Contraction morphism
In algebraic geometry, a contraction morphism is a surjective projective morphism [math]\displaystyle{ f: X \to Y }[/math] between normal projective varieties (or projective schemes) such that [math]\displaystyle{ f_* \mathcal{O}_X = \mathcal{O}_Y }[/math] 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 [math]\displaystyle{ \overline{NS}(X) }[/math] the closure of the span of irreducible curves on X in [math]\displaystyle{ N_1(X) }[/math] = the real vector space of numerical equivalence classes of real 1-cycles on X. Given a face F of [math]\displaystyle{ \overline{NS}(X) }[/math], the contraction morphism associated to F, if it exists, is a contraction morphism [math]\displaystyle{ f: X \to Y }[/math] to some projective variety Y such that for each irreducible curve [math]\displaystyle{ C \subset X }[/math], [math]\displaystyle{ f(C) }[/math] is a point if and only if [math]\displaystyle{ [C] \in F }[/math].[1] The basic question is which face F gives rise to such a contraction morphism (cf. cone theorem).
See also
References
- ↑ Kollár & Mori 1998, 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
- Robert Lazarsfeld, Positivity in Algebraic Geometry I: Classical Setting (2004)
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