Topologies and structures of the Cremona groups

Abstract

Abstract. We study the algebraic structure of the n-dimensional Cremona group and show that it is not an algebraic group of infinite dimension (indgroup) if n ≥ 2. We describe the obstruction to this, which is of a topological nature.By contrast, we show the existence of a Euclidean topology on the Cremona group which extends that of its classical subgroups and makes it a topological group.

Birational geometry of del Pezzo fibrations with terminal quotient singularities

“…In this paper we focus on G-del Pezzo fibrations. The groups which could act on del Pezzo fibrations must be embedded into Cr 2 , these are: A 6 , PSL 2 (7), and A 5 . The A 6 -del Pezzo fibrations have already been classified.…”

“…The representations are conjugate by the outer automorphism and the representations of central extensions induce the same action, hence a faithful action of PSL 2 (7) on P 2 is unique. There is also a unique del Pezzo surface S 2 of degree 2 with a faithful action of PSL 2 (7), it is a double cover r : S 2 → P 2 branched over the Klein quartic. There are no other del Pezzo surfaces admitting a faithful action of A 6 or PSL 2 (7).…”

“…Then by Lemma 3.7 the group H is cyclic. The cyclic subgroup of PSL 2 (7) of maximal size has seven elements [44], therefore |Σ| 24. On the other hand |C ∩ C 0 | C · C 0 = 4r, thus 4r 24.…”

“…Suppose f is a G-equivariant fiberpreserving birational map to a G-del Pezzo fibration. (7) and X is P 2 × P 1 or X n , then f is an isomorphism.…”

“…Recently other ways to study the Cremona group have been considered. The topology of have been studied in . We could also study subgroups of the Cremona group, for example, it has been shown that is not a simple group .…”