Contraction morphism

In algebraic geometry, a contraction morphism is a surjective projective morphism f : X → Y {\displaystyle f:X\to Y} between normal projective varieties (or projective schemes) such that f ∗ O X = O Y {\displaystyle 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.

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

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

• Castelnuovo's contraction theorem
• Flip (mathematics)

• Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, ISBN 978-0-521-63277-5, MR 1658959
• Robert Lazarsfeld, Positivity in Algebraic Geometry I: Classical Setting (2004)