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Rational connectedness and boundedness of Fano manifolds. (English) Zbl 0759.14032

A smooth projective variety is called a Fano variety if the anticanonical class is ample. The aim of this paper is to study rational curves on Fano varieties, mostly in characteristic zero. The first result asserts that any two points on a Fano variety can be connected by a connected chain of rational curves. Then using the gluing technique introduced by the authors in the paper “Rationally connected varieties” [J. Algebraic Geometry (to appear)] it is proved that two general points can be connected by an irreducible rational curve. Furthermore, we are able to bound the degree (with respect to the anticanonical class) of this rational curve by a function which depends only on the dimension. The existence of such a bound implies that in any given dimension there are only finitely many Fano varieties up to deformations.

MSC:

14J45 Fano varieties
14F45 Topological properties in algebraic geometry
14J10 Families, moduli, classification: algebraic theory
14D15 Formal methods and deformations in algebraic geometry
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