title: Contraction morphism
text: # 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).
  
  