On empty lattice simplices in dimension 4
HTML articles powered by AMS MathViewer
- by Margherita Barile, Dominique Bernardi, Alexander Borisov and Jean-Michel Kantor
- Proc. Amer. Math. Soc. 139 (2011), 4247-4253
- DOI: https://doi.org/10.1090/S0002-9939-2011-10859-1
- Published electronically: April 28, 2011
- PDF | Request permission
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.References
- Jonathan 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, DOI 10.1112/jlms/jdn078
- A. Borisov, On classification of toric singularities, J. Math. Sci. (New York) 94 (1999), no. 1, 1111–1113. Algebraic geometry, 9. MR 1703910, DOI 10.1007/BF02367251
- Alexander Borisov, Quotient singularities, integer ratios of factorials, and the Riemann hypothesis, Int. Math. Res. Not. IMRN 15 (2008), Art. ID rnn052, 19. MR 2438068, DOI 10.1093/imrn/rnn052
- N. Bourbaki, Éléments de mathématique, Livre II, Algèbre, Ch. 7, Modules sur un anneau principal, Hermann, Paris, 1964.
- William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR 1234037, DOI 10.1515/9781400882526
- Christian Haase and Günter M. Ziegler, On the maximal width of empty lattice simplices, European J. Combin. 21 (2000), no. 1, 111–119. Combinatorics of polytopes. MR 1737331, DOI 10.1006/eujc.1999.0325
- Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215
- Jean-Michel Kantor, On the width of lattice-free simplices, Compositio Math. 118 (1999), no. 3, 235–241. MR 1711323, DOI 10.1023/A:1001164317215
- Shigefumi Mori, David R. Morrison, and Ian Morrison, On four-dimensional terminal quotient singularities, Math. Comp. 51 (1988), no. 184, 769–786. MR 958643, DOI 10.1090/S0025-5718-1988-0958643-5
- David R. Morrison and Glenn Stevens, Terminal quotient singularities in dimensions three and four, Proc. Amer. Math. Soc. 90 (1984), no. 1, 15–20. MR 722406, DOI 10.1090/S0002-9939-1984-0722406-4
- Miles Reid, Decomposition of toric morphisms, Arithmetic and geometry, Vol. II, Progr. Math., vol. 36, Birkhäuser Boston, Boston, MA, 1983, pp. 395–418. MR 717617
- G. K. Sankaran, Stable quintuples and terminal quotient singularities, Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 1, 91–101. MR 1021875, DOI 10.1017/S0305004100068389
- Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Publications of the Mathematical Society of Japan, vol. 11, Princeton University Press, Princeton, NJ, 1994. Reprint of the 1971 original; Kanô Memorial Lectures, 1. MR 1291394
- V. I. Vasyunin, On a system of step functions, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 262 (1999), no. Issled. po Lineĭn. Oper. i Teor. Funkts. 27, 49–70, 231–232 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 110 (2002), no. 5, 2930–2943. MR 1734327, DOI 10.1023/A:1015331119128
- 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.
- G. K. White, Lattice tetrahedra, Canadian J. Math. 16 (1964), 389–396. MR 161837, DOI 10.4153/CJM-1964-040-2
Bibliographic Information
- Margherita Barile
- Affiliation: Dipartimento di Matematica, Università di Bari “Aldo Moro”, Via E. Orabona 4, 70125 Bari, Italy
- Email: barile@dm.uniba.it
- Dominique Bernardi
- Affiliation: Université Pierre et Marie Curie, Institut Mathématique de Jussieu, 175 Rue du Chevaleret, F-75013, Paris, France
- Email: bernardi@math.jussieu.fr
- Alexander Borisov
- Affiliation: Department of Mathematics, 301 Thackeray Hall, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
- Email: borisov@pitt.edu
- Jean-Michel Kantor
- Affiliation: Université Paris Diderot, Institut Mathématique de Jussieu, 175 Rue du Chevaleret, F-75013, Paris, France
- Email: kantor@math.jussieu.fr
- 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
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 4247-4253
- MSC (2010): Primary 14B05; Secondary 14M25, 52B20
- DOI: https://doi.org/10.1090/S0002-9939-2011-10859-1
- MathSciNet review: 2823070