# 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 (−1)-curve on (which means a smooth rational curve of self-intersection number −1), 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 (i.e., is isomorphic with ).

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.

## 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

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