Cremona group

In algebraic geometry, the Cremona group, introduced by, is the group of birational automorphisms of the $$n$$-dimensional projective space over a field $k$. It is denoted by $$Cr(\mathbb{P}^n(k))$$ or $$Bir(\mathbb{P}^n(k))$$ or $$Cr_n(k)$$.

The Cremona group is naturally identified with the automorphism group $$\mathrm{Aut}_k(k(x_1, ..., x_n)) $$ of the field of the rational functions in $$n$$ indeterminates over $$k$$, or in other words a pure transcendental extension of $$k$$, with transcendence degree $$n$$.

The projective general linear group of order $$n+1$$, of projective transformations, is contained in the Cremona group of order $$n$$. The two are equal only when $$n=0$$ or $$n=1$$, 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 Guido Castelnuovo showed that the complex Cremona group is generated by the standard quadratic transformation, along with $$\mathrm{PGL}(3,k)$$, 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.
 * showed that the Cremona group is not simple as an abstract group;
 * Blanc showed that it has no nontrivial normal subgroups that are also closed in a natural topology.
 * For the finite subgroups of the Cremona group see.

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 $$x_1, ..., x_n$$ for a field extension of $$k$$. Then a De Jonquières group is the subgroup of automorphisms of $$k(x_1, ...,x_n)$$ mapping the subfield $$k(x_1, ...,x_r)$$ into itself for some $$r\leq n$$. It has a normal subgroup given by the Cremona group of automorphisms of $$k(x_1, ..., x_n)$$ over the field $$k(x_1, ..., x_r)$$, and the quotient group is the Cremona group of $$k(x_1, ..., x_r)$$ over the field $$k$$. It can also be regarded as the group of birational automorphisms of the fiber bundle $$\mathbb{P}^r\times \mathbb{P}^{n-r} \to \mathbb{P}^r$$.

When $$ n=2$$ and $$ r=1$$ 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 $$\mathrm{PGL}_2(k)$$ and $$\mathrm{PGL}_2(k(t))$$.