In algebraic geometry, a contraction morphism is a surjective projective morphism \( f:X\to Y \) between normal projective varieties (or projective schemes) such that \( {\displaystyle f_{*}{\mathcal {O}}_{X}={\mathcal {O}}_{Y}} \) 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.
Birational perspective
The following perspective is crucial in birational geometry (in particular in Mori's minimal model program).
Let X be a projective variety and \( {\displaystyle {\overline {NS}}(X)} \) 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 \({\displaystyle {\overline {NS}}(X)} \), 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 \( [C]\in F \) .[1] The basic question is which face F gives rise to such a contraction morphism (cf. cone theorem).
See also
Castelnuovo's contraction theorem
Flip (mathematics)
References
Kollár–Mori, Definition 1.25.
Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, 134, Cambridge University Press, ISBN 978-0-521-63277-5, MR 1658959
Robert Lazarsfeld, Positivity in Algebraic Geometry I: Classical Setting (2004)
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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