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.

be a smooth projective surface over

(which means a smooth rational curve of self-intersection number −1), then there exists a morphism from

to another smooth projective surface

This contraction morphism is sometimes called a blowdown, which is the inverse operation of blowup.

is also called an exceptional curve of the first kind.