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In algebraic geometry, the Cremona group, introduced by Cremona , is the group of birational automorphisms of the n {\displaystyle n} -dimensional projective space over a field k {\displaystyle k}. It is denoted by C r ] {\displaystyle Cr]} or B i r ] {\displaystyle Bir]} or C r n {\displaystyle Cr_{n}}.
The Cremona group is naturally identified with the automorphism group A u t k ] {\displaystyle \mathrm {Aut} _{k}]} of the field of the rational functions in n {\displaystyle n} indeterminates over k {\displaystyle k} , or in other words a pure transcendental extension of k {\displaystyle k} , with transcendence degree n {\displaystyle n}.
The projective general linear group of order n + 1 {\displaystyle n+1} , of projective transformations, is contained in the Cremona group of order n {\displaystyle n}. The two are equal only when n = 0 {\displaystyle n=0} or n = 1 {\displaystyle n=1} , in which case both the numerator and the denominator of a transformation must be linear.