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

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

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Source: Wikipedia "Castelnuovo's contraction theorem" · CC BY-SA 4.0

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