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 {\displaystyle X} be a smooth projective surface over C {\displaystyle \mathbb {C} } and C {\displaystyle C} a (−1)-curve on X {\displaystyle X} (which means a smooth rational curve of self-intersection number −1), then there exists a morphism from X {\displaystyle X} to another smooth projective surface Y {\displaystyle Y} such that the curve C {\displaystyle C} has been contracted to one point P {\displaystyle P} , and moreover this morphism is an isomorphism outside C {\displaystyle C} (i.e., X ∖ C {\displaystyle X\setminus C} is isomorphic with Y ∖ P {\displaystyle Y\setminus P} ). This contraction morphism is sometimes called a blowdown, which is the inverse operation of blowup. The curve C {\displaystyle C} is also called an exceptional curve of the first kind.
References Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157 Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge: Cambridge University Press, ISBN 978-0-521-63277-5, MR 1658959