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, in which case Bir(X) is the Cremona group of rank n, or when X 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 at least 3 is not generated by linear and Jonquières elements.

Item (RF4) in main definition 3.1 was modified: many thanks to Yang He for spotting the problem! Other minor changes. To appear in Acta Mathematica

References in corpus (3)

• Connected algebraic groups acting on three-dimensional Mori fibrations
• Relations in the Cremona group over perfect fields
• Cremona maps and involutions

Cited by in corpus (8)

• Connected algebraic groups acting on three-dimensional Mori fibrations
• Boundedness results for singular Fano varieties and applications to Cremona groups
• Relations in the Cremona group over perfect fields
• Length functions on groups and rigidity
• Generators of the plane Cremona group over the field with two elements
• Sarkisov Links with centres space curves on smooth cubic surfaces
• Finite groups of birational transformations
• Rigid birational involutions of ${\mathbb{P}}^3$ and cubic threefolds