On the Picard number of divisors in Fano manifolds

Casagrande, Cinzia

doi:10.24033/asens.2168

On the Picard number of divisors in Fano manifolds
[Sur le nombre de Picard des diviseurs dans les variétés de Fano]

Casagrande, Cinzia

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 3, pp. 363-403

Abstract (VO)
Résumé

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 $ρ_{X} - ρ_{D} \leq codim 𝒩_{1} (D, X) \leq 8$ . Moreover if $codim 𝒩_{1} (D, X) \geq 3$ , then either $X ≅ S \times T$ where $S$ is a Del Pezzo surface, or $codim 𝒩_{1} (D, X) = 3$ 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 $ρ_{X} - ρ_{D} \leq codim 𝒩_{1} (D, X) \leq 8$ . De plus, si $codim 𝒩_{1} (D, X) \geq 3$ , alors soit $X ≅ S \times T$ où $S$ est une surface de Del Pezzo, soit $codim 𝒩_{1} (D, X) = 3$ et $X$ a une fibration en surfaces de Del Pezzo sur une variété de Fano lisse $T$ , telle que $ρ_{X} - ρ_{T} = 4$ .

EuDML MR Zbl | 1 citation dans Numdam

DOI : 10.24033/asens.2168

Classification : 14J45, 14E30
Keywords: Fano varieties, Mori theory, extremal rays
Mots-clés : variétés de Fano, théorie de Mori, rayons extrêmaux

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     author = {Casagrande, Cinzia},
     title = {On the {Picard} number of divisors in {Fano} manifolds},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {363--403},
     year = {2012},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 45},
     number = {3},
     doi = {10.24033/asens.2168},
     mrnumber = {3014481},
     zbl = {1267.14050},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2168/}
}

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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

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