# zbMATH — the first resource for mathematics

On contractions of extremal rays of Fano manifolds. (English) Zbl 0721.14023
Let X be a complex projective manifold and $$\phi: X\to Y$$ a morphism onto a normal projective variety Y such that all fibers of $$\phi$$ are connected. If all curves contracted by $$\phi$$ are numerically proportional and have negative intersection with the canonical divisor $$K_ X$$ then such a map is called a contraction of an extremal ray.
In the present paper the geometric structure of such contractions is examined. For example, the following inequality is proved $\dim(\text{exceptional locus of }\phi)+\dim(\text{non-trivial fiber of }\phi)\geq \dim(X)-1+\ell,$ where the integer $$\ell$$ is defined as the minimal intersection number $$-K_ X.C$$ among rational curves contracted by $$\phi$$.
Most of the results of the paper concern the case when X is a Fano manifold, i.e. $$-K_ X$$ is ample. In particular it is examined when a contraction of a Fano manifold leads to a non-Fano manifold. For example, it is shown that a conic bundle over a 3-fold is a Fano manifold only if its base is; for higher dimensions it is not true as an example shows.

##### MSC:
 14J45 Fano varieties 14C20 Divisors, linear systems, invertible sheaves
##### Keywords:
contraction of an extremal ray; Fano manifold
Full Text: