelian. In the paper by Prokhorov and Shramov [PS18] all p-subgroups of Cr 3 (C) for big prime numbers p were described.…”

Hilbert schemes of lines and conics and automorphism groups of Fano threefolds

“…One of the motivations for writing this paper was the problem of classification of finite subgroups of the Cremona group of rank 3 (cf. [Pro12], [Pro11], [Pro14], [PS16b]). This classification problem reduces to investigation of finite automorphism groups of Fano threefolds of Picard number 1 with terminal singularities and Mori fiber spaces.…”

On the number of singular points of terminal factorial Fano threefolds

“…The upper bounds of s(X) are interesting for classification problems, in particular, for the classification of finite subgroups of the space Cremona group (see., e.g., [8], [10], [11], [14]). Y. Namikawa in [7] proved the inequality s(X) ≤ 20 − rk Pic(X) + h 1,2 (X s ), which holds for any Fano threefold X with terminal Gorenstein singularities.…”