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 $$f_* \mathcal{O}_X = \mathcal{O}_Y$$ 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 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).