Castelnuovo's contraction theorem

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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 be a smooth projective surface over and a (1)-curve on (which means a smooth rational curve of self-intersection number 1), then there exists a morphism from to another smooth projective surface such that the curve has been contracted to one point , and moreover this morphism is an isomorphism outside (i.e., is isomorphic with ).

This contraction morphism is sometimes called a blowdown, which is the inverse operation of blowup. The curve is also called an exceptional curve of the first kind.

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