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 Crn(k).
The Cremona group is naturally identified with the automorphism group Autk(k(x1, ..., xn)) 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.
Read more about Cremona Group: The Cremona Group in 2 Dimensions, The Cremona Group in Higher Dimensions, De Jonquières Groups
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