Castelnuovo's contraction theorem

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 $$\mathbb{C}$$ and $$C$$ a (&minus;1)-curve on $$X$$ (which means a smooth rational curve of self-intersection number &minus;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.