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In algebraic geometry, the Cremona group, introduced by , is the group of birational automorphisms of the ''n''-dimensional projective space over a field ''k''. It is denoted by Cr(P''n''(''k'')) or Bir(P''n''(''k'')) or Cr''n''(''k''). The Cremona group is naturally identified with the automorphism group Aut''k''(''k''(x''1'', ..., 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. ==The Cremona group in 2 dimensions== In two dimensions, Max Noether and Castelnuovo showed that the complex Cremona group is generated by the standard quadratic transformation, along with PGL(3, ''k''), though there was some controversy about whether their proofs were correct, and gave a complete set of relations for these generators. The structure of this group is still not well understood, though there has been a lot of work on finding elements or subgroups of it. * showed that the Cremona group is not simple as an abstract group; *Blanc showed that it has no nontrivial normal subgroups that are also closed in a natural topology. *For the finite subgroups of the Cremona group see . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cremona group」の詳細全文を読む スポンサード リンク
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