Quotients of higher-dimensional Cremona groups

Equivalent birational embedding IV: reduced varieties

“…A reasonable set of generators is not known yet. The Cremona groups have been proved to be non simple, [CL13] [BLZ21], and their behavior is wild from many points of view, as an example one can look at the results in [BLZ21]. Already the two dimensional Cremona group has many foundational problems that are far from being solved, see [Can18] for a very nice introduction.…”

“…Therefore all Cremona equivalences of subvarieties of codimension at least 2 can be realized by transformations in the subgroup generated by de Jonquiéres transformations. This is particularly interesting when confronted with [BLZ21] where it is proven that the group of de Jonquiéres map is a proper subgroup of Cr n . That is to produce all Cremona equivalences for codimension ≥ 2 reduced schemes we do not even need the full group Cr n .…”

Relations in the Cremona group over a perfect field

“…Hence, the structure of the plane Cremona group depends fundamentally on the field. Observing similarities between the plane Cremona group over non-closed fields and the Cremona groups in high dimensions, Blanc, Lamy and Zimmermann have recently managed to construct a surjective group homomorphism from the high-dimensional Cremona group Cr n (k) to a free product of direct sums of Z/2Z, where n 3 and k ⊂ C is a subfield [3]. (More constructions in this setting can be found in [4,19].)…”

Finite groups of birational transformations

“…On the other hand, for n ≥ 2 the group Cr n (k) has extremely complicated structure. In particular, it contains linear algebraic subgoups of arbitrary dimension and has a lot of normal non-algebraic subgroups [23,17]. We refer to [2,21,22,34,44,89] for surveys, historical résumés, and introductions to the subject.…”