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In mathematics, birational geometry is a field of algebraic geometry the goal of which is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles. ==Birational maps== A rational map from one variety (understood to be irreducible) ''X'' to another variety ''Y'', written as a dashed arrow ''X'' – → ''Y'', is defined as a morphism from a nonempty open subset ''U'' of ''X'' to ''Y''. By definition of the Zariski topology used in algebraic geometry, a nonempty open subset ''U'' is always the complement of a lower-dimensional subset of ''X''. Concretely, a rational map can be written in coordinates using rational functions. A birational map from ''X'' to ''Y'' is a rational map ''f'': ''X'' – → ''Y'' such that there is a rational map ''Y'' – → ''X'' inverse to ''f''. A birational map induces an isomorphism from a nonempty open subset of ''X'' to a nonempty open subset of ''Y''. In this case, we say that ''X'' and ''Y'' are birational, or birationally equivalent. In algebraic terms, two varieties over a field ''k'' are birational if and only if their function fields are isomorphic as extension fields of ''k''. A special case is a birational morphism ''f'': ''X'' → ''Y'', meaning a morphism which is birational. That is, ''f'' is defined everywhere, but its inverse may not be. Typically, this happens because a birational morphism contracts some subvarieties of ''X'' to points in ''Y''. We say that a variety ''X'' is rational if it is birational to affine space (or equivalently, to projective space) of some dimension. Rationality is a very natural property: it means that ''X'' minus some lower-dimensional subset can be identified with affine space minus some lower-dimensional subset. For example, the circle with equation ''x''2 + ''y''2 − 1 = 0 is a rational curve, because the formulas : : define a birational map from the affine line to the circle and generates Pythagorean triples. (Explicitly, the inverse map sends (''x'',''y'') to (1 − ''y'')/''x''.) More generally, a smooth quadric (degree 2) hypersurface ''X'' of any dimension ''n'' is rational, by stereographic projection. (For ''X'' a quadric over a field ''k'', we have to assume that ''X'' has a ''k''-rational point; this is automatic if ''k'' is algebraically closed.) To define stereographic projection, let ''p'' be a point in ''X''. Then we define a birational map from ''X'' to the projective space P''n'' of lines through ''p'' by sending a point ''q'' in ''X'' to the line through ''p'' and ''q''. This is a birational equivalence but not an isomorphism of varieties, because it fails to be defined where ''q'' = ''p'' (and the inverse map fails to be defined at those lines through ''p'' which are contained in ''X''). 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「Birational geometry」の詳細全文を読む スポンサード リンク
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