Cremona group
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In mathematics, in birational geometry, the Cremona group 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 
.
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).
The problem of describing the Cremona group in three dimensions and higher has still not been settled.
