Cremona group
In birational geometry, the Cremona group, named after Luigi Cremona, is the group of birational automorphisms of the -dimensional projective space over a field, also known as Cremona transformations. It is denoted by, or.
Historical origins
The Cremona group was introduced by the Italian mathematician. In retrospect, the British mathematician Isaac Newton is considered a founder of "the theory of Cremona transformations", having developed his "organic construction" to perform birational maps of the projective plane and applied them to resolve curve singularities, nearly two centuries before Cremona. The mathematician Hilda Phoebe Hudson made contributions in the 1900s as well.Basic properties
The Cremona group is naturally identified with the automorphism group of the field of the rational functions in indeterminates over. Here, the field is a pure transcendental extension of, with transcendence degree.The projective general linear group is contained in. The two are equal only when or, in which case both the numerator and the denominator of a transformation must be linear.
A longlasting question from Federigo Enriques concerns the simplicity of the Cremona group. It has been now mostly answered.
The Cremona group in 2 dimensions
In two dimensions, Max Noether and Guido Castelnuovo showed that the complex Cremona group is generated by the standard quadratic transformation, along with, though there was some controversy about whether their proofs were correct. gave a complete set of relations for these generators. The structure of this group is still not well understood, though there has been a lot of work on finding elements or subgroups of it.- showed that for an algebraically closed field, the group is not simple.
- showed that it is topologically simple for the Zariski topology.
- For the finite subgroups of the Cremona group see.
- computed the abelianization of. From this, she deduces that there is no analogue of Noether–Castelnuovo theorem in this context.
The Cremona group in higher dimensions
There is no easy analogue of the Noether–Castelnouvo theorem, as showed that the Cremona group in dimension at least 3 is not generated by its elements of degree bounded by any fixed integer.
showed that it is connected, answering a question of. Later, showed that for any infinite field, the group is topologically simple for the Zariski topology, and even for the euclidean topology when is a local field.
proved that when is a subfield of the complex numbers and, then is not a simple group.
De Jonquières groups
A De Jonquières group is a subgroup of a Cremona group of the following form. Pick a transcendence basis for a field extension of. Then a De Jonquières group is the subgroup of automorphisms of mapping the subfield into itself for some. It has a normal subgroup given by the Cremona group of automorphisms of over the field, and the quotient group is the Cremona group of over the field. It can also be regarded as the group of birational automorphisms of the fiber bundle.When and the De Jonquières group is the group of Cremona transformations fixing a pencil of lines through a given point, and is the semidirect product of and.