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

This contraction morphism is sometimes called a blowdown, which is the inverse operation of blowup. The curve ${\displaystyle C}$ 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, MR 0463157
• 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, MR 1658959