In algebraic geometry, a contraction morphism is a surjective

projective morphism This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. ...

f: X \to Y

between normal projective varieties (or projective schemes) such that

f_* \mathcal_X = \mathcal_Y

or, equivalently, the geometric fibers are all connected (

Zariski's connectedness theorem In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for ...

). It is also commonly called an algebraic fiber space, as it is an analog of a

fiber space In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion \pi : E \to B\, that is, a surjective differentiable mapping such that at each point y \in U the tangent mapping T_y \pi : T_ ...

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classif ...

. By the

Stein factorization In algebraic geometry, the Stein factorization, introduced by for the case of complex spaces, states that a proper morphism can be factorized as a composition of a finite mapping and a proper morphism with connected fibers. Roughly speaking, Stein ...

, any surjective projective morphism is a contraction morphism followed by a finite morphism. Examples include

ruled surface In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directri ...

s and

Mori fiber space In algebraic geometry, a Fano fibration or Fano fiber space, named after Gino Fano, is a morphism of varieties whose general fiber is a Fano variety (in other words has ample anticanonical bundle) of positive dimension. The ones arising from extrem ...

Birational perspective

The following perspective is crucial in

birational geometry In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rationa ...

(in particular in

Mori's minimal model program In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its ori ...

). Let ''X'' be a projective variety and

\overline(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(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

\in F

. The basic question is which face ''F'' gives rise to such a contraction morphism (cf.

cone theorem In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety X is a combinatorial invariant of importance to the birational geometry of X. Definition Let X be a proper variety. By definition, a (real) ''1-cycle'' ...

References

* * Robert Lazarsfeld, ''Positivity in Algebraic Geometry I: Classical Setting'' (2004) Algebraic geometry {{algebraic-geometry-stub