Cremona Group

In algebraic geometry, the Cremona group, introduced by Cremona (1863, 1865), is the group of birational automorphisms of the n-dimensional projective space over a field k. It is denoted by Cr(Pn(k)) or Bir(Pn(k)) or Cr_n(k).

The Cremona group is naturally identified with the automorphism group Aut_k(k(x₁, ..., 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.