Generalized Mukai conjecture for special Fano varieties

Marco Andreatta; Elena Chierici; Gianluca Occhetta

Open Mathematics (2004)

  • Volume: 2, Issue: 2, page 272-293
  • ISSN: 2391-5455

Abstract

top
Let X be a Fano variety of dimension n, pseudoindex i X and Picard number ρX. A generalization of a conjecture of Mukai says that ρX(i X−1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i X≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρX> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.

How to cite

top

Marco Andreatta, Elena Chierici, and Gianluca Occhetta. "Generalized Mukai conjecture for special Fano varieties." Open Mathematics 2.2 (2004): 272-293. <http://eudml.org/doc/268759>.

@article{MarcoAndreatta2004,
abstract = {Let X be a Fano variety of dimension n, pseudoindex i X and Picard number ρX. A generalization of a conjecture of Mukai says that ρX(i X−1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i X≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρX> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.},
author = {Marco Andreatta, Elena Chierici, Gianluca Occhetta},
journal = {Open Mathematics},
keywords = {14J45; 14E30},
language = {eng},
number = {2},
pages = {272-293},
title = {Generalized Mukai conjecture for special Fano varieties},
url = {http://eudml.org/doc/268759},
volume = {2},
year = {2004},
}

TY - JOUR
AU - Marco Andreatta
AU - Elena Chierici
AU - Gianluca Occhetta
TI - Generalized Mukai conjecture for special Fano varieties
JO - Open Mathematics
PY - 2004
VL - 2
IS - 2
SP - 272
EP - 293
AB - Let X be a Fano variety of dimension n, pseudoindex i X and Picard number ρX. A generalization of a conjecture of Mukai says that ρX(i X−1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i X≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρX> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.
LA - eng
KW - 14J45; 14E30
UR - http://eudml.org/doc/268759
ER -

References

top
  1. [1] M. Andreatta and J.A. Wiśniewski: “On manifolds whose tangent bundle contains an ample subbundle”, Invent. Math., Vol. 146, (2001), pp. 209–217. http://dx.doi.org/10.1007/PL00005808 Zbl1081.14060
  2. [2] L. Bonavero, C. Casagrande, O. Debarre and S. Druel: “Sur une conjecture de Mukai”, Comment. Math. Helv., Vol. 78, (2003), pp. 601–626. http://dx.doi.org/10.1007/s00014-003-0765-x Zbl1044.14019
  3. [3] L. Bonavero, F. Campana and J.A. Wiśniewski: “Variétés complexes dont l'éclat'ee en un point est de Fano”, C.R. Math. Acad. Sci. Paris, Vol. 334, (2002), pp. 463–468. Zbl1036.14020
  4. [4] F. Campana: “Connexité rationnelle des variétés de Fano”, Ann. Sci. École Norm. Sup., Vol. 25, (1992), pp. 539–545. 
  5. [5] K. Cho, Y. Miyaoka and N.I. Shepherd-Barron: “Characterizations of projective space and applications to complex symplectic manifolds”, in: Higher dimensional birational geometry (Kyoto, 1997) Adv. Stud. Pure Math., Vol. 35, Math. Soc. Japan, Tokyo, 2002, pp. 1–88. Zbl1063.14065
  6. [6] O. Debarre: Higher-Dimensional Algebraic Geometry, Universitext Springer-Verlag, New York, 2001. 
  7. [7] S. Kebekus: “Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron”, In: Complex geometry (Göttingen, 2000), Springer, Berlin, 2002, pp. 147–155. Zbl1046.14028
  8. [8] J. Kollár: Rational Curves on Algebraic Varieties, Ergebnisse der Math. Vol. 32, Springer-Verlag, 1996. 
  9. [9] J. Kollár, Y. Miyaoka and S. Mori: “Rational connectedness and boundedness of Fano manifolds”, J. Diff. Geom. Vol. 36, (1992), pp. 765–779. Zbl0759.14032
  10. [10] S. Mori: “Projective manifolds with ample tangent bundle”, Ann. Math., Vol. 110, (1979), pp. 595–606. http://dx.doi.org/10.2307/1971241 Zbl0423.14006
  11. [11] S. Mukai: “Open problems”, In: Birational geometry of algebraic varieties, Taniguchi Foundation, Katata, 1988. 
  12. [12] G. Occhetta: A characterization of products of projective spaces, preprint, February 2003, http://www.science.unitn.it/∼occhetta. 
  13. [13] J.A. Wiśniewski: “On a conjecture of Mukai”, Manuscripta Math., Vol. 68, (1990), pp. 135–141. Zbl0715.14033

NotesEmbed ?

top

You must be logged in to post comments.