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 ∖ C is isomorphic with Y ∖ P ).
This contraction morphism is sometimes called a blowdown, which is the inverse operation of blowup. We also call curve C exceptional curve of the first kind.
