Connexité rationnelle des variétés de Fano

Abstract

<http://www.numdam.org/item?id=ASENS_1992_4_25_5_539_0> © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1992, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Degenerations of rationally connected varieties

“…The notions of rational connectedness and rational chain connectedness are known to coincide when X/k is smooth and proper. A result of Campana and Kollár-Miyaoka-Mori says that all smooth Fano varieties are rationally connected (see [Ca92], [KMM92a] or [Kol96]). More generally, log Q-Fano varieties are rationally connected ([Zh06] or [HM07]).…”

Rationally connected varieties

“…We make this precise in the next theorem, which was proved in [Cam92] and [KMM92b]. Moreover, this morphism is unique up to birational equivalence.…”

“…As a substitute, the more general concept of rationally connected varieties was introduced in [Cam92] and [KMM92b]. As we shall see in the next section, this bigger class of varieties is very well behaved.…”

“…We have proved that H 0 (X, (Ω 1 X ) ⊗m ) = 0. The fact that Fano varieties are rationally connected was established in [Cam92] and [KMM92a].…”

Characterizations of projective spaces and hyperquadrics

“…, H k )-equivalent if they can be connected by a chain of 1-cycles from H 1 ∪ · · · ∪ H k . By [Cam92] (see also [Kol96,IV.4.16]), there exists a proper surjective morphism π 0 : X 0 → Y 0 from a dense open subset of X onto a normal variety whose fibers are (H 1 , . .…”