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 $f_* \mathcal_X = \mathcal_Y$ 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 $\overline(X)$ 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).

See also

*Castelnuovo's contraction theorem
* Flip (mathematics)

References

*
* Robert Lazarsfeld, ''Positivity in Algebraic Geometry I: Classical Setting'' (2004)

Algebraic geometry

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