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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 Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
projective surface over \mathbb and C a (−1)-
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
on X (which means a smooth
rational curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
of
self-intersection In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
number −1), then there exists a
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphis ...
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 In algebraic geometry, a contraction morphism is a surjective projective morphism f: X \to Y between normal projective varieties (or projective schemes) such that f_* \mathcal_X = \mathcal_Y or, equivalently, the geometric fibers are all connected ...
is sometimes called a
blowdown Blowdown may refer to: * Windthrow or forest blowdown, a felling of trees by windstorm * Blowdown stack A blowdown stack is an elevated vent or vertical stack that is used to vent the pressure of components of a chemical, refinery or other plan ...
, which is the inverse operation of
blowup ''Blowup'' (sometimes styled as ''Blow-up'' or ''Blow Up'') is a 1966 mystery drama thriller film directed by Michelangelo Antonioni and produced by Carlo Ponti. It was Antonioni's first entirely English-language film, and stars David Hemmi ...
. The curve C is also called an exceptional curve of the first kind.


References

* *{{Citation , last1=Kollár , first1=János , last2=Mori , first2=Shigefumi , author2-link=Shigefumi Mori , author1-link=János Kollár , title=Birational geometry of algebraic varieties , publisher=
Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambr ...
, series=Cambridge Tracts in Mathematics , location=Cambridge , isbn=978-0-521-63277-5 , mr=1658959 , year=1998 , volume=134 Algebraic surfaces Theorems in geometry