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 $X$ be a smooth projective surface over $\mathbb{C}$ 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., $X\setminus C$ is isomorphic with $Y\setminus P$).

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.

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