Castelnuovo's contraction theorem

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Short description: Constructs the minimal model of a given smooth algebraic surface

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

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

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