Castelnuovo's contraction theorem

Short description: Constructs the minimal model of a given smooth algebraic surface

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 [math]\displaystyle{ X }[/math] be a smooth projective surface over [math]\displaystyle{ \mathbb{C} }[/math] and [math]\displaystyle{ C }[/math] a (−1)-curve on [math]\displaystyle{ X }[/math] (which means a smooth rational curve of self-intersection number −1), then there exists a morphism from [math]\displaystyle{ X }[/math] to another smooth projective surface [math]\displaystyle{ Y }[/math] such that the curve [math]\displaystyle{ C }[/math] has been contracted to one point [math]\displaystyle{ P }[/math], and moreover this morphism is an isomorphism outside [math]\displaystyle{ C }[/math] (i.e., [math]\displaystyle{ X\setminus C }[/math] is isomorphic with [math]\displaystyle{ Y\setminus P }[/math]).

This contraction morphism is sometimes called a blowdown, which is the inverse operation of blowup. The curve [math]\displaystyle{ C }[/math] is also called an exceptional curve of the first kind.

References

• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90244-9 
• Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge: Cambridge University Press, ISBN 978-0-521-63277-5