Castelnuovo's contraction theorem

Template: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 $X$ be a smooth projective surface over $ℂ$ 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., $X\setminus C$ is isomorphic with $Y\setminus P$).

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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