Contraction morphism

In algebraic geometry, a contraction morphism is a surjective projective morphism formula_1 between normal projective varieties (or projective schemes) such that formula_2 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 formula_3 the closure of the span of irreducible curves on "X" in formula_4 = the real vector space of numerical equivalence classes of real 1-cycles on "X". Given a face "F" of formula_3, the contraction morphism associated to "F", if it exists, is a contraction morphism formula_1 to some projective variety "Y" such that for each irreducible curve formula_7, formula_8 is a point if and only if formula_9. The basic question is which face "F" gives rise to such a contraction morphism (cf. cone theorem).