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Contraction morphism

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated August 30, 2026

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 {O}}_{X}={\mathcal {O}}_{Y} or, equivalently, the geometric fibers are all connected (Zariski's connectedness theorem). It is also commonly called an algebraic fiber space, as it is an analog of a fiber space in algebraic topology.

By the Stein factorization, any surjective projective morphism is a contraction morphism followed by a finite morphism.

Examples include ruled surfaces and Mori fiber spaces.

01Birational perspective

The following perspective is crucial in birational geometry (in particular in Mori's minimal model program).

Let X be a projective variety and {\overline {NS}}(X) the closure of the span of irreducible curves on X in N_{1}(X) = the real vector space of numerical equivalence classes of real 1-cycles on X. Given a face F of {\overline {NS}}(X), the contraction morphism associated to F, if it exists, is a contraction morphism f:X\to Y to some projective variety Y such that for each irreducible curve C\subset X, f(C) is a point if and only if [C]\in F. The basic question is which face F gives rise to such a contraction morphism (cf. cone theorem).

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This article is adapted from the Wikipedia article “Contraction morphism”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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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-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} ).

Flip (algebraic geometry)

In algebraic geometry, flips and flops are codimension-2 surgery operations arising in the minimal model program, given by blowing up along a relative canonical ring. In dimension 3 flips are used to construct minimal models, and any two birationally equivalent minimal models are connected by a sequence of flops.