Quotients of higher dimensional Cremona groups

Abstract

We study large groups of birational transformations Bir(X), where X is a variety of dimension at least 3, defined over C or a subfield of C. Two prominent cases are when X is the projective space P n , in which case Bir(X) is the Cremona group of rank n, or when X ⊂ P n+1 is a smooth cubic hypersurface. In both cases, and more generally when X is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from Bir(X) to Z/2, showing in particular that the group Bir(X) is not perfect and thus not simple. As a consequence we also obtain that the Cremona group of rank n 3 is not generated by linear and Jonquières elements.

Length functions on groups and rigidity

“…The Cremona group Bir(P 2 k ) is sub-quotient universal: every countable group can be embedded in a quotient group of Bir(P 2 k ) (see [21], Theorem 4.7). Moreover, Blanc-Lamy-Zimmermann [10] (Theorem E) proves that when n ≥ 3, there is a surjection from Bir(P n k ) onto a free product of two-element groups Z/2. This means that Corrolary 0.9 can never hold for higher dimensional Cremona groups.…”

Relations in the Cremona group over perfect fields

“…For many perfect fields, however, Lamy and Zimmermann constructed a surjective group homomorphism from the plane Cremona group to a free product of Z{2Z [LZ19, Theorem C], implying non-perfectness and thus reproving non-simplicity of the Cremona group in these cases. Recently, Blanc, Lamy and Zimmermann managed to construct a surjective group homomorphism from the high-dimensional Cremona group Cr n pkq to a free product of direct sums of Z{2Z, where n ě 3 and k Ă C is a subfield [BLZ19]. For the high-dimensional case, it turned out that it is more suitable not to use the high-dimensional analogy of [LZ19] but to take a different construction.…”

“…For the high-dimensional case, it turned out that it is more suitable not to use the high-dimensional analogy of [LZ19] but to take a different construction. The goal of this article is to adapt the strategy of [BLZ19] back to dimension two over perfect fields and find new normal subgroups of Cr 2 pkq. No knowledge of [BLZ19] is required to read our paper but we will highlight the connections to their proof.…”

“…The goal of this article is to adapt the strategy of [BLZ19] back to dimension two over perfect fields and find new normal subgroups of Cr 2 pkq. No knowledge of [BLZ19] is required to read our paper but we will highlight the connections to their proof. In fact, only classical algebraic geometry is used, and the well-established decomposition of birational maps into Sarkisov links (proven in [Isk96, Theorem 2.5] and [Cor95,Appendix], see Theorem 4 below).…”

“…In [LZ19] the focus lies on Bertini involutions (the blow-up of an orbit of size 8 in P 2 , followed by the contraction of an orbit of curves of size 8). For higher dimensions, the focus lies on links of type II between conic bundles that have a large covering gonality (see [BLZ19] for definitions). Translating this back to the 2-dimensional case, we focus on links of type II between conic bundles (that is, a Mori fiber space X Ñ B where B is a curve) that have a large cardinality and find the following generating relations:…”