Cremona group

In algebraic geometry, the Cremona group, introduced by , is the group of birational automorphisms of the formula_1-dimensional projective space over a field It is denoted by formula_2
or formula_3 or formula_4.
The Cremona group is naturally identified with the automorphism group formula_5 of the field of the rational functions in formula_1 indeterminates over formula_7, or in other words a pure transcendental extension of formula_7, with transcendence degree formula_1.
The projective general linear group of order formula_10, of projective transformations, is contained in the Cremona group of order formula_1. The two are equal only when formula_12 or formula_13, in which case both the numerator and the denominator of a transformation must be linear.
The Cremona group in 2 dimensions.
In two dimensions, Max Noether and Castelnuovo showed that the complex Cremona group is generated by the standard quadratic transformation, along with formula_14, though there was some controversy about whether their proofs were correct, and 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. 
The Cremona group in higher dimensions.
There is little known about the structure of the Cremona group in three dimensions and higher though many elements of it have been described. showed that it is (linearly) connected, answering a question of . 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.
De Jonquières groups.
A De Jonquières group is a subgroup of a Cremona group of the following form . Pick a transcendence basis
formula_15 for a field extension of formula_7. Then a De Jonquières group is the subgroup of automorphisms of formula_17 mapping the subfield formula_18 into itself for some formula_19. It has a normal subgroup given by the Cremona group of automorphisms of formula_20 over the field formula_21, and the quotient group is the Cremona group of formula_21 over the field formula_7. It can also be regarded as the group of birational automorphisms of the fiber bundle formula_24.
When formula_25 and formula_26 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 
formula_27 and formula_28.