On some numerical properties of Fano varieties

Cinzia Casagrande

Bollettino dell'Unione Matematica Italiana (2004)

  • Volume: 7-B, Issue: 3, page 663-671
  • ISSN: 0392-4041

Abstract

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This is the text of a talk given at the XVII Convegno dell’Unione Matematica Italiana held at Milano, September 8-13, 2003. I would like to thank Angelo Lopez and Ciro Ciliberto for the kind invitation to the conference. I survey some numerical conjectures and theorems concerning relations between the index, the pseudo-index and the Picard number of a Fano variety. The results I refer to are contained in the paper [3], wrote in collaboration with Bonavero, Debarre and Druel.

How to cite

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Casagrande, Cinzia. "On some numerical properties of Fano varieties." Bollettino dell'Unione Matematica Italiana 7-B.3 (2004): 663-671. <http://eudml.org/doc/195885>.

@article{Casagrande2004,
abstract = {This is the text of a talk given at the XVII Convegno dell’Unione Matematica Italiana held at Milano, September 8-13, 2003. I would like to thank Angelo Lopez and Ciro Ciliberto for the kind invitation to the conference. I survey some numerical conjectures and theorems concerning relations between the index, the pseudo-index and the Picard number of a Fano variety. The results I refer to are contained in the paper [3], wrote in collaboration with Bonavero, Debarre and Druel.},
author = {Casagrande, Cinzia},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {663-671},
publisher = {Unione Matematica Italiana},
title = {On some numerical properties of Fano varieties},
url = {http://eudml.org/doc/195885},
volume = {7-B},
year = {2004},
}

TY - JOUR
AU - Casagrande, Cinzia
TI - On some numerical properties of Fano varieties
JO - Bollettino dell'Unione Matematica Italiana
DA - 2004/10//
PB - Unione Matematica Italiana
VL - 7-B
IS - 3
SP - 663
EP - 671
AB - This is the text of a talk given at the XVII Convegno dell’Unione Matematica Italiana held at Milano, September 8-13, 2003. I would like to thank Angelo Lopez and Ciro Ciliberto for the kind invitation to the conference. I survey some numerical conjectures and theorems concerning relations between the index, the pseudo-index and the Picard number of a Fano variety. The results I refer to are contained in the paper [3], wrote in collaboration with Bonavero, Debarre and Druel.
LA - eng
UR - http://eudml.org/doc/195885
ER -

References

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  1. ANDREATTA, MARCO- CHIERICI, ELENA- OCCHETTA, GIANLUCA, Generalized Mukai conjecture for special Fano varieties, preprint math.AG/0309473, 2003. To appear on Central European Journal of Mathematics. Zbl1068.14049MR2113552
  2. BATYREV, VICTOR V., On the classification of toric Fano 4-folds, Journal of Mathematical Sciences (New York), 94 (1999), 1021-1050. Zbl0929.14024MR1703904
  3. BONAVERO, LAURENT- CASAGRANDE, CINZIA- DEBARRE, OLIVIER- DRUEL, STÉPHANE, Sur une conjecture de Mukai, Commentarii Mathematici Helvetici, 78 (2003), 601-626. Zbl1044.14019MR1998396
  4. BONAVERO, LAURENT, Pseudo-index of Fano manifolds and smooth blowups, preprint math.AG/0309460, 2003. Zbl1081.14056
  5. CASAGRANDE, CINZIA, Contractible classes in toric varieties, Mathematische Zeitschrift, 243 (2003), 99-126. Zbl1077.14070MR1953051
  6. CASAGRANDE, CINZIA, Toric Fano varieties and birational morphisms, International Mathematics Research Notices, 27 (2003), 1473-1505. Zbl1083.14516MR1976232
  7. CHO, KOJI- MIYAOKA, YOICHI- SHEPHERD-BARRON, NICK, Characterizations of projective space and applications to complex symplectic geometry. In Higher Dimensional Birational Geometry, volume 35 of Advanced Studies in Pure Mathematics, 1-89, Mathematical Society of Japan, 2002. Zbl1063.14065MR1929791
  8. DEBARRE, OLIVIER, Fano varieties. In Higher Dimensional Varieties and Rational Points (Budapest, 2001), volume 12 of Bolyai Society Mathematical Studies, 93-132, Springer-Verlag, 2003. Zbl0939.14022MR2011745
  9. ISKOVSKIKH, VASILII A.- PROKHOROV, YURI G., Algebraic Geometry V - Fano Varieties, volume 47 of Encyclopaedia of Mathematical Sciences, Springer-Verlag, 1999. MR1668579
  10. KOLLÁR, JÁNOS, Rational Curves on Algebraic Varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, 1996. Zbl0877.14012MR1440180
  11. MUKAI, SHIGERU, Problems on characterization of the complex projective space. In Birational Geometry of Algebraic Varieties, Open Problems, Proceedings of the 23rd Symposium of the Taniguchi Foundation at Katata, Japan, 57-60, 1988. 
  12. OCCHETTA, GIANLUCA, A characterization of products of projective spaces, Preprint, available at the author's web page http://www.science.unitn.it/~occhetta/mainh.hmtl, 2003. Zbl1115.14034MR2226250
  13. ODA, TADAO, Convex Bodies and Algebraic Geometry - An Introduction to the Theory of Toric Varieties, volume 15 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, 1988. Zbl0628.52002MR922894
  14. SATO, HIROSHI, Toward the classification of higher-dimensional toric Fano varieties, Tôhoku Mathematical Journal, 52 (2000), 383-413. Zbl1028.14015MR1772804
  15. VOSKRESENSKĬ, V. E.- KLYACHKO, ALEXANDER, Toroidal Fano varieties and roots systems, Mathematics of the USSR Izvestiya, 24 (1985), 221-244. Zbl0572.14029
  16. WIŚNIEWSKI, JAROSLAW A., On a conjecture of Mukai, Manuscripta Mathematica, 68 (1990), 135-141. Zbl0715.14033MR1063222
  17. WIŚNIEWSKI, JAROSLAW A., Toric Mori theory and Fano manifolds. In Geometry of Toric Varieties, volume 6 of Séminaires et Congrés, pages 249-272. Société Mathématique de France, 2002. Zbl1053.14002MR2063740

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