Skip to main content
SpringerLink
Log in

Finitely many smooth d-polytopes with n lattice points

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We prove that for fixed n there are only finitely many embeddings of ℚ-factorial toric varieties X into ℙn 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. 4ti2 team, 4ti2 — A software package for algebraic, geometric and combinatorial problems on linear spaces which is Available at www.4ti2.de (See http://www.4ti2.de/cite.html).

  2. V. V. Batyrev, On the classification of smooth projective toric varieties, Tôhoku Mathematical Journal 43 (1991), 569–585.

    Article  MATH  MathSciNet  Google Scholar 

  3. V. V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, Journal of Algebraic Geometry 3 (1994), 493–535.

    MATH  MathSciNet  Google Scholar 

  4. V. V. Batyrev, On the classification of toric Fano 4-folds, Journal of Matehmatical Sciences (New York) 94 (1999), 1021–1050.

    Article  MATH  MathSciNet  Google Scholar 

  5. 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.

    MATH  Google Scholar 

  6. 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 http://www.mathematik.uni-osnabrueck.de/normaliz/.

  7. W. Bruns and R. Koch, Computing the integral closure of an affine semigroup, Universitatis Iagellonicae. Acta Mathematica 39 (2001), 59–70.

    MathSciNet  Google Scholar 

  8. I. Bárány and N. Tokushige, The minimum area of convex lattice n-gons, Combinatorica 24 (2004), 171–185.

    Article  MATH  MathSciNet  Google Scholar 

  9. D. A. Cox, J. B. Little and H. K. Schenck, Toric Varieties, Graduate Studies in Mathematics, Vol. 124 American Mathematical Society, Providence, RI, 2011.

    MATH  Google Scholar 

  10. J. A. De Loera, R. Hemmecke, J. Tauzer and R. Yoshida, LattE, Software for lattice point enumeration in polyhedra, available at http://www.math.ucdavis.edu/~latte/.

  11. 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.

    Article  MathSciNet  Google Scholar 

  12. G. Ewald, Combinatorial Convexity and Algebraic Geometry, Graduate Texts in Mathematics, Vol. 168, Springer-Verlag, Berlin, 1996.

    MATH  Google Scholar 

  13. W. Fulton, Introduction to Toric Varieties, Annals of Mathematics Studies, Vol. 131, Princeton University Press, Princeton, NJ, 1993.

    MATH  Google Scholar 

  14. E. Gawrilow and M. Joswig, polymake, software tool for studying the combinatorics and geometry of convex polytopes, available at http://www.polymake.de.

  15. 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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  17. D. Hensley, Lattice vertex polytopes with interior lattice points, Pacific Journal of Mathematics 105 (1983), 183–191.

    Article  MATH  MathSciNet  Google Scholar 

  18. T. Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw Mathematics Texts, Carslaw Publications, Glebe, 1992.

    Google Scholar 

  19. 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. M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes. Annals of Mathematics 96 (1972), 318–337.

    Article  MATH  MathSciNet  Google Scholar 

  21. 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. A. M. Kasprzyk, Toric Fano three-folds with terminal singularities, Tôhoku Mathematical Journal 58 (2006), 101–121.

    Article  MATH  MathSciNet  Google Scholar 

  23. A. M. Kasprzyk, Canonical toric Fano threefolds, Canadian Journal of Mathematics 62 (2010), 1293–1309.

    Article  MATH  MathSciNet  Google Scholar 

  24. 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.

    Article  MATH  MathSciNet  Google Scholar 

  25. M. Kreuzer and B. Nill, Classification of toric Fano 5-folds, Advances in Geometry 9 (2009), 85–97.

    Article  MATH  MathSciNet  Google Scholar 

  26. P. Kleinschmidt and B. Sturmfels, Smooth toric varieties with small Picard number are projective, Topology 30 (1991), 289–299.

    Article  MATH  MathSciNet  Google Scholar 

  27. M. Kreuzer and H. Skarke, Classification of reflexive polyhedra in three dimensions, Advances in Theoretical and Mathematical Physics 2 (1998), 853–871.

    MATH  MathSciNet  Google Scholar 

  28. M. Kreuzer and H. Skarke, Complete classification of reflexive polyhedra in four dimensions, Advances in Theoretical and Mathematical Physics 4 (2000), 1209–1230.

