Английская Википедия:Castelnuovo's contraction theorem
Шаблон:Short description In mathematics, Castelnuovo's contraction theorem is used in the classification theory of algebraic surfaces to construct the minimal model of a given smooth algebraic surface.
More precisely, let <math>X</math> be a smooth projective surface over <math>\mathbb{C}</math> and <math>C</math> a (−1)-curve on <math>X</math> (which means a smooth rational curve of self-intersection number −1), then there exists a morphism from <math>X</math> to another smooth projective surface <math>Y</math> such that the curve <math>C</math> has been contracted to one point <math>P</math>, and moreover this morphism is an isomorphism outside <math>C</math> (i.e., <math>X\setminus C</math> is isomorphic with <math>Y\setminus P</math>).
This contraction morphism is sometimes called a blowdown, which is the inverse operation of blowup. The curve <math>C</math> is also called an exceptional curve of the first kind.
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