Israel Journal of Mathematics

, Volume 207, Issue 1, pp 301–329 | Cite as

Finitely many smooth d-polytopes with n lattice points

  • Tristram BogartEmail author
  • Christian Haase
  • Milena Hering
  • Benjamin Lorenz
  • Benjamin Nill
  • Andreas Paffenholz
  • Günter Rote
  • Francisco Santos
  • Hal Schenck


We prove that for fixed n there are only finitely many embeddings of ℚ-factorial toric varieties X into ℙ n that are induced by a complete linear system. The proof is based on a combinatorial result that implies that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with ≤ 12 lattice points.


Line Bundle Lattice Point Toric Variety Tangent Cone Ample Line Bundle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [4ti2]
    4ti2 team, 4ti2 — A software package for algebraic, geometric and combinatorial problems on linear spaces which is Available at (See
  2. [Bat91]
    V. V. Batyrev, On the classification of smooth projective toric varieties, Tôhoku Mathematical Journal 43 (1991), 569–585.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [Bat94]
    V. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, Journal of Algebraic Geometry 3 (1994), 493–535.zbMATHMathSciNetGoogle Scholar
  4. [Bat99]
    V. V. Batyrev, On the classification of toric Fano 4-folds, Journal of Matehmatical Sciences (New York) 94 (1999), 1021–1050.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [BGT97]
    W. Bruns, J. Gubeladze and N. V. Trung, Normal polytopes, triangulations, and Koszul algebras, Journal für die Reine und Angewandte Mathematik 485 (1997), 123–160.zbMATHGoogle Scholar
  6. [BIS]
    W. Bruns, B. Ichim, C. Söger and T. Römer, normaliz 2, Software for computations in affine monoids, vector configurations, lattice polytopes, and rational cones, available at
  7. [BK01]
    W. Bruns and R. Koch, Computing the integral closure of an affine semigroup, Universitatis Iagellonicae. Acta Mathematica 39 (2001), 59–70.MathSciNetGoogle Scholar
  8. [BT04]
    I. Bárány and N. Tokushige, The minimum area of convex lattice n-gons, Combinatorica 24 (2004), 171–185.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [CLS11]
    D. A. Cox, J. B. Little and H. K. Schenck, Toric Varieties, Graduate Studies in Mathematics, Vol. 124 American Mathematical Society, Providence, RI, 2011.zbMATHGoogle Scholar
  10. [LattE]
    J. A. De Loera, R. Hemmecke, J. Tauzer and R. Yoshida, LattE, Software for lattice point enumeration in polyhedra, available at
  11. [LHTY04]
    J. A. De Loera, R. Hemmecke, J. Tauzer and R. Yoshida, Effective lattice point counting in rational convex polytopes, Journal of Symbolic Computation 38 (2004), 1273–1302.MathSciNetCrossRefGoogle Scholar
  12. [Ewa96]
    G. Ewald, Combinatorial Convexity and Algebraic Geometry, Graduate Texts in Mathematics, Vol. 168, Springer-Verlag, Berlin, 1996.zbMATHGoogle Scholar
  13. [Ful93]
    W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, Vol. 131, Princeton University Press, Princeton, NJ, 1993.zbMATHGoogle Scholar
  14. [GJ]
    E. Gawrilow and M. Joswig, polymake, software tool for studying the combinatorics and geometry of convex polytopes, available at
  15. [GJ00]
    E. Gawrilow and M. Joswig, Polymake: a framework for analyzing convex polytopes, in Polytopes—Combinatorics and Computation (Obervofach, 1997), DVM Seminar, Vol. 29, Birkhäuser, Basel, 2000, pp. 43–74.Google Scholar
  16. [Gub09]
    J. Gubeladze, Convex normality of rational polytopes with long edges, Advances in Mathematics 230 (2012), 372–389. DOI:10.1016/j.aim.2011.12.003.zbMATHMathSciNetCrossRefGoogle Scholar
  17. [Hen83]
    D. Hensley, Lattice vertex polytopes with interior lattice points, Pacific Journal of Mathematics 105 (1983), 183–191.zbMATHMathSciNetCrossRefGoogle Scholar
  18. [Hib92]
    T. Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw Mathematics Texts, Carslaw Publications, Glebe, 1992.Google Scholar
  19. [HLP10]
    C. Haase, B. Lorenz and A. Paffenholz, Generating smooth lattice polytopes, in Proceedings of the Third International Congress on Mathematical Software 2010, Lecture Notes in Computer Sience, Vol. 6327, Springer-Verlag, Berlin-Heidelberg, 2010, pp. 315–328.Google Scholar
  20. [Hoc72]
    M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes. Annals of Mathematics 96 (1972), 318–337.zbMATHMathSciNetCrossRefGoogle Scholar
  21. [JMP09]
    M. Joswig, B. Müller and A. Paffenholz, Polymake and lattice polytopes, in Proceedings of the 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), Discrete Mathematics & Theoretical Computer Sience Proceedings, Nancy, 2009, pp. 492–502, arxiv:0902.2919.Google Scholar
  22. [Kas06]
    A. M. Kasprzyk, Toric Fano three-folds with terminal singularities, Tôhoku Mathematical Journal 58 (2006), 101–121.zbMATHMathSciNetCrossRefGoogle Scholar
  23. [Kas10]
