On the Picard number of divisors in Fano manifolds
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 3, p. 363-403

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.

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.

DOI : https://doi.org/10.24033/asens.2168
Classification:  14J45,  14E30
Keywords: Fano varieties, Mori theory, extremal rays
@article{ASENS_2012_4_45_3_363_0,
     author = {Casagrande, Cinzia},
     title = {On the Picard number of divisors in Fano manifolds},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 45},
     number = {3},
     year = {2012},
     pages = {363-403},
     doi = {10.24033/asens.2168},
     zbl = {1267.14050},
     mrnumber = {3014481},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2012_4_45_3_363_0}
}
Casagrande, Cinzia. On the Picard number of divisors in Fano manifolds. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 3, pp. 363-403. doi : 10.24033/asens.2168. http://www.numdam.org/item/ASENS_2012_4_45_3_363_0/

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