The number of vertices of a Fano polytope
Annales de l'Institut Fourier, Volume 56 (2006) no. 1, pp. 121-130.

Let X be a Gorenstein, -factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of X in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the maximal number of vertices of a simplicial reflexive polytope.

Soit X une variété de Fano torique, Gorenstein et -factorielle. Nous démontrons deux conjectures sur le nombre de Picard maximal de X en fonction de sa dimension et de son pseudo-indice, et nous caractérisons les cas limites. De façon équivalente, nous déterminons le nombre maximal de sommets d’un polytope réflexif simplicial.

DOI: 10.5802/aif.2175
Classification: 52B20,  14M25,  14J45
Keywords: toric varieties, Fano varieties, reflexive polytopes, Fano polytopes
Casagrande, Cinzia 1

1 Università di Pisa Dipartimento di Matematica “L. Tonelli” Largo Bruno Pontecorvo, 5 56127 Pisa (Italy)
@article{AIF_2006__56_1_121_0,
     author = {Casagrande, Cinzia},
     title = {The number of vertices of a {Fano} polytope},
     journal = {Annales de l'Institut Fourier},
     pages = {121--130},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {56},
     number = {1},
     year = {2006},
     doi = {10.5802/aif.2175},
     mrnumber = {2228683},
     zbl = {1095.52005},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2175/}
}
TY  - JOUR
AU  - Casagrande, Cinzia
TI  - The number of vertices of a Fano polytope
JO  - Annales de l'Institut Fourier
PY  - 2006
DA  - 2006///
SP  - 121
EP  - 130
VL  - 56
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2175/
UR  - https://www.ams.org/mathscinet-getitem?mr=2228683
UR  - https://zbmath.org/?q=an%3A1095.52005
UR  - https://doi.org/10.5802/aif.2175
DO  - 10.5802/aif.2175
LA  - en
ID  - AIF_2006__56_1_121_0
ER  - 
%0 Journal Article
%A Casagrande, Cinzia
%T The number of vertices of a Fano polytope
%J Annales de l'Institut Fourier
%D 2006
%P 121-130
%V 56
%N 1
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2175
%R 10.5802/aif.2175
%G en
%F AIF_2006__56_1_121_0
Casagrande, Cinzia. The number of vertices of a Fano polytope. Annales de l'Institut Fourier, Volume 56 (2006) no. 1, pp. 121-130. doi : 10.5802/aif.2175. https://aif.centre-mersenne.org/articles/10.5802/aif.2175/

[1] Andreatta, Marco; Chierici, Elena; Occhetta, Gianluca Generalized Mukai conjecture for special Fano varieties, Central European Journal of Mathematics, Volume 2 (2004) no. 2, pp. 272-293 | DOI | MR | Zbl

[2] Batyrev, Victor V. Toric Fano threefolds, Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, Volume 45 (1981) no. 4, pp. 704-717 (in Russian). English translation: Mathematics of the USSR Izvestiya, 19 (1982), p. 13-25 | MR | Zbl

[3] Batyrev, Victor V. Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, Journal of Algebraic Geometry, Volume 3 (1994), pp. 493-535 | MR | Zbl

[4] Batyrev, Victor V. On the classification of toric Fano 4-folds, Journal of Mathematical Sciences (New York), Volume 94 (1999), pp. 1021-1050 | DOI | MR | Zbl

[5] Bonavero, Laurent; Casagrande, Cinzia; Debarre, Olivier; Druel, Stéphane Sur une conjecture de Mukai, Commentarii Mathematici Helvetici, Volume 78 (2003), pp. 601-626 | DOI | MR | Zbl

[6] Casagrande, Cinzia Toric Fano varieties and birational morphisms, International Mathematics Research Notices, Volume 27 (2003), pp. 1473-1505 | DOI | MR | Zbl

[7] Cho, Koji; Miyaoka, Yoichi; Shepherd-Barron, Nick Characterizations of projective space and applications to complex symplectic geometry, Higher Dimensional Birational Geometry (Advanced Studies in Pure Mathematics), Volume 35, Mathematical Society of Japan, 2002, pp. 1-89 | MR | Zbl

[8] Debarre, Olivier Higher-Dimensional Algebraic Geometry, Universitext, Springer Verlag, 2001 | MR | Zbl

[9] Debarre, Olivier Fano varieties, Higher Dimensional Varieties and Rational Points (Bolyai Society Mathematical Studies), Volume 12 (2003), pp. 93-132 | MR

[10] Ewald, Günter Combinatorial Convexity and Algebraic Geometry, Graduate Texts in Mathematics, 168, Springer Verlag, 1996 | MR | Zbl

[11] Grünbaum, Branko Convex Polytopes, Graduate Texts in Mathematics, 221, Springer Verlag, 2003 (first edition 1967) | MR | Zbl

[12] Nill, Benjamin Complete toric varieties with reductive automorphism group (2004) (preprint math.AG/0407491)

[13] Nill, Benjamin Gorenstein toric Fano varieties, Manuscripta Mathematica, Volume 116 (2005) no. 2, pp. 183-210 | DOI | MR | Zbl

[14] Occhetta, Gianluca A characterization of products of projective spaces (2003) (preprint, available at the author’s web page http://www.science.unitn.it/~occhetta/)

[15] Sato, Hiroshi Toward the classification of higher-dimensional toric Fano varieties, Tôhoku Mathematical Journal, Volume 52 (2000), pp. 383-413 | DOI | MR | Zbl

[16] Voskresenskiĭ, V. E.; Klyachko, Alexander Toric Fano varieties and systems of roots, Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya, Volume 48 (1984) no. 2, pp. 237-263 (in Russian). English translation: Mathematics of the USSR Izvestiya, 24 (1985), p. 221-244 | MR | Zbl

[17] Watanabe, Keiichi; Watanabe, Masayuki The classification of Fano 3-folds with torus embeddings, Tokyo Journal of Mathematics, Volume 5 (1982), pp. 37-48 | DOI | MR | Zbl

[18] Wiśniewski, Jarosław A. On a conjecture of Mukai, Manuscripta Mathematica, Volume 68 (1990), pp. 135-141 | DOI | MR | Zbl

[19] Wiśniewski, Jarosław A. Toric Mori theory and Fano manifolds, Geometry of Toric Varieties (Séminaires et Congrès), Volume 6, Société Mathématique de France, 2002, pp. 249-272 | MR | Zbl

Cited by Sources: