On the Picard number of divisors in Fano manifolds
[Sur le nombre de Picard des diviseurs dans les variétés de Fano]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 3, pp. 363-403.

Soient X une variété de Fano lisse et complexe de dimension arbitraire, et D un diviseur premier dans X. Nous considérons l’image 𝒩 1 (D,X) de 𝒩 1 (D) dans 𝒩 1 (X) par l’application naturelle de push-forward de 1-cycles. Nous démontrons que ρ X -ρ D codim𝒩 1 (D,X)8. De plus, si codim𝒩 1 (D,X)3, alors soit XS×TS est une surface de Del Pezzo, soit codim𝒩 1 (D,X)=3 et X a une fibration en surfaces de Del Pezzo sur une variété de Fano lisse T, telle que ρ X -ρ T =4.

Let X be a complex Fano manifold of arbitrary dimension, and D a prime divisor in X. We consider the image 𝒩 1 (D,X) of 𝒩 1 (D) in 𝒩 1 (X) under the natural push-forward of 1-cycles. We show that ρ X -ρ D codim𝒩 1 (D,X)8. Moreover if codim𝒩 1 (D,X)3, then either XS×T where S is a Del Pezzo surface, or codim𝒩 1 (D,X)=3 and X has a fibration in Del Pezzo surfaces onto a Fano manifold T such that ρ X -ρ T =4.

DOI : 10.24033/asens.2168
Classification : 14J45, 14E30
Keywords: Fano varieties, Mori theory, extremal rays
Mot clés : variétés de Fano, théorie de Mori, rayons extrêmaux
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     title = {On the {Picard} number of divisors in {Fano} manifolds},
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Casagrande, Cinzia. On the Picard number of divisors in Fano manifolds. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 3, pp. 363-403. doi : 10.24033/asens.2168. http://www.numdam.org/articles/10.24033/asens.2168/

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