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.

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).