- Cremona group
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In mathematics, in birational geometry, the Cremona group (named for Luigi Cremona) of order n over a field k is the group of birational automorphisms of the n-dimensional projective space over 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 of the function field over k in n indeterminates, 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 = 1, in which case both the numerator and the denominator of a transformation must be linear.
In two dimensions, the Cremona group is generated by the standard quadratic transformation, along with PGL(3, k), provided that k is an algebraically closed field.
The problem of describing the Cremona group in three dimensions and higher has still not been settled.
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