    A. M. Kasprzyk, Canonical toric Fano threefolds, Canadian Journal of Mathematics 62 (2010), 1293–1309.zbMATHMathSciNetCrossRefGoogle Scholar
  24. [KKN10]
    A. M. Kasprzyk, M. Kreuzer and B. Nill, On the combinatorial classification of toric log Del Pezzo surfaces, LMS Journal of Computation and Mathematics 13 (2010), 33–46.zbMATHMathSciNetCrossRefGoogle Scholar
  25. [KN09]
    M. Kreuzer and B. Nill, Classification of toric Fano 5-folds, Advances in Geometry 9 (2009), 85–97.zbMATHMathSciNetCrossRefGoogle Scholar
  26. [KS91]
    P. Kleinschmidt and B. Sturmfels, Smooth toric varieties with small Picard number are projective, Topology 30 (1991), 289–299.zbMATHMathSciNetCrossRefGoogle Scholar
  27. [KS98]
    M. Kreuzer and H. Skarke, Classification of reflexive polyhedra in three dimensions, Advances in Theoretical and Mathematical Physics 2 (1998), 853–871.zbMATHMathSciNetGoogle Scholar
  28. [KS00]
    M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four dimensions, Advances in Theoretical and Mathematical Physics 4 (2000), 1209–1230.zbMATHMathSciNetGoogle Scholar
  29. [Lat96]
    R. Laterveer, Linear systems on toric varieties, Tôhoku Mathemaical Journal 2 (1996), 451–458.MathSciNetCrossRefGoogle Scholar
  30. [Lor09]
    B. Lorenz, Classification of smooth lattice polytopes with few lattice points, Master’s thesis, Freie Universität Berlin, 2009, arXiv:1001.0514.Google Scholar
  31. [Lun13]
    A. Lundman, A classification of smooth convex 3-polytopes with at most 16 lattice points, Journal of Algebraic Combinatorics 37 (2013), 139–165.zbMATHMathSciNetCrossRefGoogle Scholar
  32. [LZ91]
    J. C. Lagarias and G. M. Ziegler, Bounds for lattice polytopes containing a fixed number of interior points in a sublattice, Canadian Journal of Mathematics 43 (1991), 1022–1035.zbMATHMathSciNetCrossRefGoogle Scholar
  33. [MFO07]
    Mini-workshop: Projective normality of smooth toric varieties, Oberwolfach Reports 4 (2007), 2283–2320. Abstracts from the mini-workshop held August 12–18, 2007. Organized by Christian Haase, Takayuki Hibi and Diane MacLagan.MathSciNetGoogle Scholar
  34. [Nil06]
    B. Nill, Classification of pseudo-symmetric simplicial reflexive polytopes, in Algebraic and Geometric Combinatorics, Contemporary Mathematics, Vol. 423, American Mathematical Society, Providence, RI, 2006, pp. 269–282.CrossRefGoogle Scholar
  35. [Øbr07]
    M. Øbro, An algorithm for the classification of smooth Fano polytopes, 2007 arxiv:0704.0049.Google Scholar
  36. [Oda88]
    T. Oda, Convex Bodies and Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 15, Springer-Verlag, Berlin, 1988.zbMATHGoogle Scholar
  37. [Oda08]
    T. Oda, Problems on Minkowski sums of convex lattice polytopes, 2008, arXiv: 0812.1418.Google Scholar
  38. [ON02]
    S. Ogata and K. Nakagawa, On generators of ideals defining projective toric varieties, Manuscripta Mathematica 108 (2002), 33–42.zbMATHMathSciNetCrossRefGoogle Scholar
  39. [Pic99]
    G. A. Pick, Geometrisches zur Zahlenlehre, Sitzenber. Lotos (Prague) 19 (1899), 311–319.Google Scholar
  40. [Pik01]
    O. Pikhurko, Lattice points in lattice polytopes, Mathematika 48 (2001), 15–24.zbMATHMathSciNetCrossRefGoogle Scholar
  41. [Ree57]
    J. E. Reeve, On the volume of lattice polyhedra, Proceedings of the London Mathematical Society 7 (1957), 378–395.zbMATHMathSciNetCrossRefGoogle Scholar
  42. [Sat00]
    H. Sato, Toward the classification of higher-dimensional toric Fano varieties, Tôhoku Mathematical Journal 52 (2000), 383–413.zbMATHCrossRefGoogle Scholar
  43. [Sta99]
    R. P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, Vol. 62, Cambridge University Press, Cambridge, 1999.CrossRefGoogle Scholar
  44. [VK85]
    V. E. Voskresenskiĭ and A. A. Klyachko, Toroidal Fano varieties and root systems, Mathematics of the USSR-Izvestiya 24 (1985), 221–244.CrossRefGoogle Scholar
  45. [Zie95]
    G. M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, Vol. 152, Springer-Verlag, New York, 1995.zbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2015

Authors and Affiliations

  • Tristram Bogart
    • 1
    Email author
  • Christian Haase
    • 2
  • Milena Hering
    • 3
  • Benjamin Lorenz
    • 4
  • Benjamin Nill
    • 5
  • Andreas Paffenholz
    • 6
  • Günter Rote
    • 7
  • Francisco Santos
    • 8
  • Hal Schenck
    • 9
  1. 1.Universidad de los AndesBogotáColombia
  2. 2.Goethe-UniversitätFrankfurt am MainGermany
  3. 3.School of MathematicsThe University of EdinburghEdinburghRussia
  4. 4.Institut für Mathematik, Sekretariat MATechnische Universität BerlinBerlinGermany
  5. 5.Matematiska institutionenStockholms UniversitetStockholmSweden
  6. 6.Technische Universität DarmstadtDarmstadtGermany
  7. 7.Institut für InformatikFreie Universität BerlinBerlinGermany
  8. 8.Dept. Matematicas, Estad. y Comp.Universidad de CantabriaSantanderSpain
  9. 9.Mathematics DepartmentUniversity of IllinoisUrbanaUSA

Personalised recommendations