    MATH  MathSciNet  Google Scholar 

  29. R. Laterveer, Linear systems on toric varieties, Tôhoku Mathemaical Journal 2 (1996), 451–458.

    Article  MathSciNet  Google Scholar 

  30. B. Lorenz, Classification of smooth lattice polytopes with few lattice points, Master’s thesis, Freie Universität Berlin, 2009, arXiv:1001.0514.

  31. A. Lundman, A classification of smooth convex 3-polytopes with at most 16 lattice points, Journal of Algebraic Combinatorics 37 (2013), 139–165.

    Article  MATH  MathSciNet  Google Scholar 

  32. 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.

    Article  MATH  MathSciNet  Google Scholar 

  33. 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.

    MathSciNet  Google Scholar 

  34. 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.

    Chapter  Google Scholar 

  35. M. Øbro, An algorithm for the classification of smooth Fano polytopes, 2007 arxiv:0704.0049.

  36. T. Oda, Convex Bodies and Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 15, Springer-Verlag, Berlin, 1988.

    MATH  Google Scholar 

  37. T. Oda, Problems on Minkowski sums of convex lattice polytopes, 2008, arXiv: 0812.1418.

  38. S. Ogata and K. Nakagawa, On generators of ideals defining projective toric varieties, Manuscripta Mathematica 108 (2002), 33–42.

    Article  MATH  MathSciNet  Google Scholar 

  39. G. A. Pick, Geometrisches zur Zahlenlehre, Sitzenber. Lotos (Prague) 19 (1899), 311–319.

    Google Scholar 

  40. O. Pikhurko, Lattice points in lattice polytopes, Mathematika 48 (2001), 15–24.

    Article  MATH  MathSciNet  Google Scholar 

  41. J. E. Reeve, On the volume of lattice polyhedra, Proceedings of the London Mathematical Society 7 (1957), 378–395.

    Article  MATH  MathSciNet  Google Scholar 

  42. H. Sato, Toward the classification of higher-dimensional toric Fano varieties, Tôhoku Mathematical Journal 52 (2000), 383–413.

    Article  MATH  Google Scholar 

  43. R. P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, Vol. 62, Cambridge University Press, Cambridge, 1999.

    Book  Google Scholar 

  44. V. E. Voskresenskiĭ and A. A. Klyachko, Toroidal Fano varieties and root systems, Mathematics of the USSR-Izvestiya 24 (1985), 221–244.

    Article  Google Scholar 

  45. G. M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, Vol. 152, Springer-Verlag, New York, 1995.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universidad de los Andes, Cra 1, No. 18A-10, Edificio H, Bogotá, 111711, Colombia

    Tristram Bogart

  2. Goethe-Universität, Robert-Mayer-Straße 10, 60325, Frankfurt am Main, Germany

    Christian Haase

  3. School of Mathematics, The University of Edinburgh, The King’s Buildings, Edinburgh, EH9 3JZ, Russia

    Milena Hering

  4. Institut für Mathematik, Sekretariat MA, Technische Universität Berlin, 6-2 Straße des 17. Juni 136, 10623, Berlin, Germany

    Benjamin Lorenz

  5. Matematiska institutionen, Stockholms Universitet, 106 91, Stockholm, Sweden

    Benjamin Nill

  6. Technische Universität Darmstadt, FB Mathematik, Dolivostr. 15, 64293, Darmstadt, Germany

    Andreas Paffenholz

  7. Institut für Informatik, Freie Universität Berlin, Takustr. 9, 14195, Berlin, Germany

    Günter Rote

  8. Dept. Matematicas, Estad. y Comp., Universidad de Cantabria, E-39005, Santander, Spain

    Francisco Santos

  9. Mathematics Department, University of Illinois, Urbana, Illinois, 61801, USA

    Hal Schenck

Authors
  1. Tristram Bogart
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Haase
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Milena Hering
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Benjamin Lorenz
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Benjamin Nill
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Andreas Paffenholz
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Günter Rote
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Francisco Santos
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. Hal Schenck
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tristram Bogart.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bogart, T., Haase, C., Hering, M. et al. Finitely many smooth d-polytopes with n lattice points. Isr. J. Math. 207, 301–329 (2015). https://doi.org/10.1007/s11856-015-1175-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-015-1175-7

Keywords

Search

Navigation