Contraction morphism explained

f: X \to Y between normal projective varieties (or projective schemes) such that f _{*} l{O} _{X} = l{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

• • Robert Lazarsfeld, Positivity in Algebraic Geometry I: Classical Setting (2004)