Contraction morphism

In algebraic geometry, a contraction morphism is a surjective projective morphism $f:X\to Y$ between normal projective varieties (or projective schemes) such that $f_{*}{{𝒪}}_X={{𝒪}}_Y$ 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 $\overline{NS}(X)$ 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).

See also

• Castelnuovo's contraction theorem
• Flip (mathematics)

References

• 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)