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 be a smooth projective surface over and a -curve on , then there exists a morphism from to another smooth projective surface such that the curve has been contracted to one point, and moreover this morphism is an isomorphism outside .
This contraction morphism is sometimes called a blowdown, which is the inverse operation of blowup. The curve is also called an exceptional curve of the first kind.
