In algebraic geometry, a contraction morphism is a surjective projective morphism
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.
The following perspective is crucial in birational geometry (in particular in Mori's minimal model program).
the closure of the span of irreducible curves on
= the real vector space of numerical equivalence classes of real 1-cycles on
, the contraction morphism associated to F, if it exists, is a contraction morphism
[1] The basic question is which face
gives rise to such a contraction morphism (cf.
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