On the Picard number of divisors in Fano manifolds

Casagrande, Cinzia

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, Tome 45 (2012), p. 363-403 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

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 $ρ_{X} - ρ_{T} = 4$ .

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 $ρ_{X} - ρ_{T} = 4$ .

Publié le : 2012-01-01

MR 3014481 | Zbl 1267.14050

DOI : https://doi.org/10.24033/asens.2168
Classification: 14J45, 14E30
Mots clés: variétés de Fano, théorie de Mori, rayons extrêmaux

@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},
     volume = {45},
     year = {2012},
     pages = {363-403},
     doi = {10.24033/asens.2168},
     mrnumber = {3014481},
     zbl = {1267.14050},
     language = {en},
     url = {http://dml.mathdoc.fr/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, Tome 45 (2012) pp. 363-403. doi : 10.24033/asens.2168. http://gdmltest.u-ga.fr/item/ASENS_2012_4_45_3_363_0/

Bibliographie

[1] T. Ando, On extremal rays of the higher-dimensional varieties, Invent. Math. 81 (1985), 347-357. | MR 799271 | Zbl 0554.14001

[2] M. Andreatta, E. Ballico & J. A. Wiśniewski, Vector bundles and adjunction, Internat. J. Math. 3 (1992), 331-340. | MR 1163727 | Zbl 0770.14008

[3] M. Andreatta & J. A. Wiśniewski, A view on contractions of higher-dimensional varieties, in Algebraic geometry-Santa Cruz 1995, Proc. Sympos. Pure Math. 62, Amer. Math. Soc., 1997, 153-183. | MR 1492522 | Zbl 0948.14014

[4] A. Beauville, Prym varieties and the Schottky problem, Invent. Math. 41 (1977), 149-196. | MR 572974 | Zbl 0333.14013

[5] C. Birkar, P. Cascini, C. D. Hacon & J. Mckernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405-468. | MR 2601039 | Zbl 1210.14019

[6] L. Bonavero, F. Campana & J. A. Wiśniewski, Variétés complexes dont l'éclatée en un point est de Fano, C. R. Math. Acad. Sci. Paris 334 (2002), 463-468. | MR 1890634 | Zbl 1036.14020

[7] L. Bonavero, C. Casagrande, O. Debarre & S. Druel, Sur une conjecture de Mukai, Comment. Math. Helv. 78 (2003), 601-626. | MR 1998396 | Zbl 1044.14019

[8] L. Bonavero, C. Casagrande & S. Druel, On covering and quasi-unsplit families of curves, J. Eur. Math. Soc. (JEMS) 9 (2007), 45-57. | MR 2283102 | Zbl 1107.14015

[9] C. Casagrande, Toric Fano varieties and birational morphisms, Int. Math. Res. Not. 27 (2003), 1473-1505. | MR 1976232 | Zbl 1083.14516

[10] C. Casagrande, Quasi-elementary contractions of Fano manifolds, Compos. Math. 144 (2008), 1429-1460. | MR 2474316 | Zbl 1158.14037

[11] C. Casagrande, On Fano manifolds with a birational contraction sending a divisor to a curve, Michigan Math. J. 58 (2009), 783-805. | MR 2595565 | Zbl 1184.14072

[12] O. Debarre, Higher-dimensional algebraic geometry, Universitext, Springer, 2001. | MR 1841091 | Zbl 0978.14001

[13] Y. Hu & S. Keel, Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331-348. | MR 1786494 | Zbl 1077.14554

[14] S. Ishii, Quasi-Gorenstein Fano $3$ -folds with isolated nonrational loci, Compositio Math. 77 (1991), 335-341. | Numdam | MR 1092773 | Zbl 0738.14025

[15] J. Kollár, Rational curves on algebraic varieties, Ergebn. Math. Grenzg. 32, Springer, 1996. | Zbl 0877.14012

[16] J. Kollár & S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge Univ. Press, 1998. | Zbl 0926.14003

[17] R. Lazarsfeld, Positivity in algebraic geometry. I, Ergebn. Math. Grenzg. 48, Springer, 2004. | MR 2095471 | Zbl 1093.14500

[18] G. Occhetta, A characterization of products of projective spaces, Canad. Math. Bull. 49 (2006), 270-280. | Zbl 1115.14034

[19] V. G. Sarkisov, On conic bundle structures, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 371-408; English translation: Math. USSR Izvestia 20 (1982), 355-390. | Zbl 0593.14034

[20] T. Tsukioka, Classification of Fano manifolds containing a negative divisor isomorphic to projective space, Geom. Dedicata 123 (2006), 179-186. | Zbl 1121.14036

[21] J. A. Wiśniewski, On contractions of extremal rays of Fano manifolds, J. reine angew. Math. 417 (1991), 141-157. | Zbl 0721.14023