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In algebraic geometry, a Fano variety, introduced in , is a complete variety ''X'' whose anticanonical bundle ''K''X * is ample. In this definition, one could assume that ''X'' is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. ==Examples== *The fundamental example of Fano varieties are the projective spaces: the anticanonical line bundle of P''n'' over a field ''k'' is ''O''(''n''+1), which is very ample (over the complex numbers, its curvature is ''n+1'' times the Fubini–Study symplectic form). *Let ''D'' be a smooth codimension-1 subvariety in Pn. From the adjunction formula, we infer that ''K''''D'' = (''K''''X'' + ''D'')|''D'' = (−(''n''+1)''H'' + deg(''D'')H)|''D'', where ''H'' is the class of a hyperplane. The hypersurface ''D'' is therefore Fano if and only if deg(''D'') < ''n''+1. *More generally, a smooth complete intersection of hypersurfaces in ''n''-dimensional projective space is Fano if and only if the sum of their degrees is at most ''n''. *Weighted projective space P(''a''0,...,''a''''n'') is a singular (klt) Fano variety. This is the projective scheme associated to a graded polynomial ring whose generators have degrees ''a''0,...,''a''''n''. If this is well formed, in the sense that no ''n'' of the numbers ''a'' have a common factor greater than 1, then any complete intersection of hypersurfaces such that the sum of their degrees is less than ''a''0+...+''a''''n'' is a Fano variety. *Every projective variety in characteristic zero that is homogeneous under a linear algebraic group is Fano. 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「Fano variety」の詳細全文を読む スポンサード リンク
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