On the Picard number of divisors in Fano manifolds

Cinzia Casagrande

Annales scientifiques de l'École Normale Supérieure (2012)

  • Volume: 45, Issue: 3, page 363-403
  • ISSN: 0012-9593

Abstract

top
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 ≤ codim 𝒩 1 ( D , X ) ≤ 8 . Moreover if codim 𝒩 1 ( D , X ) ≥ 3 , then either X ≅ S × 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 .

How to cite

top

Casagrande, Cinzia. "On the Picard number of divisors in Fano manifolds." Annales scientifiques de l'École Normale Supérieure 45.3 (2012): 363-403. <http://eudml.org/doc/272206>.

@article{Casagrande2012,
abstract = {Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$. We consider the image $\mathcal \{N\}_1(D,X)$ of $\mathcal \{N\}_1(D)$ in $\mathcal \{N\}_1(X)$ under the natural push-forward of $1$-cycles. We show that $\rho _X-\rho _D\le \operatorname\{codim\}\mathcal \{N\}_1(D,X)\le 8$. Moreover if $\operatorname\{codim\}\mathcal \{N\}_1(D,X)\ge 3$, then either $X\cong S\times T$ where $S$ is a Del Pezzo surface, or $\operatorname\{codim\}\mathcal \{N\}_1(D,X)=3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that $\rho _X-\rho _T=4$.},
author = {Casagrande, Cinzia},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Fano varieties; Mori theory; extremal rays},
language = {eng},
number = {3},
pages = {363-403},
publisher = {Société mathématique de France},
title = {On the Picard number of divisors in Fano manifolds},
url = {http://eudml.org/doc/272206},
volume = {45},
year = {2012},
}

TY - JOUR
AU - Casagrande, Cinzia
TI - On the Picard number of divisors in Fano manifolds
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2012
PB - Société mathématique de France
VL - 45
IS - 3
SP - 363
EP - 403
AB - Let $X$ be a complex Fano manifold of arbitrary dimension, and $D$ a prime divisor in $X$. We consider the image $\mathcal {N}_1(D,X)$ of $\mathcal {N}_1(D)$ in $\mathcal {N}_1(X)$ under the natural push-forward of $1$-cycles. We show that $\rho _X-\rho _D\le \operatorname{codim}\mathcal {N}_1(D,X)\le 8$. Moreover if $\operatorname{codim}\mathcal {N}_1(D,X)\ge 3$, then either $X\cong S\times T$ where $S$ is a Del Pezzo surface, or $\operatorname{codim}\mathcal {N}_1(D,X)=3$ and $X$ has a fibration in Del Pezzo surfaces onto a Fano manifold $T$ such that $\rho _X-\rho _T=4$.
LA - eng
KW - Fano varieties; Mori theory; extremal rays
UR - http://eudml.org/doc/272206
ER -

References

top
  1. [1] T. Ando, On extremal rays of the higher-dimensional varieties, Invent. Math.81 (1985), 347–357. Zbl0554.14001MR799271
  2. [2] M. Andreatta, E. Ballico & J. A. Wiśniewski, Vector bundles and adjunction, Internat. J. Math.3 (1992), 331–340. Zbl0770.14008MR1163727
  3. [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. Zbl0948.14014MR1492522
  4. [4] A. Beauville, Prym varieties and the Schottky problem, Invent. Math.41 (1977), 149–196. Zbl0333.14013MR572974
  5. [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. Zbl1210.14019MR2601039
  6. [6] L. Bonavero, F. Campana & J. A. Wiśniewski, Variétés complexes dont l’éclatée en un point est de Fano, C. R. Math. Acad. Sci. Paris334 (2002), 463–468. Zbl1036.14020MR1890634
  7. [7] L. Bonavero, C. Casagrande, O. Debarre & S. Druel, Sur une conjecture de Mukai, Comment. Math. Helv.78 (2003), 601–626. Zbl1044.14019MR1998396
  8. [8] L. Bonavero, C. Casagrande & S. Druel, On covering and quasi-unsplit families of curves, J. Eur. Math. Soc. (JEMS) 9 (2007), 45–57. Zbl1107.14015MR2283102
  9. [9] C. Casagrande, Toric Fano varieties and birational morphisms, Int. Math. Res. Not.27 (2003), 1473–1505. Zbl1083.14516MR1976232
  10. [10] C. Casagrande, Quasi-elementary contractions of Fano manifolds, Compos. Math.144 (2008), 1429–1460. Zbl1158.14037MR2474316
  11. [11] C. Casagrande, On Fano manifolds with a birational contraction sending a divisor to a curve, Michigan Math. J.58 (2009), 783–805. Zbl1184.14072MR2595565
  12. [12] O. Debarre, Higher-dimensional algebraic geometry, Universitext, Springer, 2001. Zbl0978.14001MR1841091
  13. [13] Y. Hu & S. Keel, Mori dream spaces and GIT, Michigan Math. J.48 (2000), 331–348. Zbl1077.14554MR1786494
  14. [14] S. Ishii, Quasi-Gorenstein Fano 3 -folds with isolated nonrational loci, Compositio Math.77 (1991), 335–341. Zbl0738.14025MR1092773
  15. [15] J. Kollár, Rational curves on algebraic varieties, Ergebn. Math. Grenzg. 32, Springer, 1996. Zbl0877.14012
  16. [16] J. Kollár & S. Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics 134, Cambridge Univ. Press, 1998. Zbl0926.14003
  17. [17] R. Lazarsfeld, Positivity in algebraic geometry. I, Ergebn. Math. Grenzg. 48, Springer, 2004. Zbl1093.14500MR2095471
  18. [18] G. Occhetta, A characterization of products of projective spaces, Canad. Math. Bull.49 (2006), 270–280. Zbl1115.14034
  19. [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. Zbl0593.14034
  20. [20] T. Tsukioka, Classification of Fano manifolds containing a negative divisor isomorphic to projective space, Geom. Dedicata123 (2006), 179–186. Zbl1121.14036
  21. [21] J. A. Wiśniewski, On contractions of extremal rays of Fano manifolds, J. reine angew. Math. 417 (1991), 141–157. Zbl0721.14023

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.