Contraction morphism

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

See also

• Castelnuovo's contraction theorem
• Flip (mathematics)

References

1. Kollár & Mori 1998, Definition 1.25.

• 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)