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On empty lattice simplices in dimension 4

Authors: Margherita Barile, Dominique Bernardi, Alexander Borisov and Jean-Michel Kantor
Journal: Proc. Amer. Math. Soc. 139 (2011), 4247-4253
MSC (2010): Primary 14B05; Secondary 14M25, 52B20
Published electronically: April 28, 2011
MathSciNet review: 2823070
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Abstract: We give an almost complete classification of empty lattice simplices in dimension 4 using the conjectural results of Mori-Morrison-Morrison, which were later proved by Sankaran and Bober. In particular, all of these simplices correspond to cyclic quotient singularities, and all but finitely many of them have width bounded by 2.

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  • 1. J. W. Bober, Factorial ratios, hypergeometric series, and a family of step functions, J. Lond. Math. Soc. (2) 79 (2009), no. 2, 422-444. MR 2496522 (2010f:11138)
  • 2. A. Borisov. On classification of toric singularities, Algebraic geometry, 9. J. Math. Sci. (New York) 94 (1999), no. 1, 1111-1113. MR 1703910 (2000e:14003)
  • 3. A. Borisov. Quotient singularities, integer ratios of factorials, and the Riemann hypothesis, Int. Math. Res. Not. IMRN 2008, no. 15, Art. ID rnn052. MR 2438068 (2009f:11108)
  • 4. N. Bourbaki, Éléments de mathématique, Livre II, Algèbre, Ch. 7, Modules sur un anneau principal, Hermann, Paris, 1964.
  • 5. W. Fulton, Introduction to toric varieties, Princeton University Press, Princeton, 1993. MR 1234037 (94g:14028)
  • 6. Chr. Haase, G. M. Ziegler, On the maximal width of empty lattice simplices. Europ. J. Combinatorics 21 (2000), 111-119. MR 1737331 (2001a:52006)
  • 7. M. Hall, Jr., The theory of groups, The Macmillan Co., New York, NY, 1959. MR 0103215 (21:1996)
  • 8. J.-M. Kantor, On the width of lattice-free simplices, Compos. Math. 118 (1999), 235-241. MR 1711323 (2001h:52011)
  • 9. Sh. Mori, D. Morrison, I. Morrison, On four-dimensional terminal quotient singularities, Math. Comp. 51 (1988), 769-786. MR 958643 (89i:14036)
  • 10. D. Morrison, G. Stevens, Terminal quotient singularities in dimensions three and four, Proc. Amer. Math. Soc. 90 (1984), 15-20. MR 722406 (85a:14004)
  • 11. M. Reid, Decomposition of toric morphisms. in: Arithmetic and geometry. Pap. dedic. I. R. Shafarevich, Vol. II, Progr. Math. 36, Birkhäuser Boston, 1983, 395-418. MR 717617 (85e:14071)
  • 12. G. K. Sankaran, Stable quintuples and terminal quotient singularities, Math. Proc. Cambridge Philos. Soc. 107 (1990), 91-101. MR 1021875 (90j:14017)
  • 13. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, 11, Iwanami Shoten and Princeton University Press, 1971. MR 1291394 (95e:11048)
  • 14. V. I. Vasyunin, On a system of step functions. (Russian. English, Russian summary) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 262 (1999), Issled. po Linein. Oper. i Teor. Funkts. 27, 49-70, 231-232; translation in J. Math. Sci. (New York) 110 (2002), no. 5, 2930-2943. MR 1734327 (2001i:11112)
  • 15. U. Wessels, Die Sätze von White und Mordell über kritische Gitter von Polytopen in den Dimensionen 4 und 5. Master Thesis, Ruhr-Universität Bochum, 1989.
  • 16. G. K. White, Lattice tetrahedra, Canadian J. Math. 16 (1964), 389-396. MR 0161837 (28:5041)

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Additional Information

Margherita Barile
Affiliation: Dipartimento di Matematica, Università di Bari “Aldo Moro”, Via E. Orabona 4, 70125 Bari, Italy

Dominique Bernardi
Affiliation: Université Pierre et Marie Curie, Institut Mathématique de Jussieu, 175 Rue du Chevaleret, F-75013, Paris, France

Alexander Borisov
Affiliation: Department of Mathematics, 301 Thackeray Hall, University of Pittsburgh, Pittsburgh, Pennsylvania 15260

Jean-Michel Kantor
Affiliation: Université Paris Diderot, Institut Mathématique de Jussieu, 175 Rue du Chevaleret, F-75013, Paris, France

Keywords: Lattice polytopes, terminal singularities, width
Received by editor(s): May 6, 2010
Received by editor(s) in revised form: October 22, 2010
Published electronically: April 28, 2011
Additional Notes: The research of the first author has been co-financed by the Italian Ministry of Education, University and Research (PRIN “Algebra Commutativa, Combinatoria e Computazionale”).
The research of the third author has been supported by NSA, grant H98230-08-1-0129
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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