Journal of Algebraic Combinatorics

, Volume 42, Issue 1, pp 165–182 | Cite as

Very ample and Koszul segmental fibrations

  • Matthias Beck
  • Jessica Delgado
  • Joseph Gubeladze
  • Mateusz Michałek


In the hierarchy of structural sophistication for lattice polytopes, normal polytopes mark a point of origin; very ample and Koszul polytopes occupy bottom and top spots in this hierarchy, respectively. In this paper we explore a simple construction for lattice polytopes with a twofold aim. On the one hand, we derive an explicit series of very ample 3-dimensional polytopes with arbitrarily large deviation from the normality property, measured via the highest discrepancy degree between the corresponding Hilbert functions and Hilbert polynomials. On the other hand, we describe a large class of Koszul polytopes of arbitrary dimensions, containing many smooth polytopes and extending the previously known class of Nakajima polytopes.


Normal polytope Very ample polytope Koszul polytope  Regular unimodular triangulation 

Mathematics Subject Classification

Primary 52B20 Secondary 13P10 14M25 



We thank Winfried Bruns and Serkan Hoşten for helpful comments and providing us with an invaluable set of examples of very ample polytopes. We also thank Milena Hering for pointing out the overlap of our work with [12] and two anonymous referees for helpful comments. The last author also thanks Mathematisches Forschungsinstitut Oberwolfach for the great working atmosphere and hosting. This research was partially supported by the U. S. National Science Foundation through the Grants DMS-1162638 (Beck), DGE-0841164 (Delgado), DMS-1000641 & DMS-1301487 (Gubeladze), and the Polish National Science Centre Grant No. 2012/05/D/ST1/01063 (Michałek).


  1. 1.
    Alfonsín, J.L.R.: The Diophantine, Frobenius Problem. Oxford Lecture Series in Mathematics and its Applications. vol. 30, Oxford University Press, Oxford (2005)Google Scholar
  2. 2.
    Beck, M., Robins, S.: Computing the Continuous Discretely. Undergraduate Texts in Mathematics. Springer, New York (2007)MATHGoogle Scholar
  3. 3.
    Bogart, T., Haase, C., Hering, M., Lorenz, B., Nill, B., Paffenholz, A., Santos, F., Schenck, H.: Few smooth \(d\)-polytopes with \({N}\) lattice points. Israel J. Math. (2010). arXiv:1010.3887v1
  4. 4.
    Bruns, W., Herzog, J., Vetter, U.: Syzygies and walks. In: Commutative Algebra (Trieste, 1992), pp. 36–57. World Scientific Publishing, River Edge (1994)Google Scholar
  5. 5.
    Bruns, W., Gubeladze, J., Trung, N.V.: Normal polytopes, triangulations, and Koszul algebras. J. Reine Angew. Math. 485, 123–160 (1997)MATHMathSciNetGoogle Scholar
  6. 6.
    Bruns, W., Gubeladze, J.: Normality and covering properties of affine semigroups. J. Reine Angew. Math. 510, 161–178 (1999)MATHMathSciNetGoogle Scholar
  7. 7.
    Bruns, W., Gubeladze, J.: Polytopes, Rings, and \(K\)-theory. Springer Monographs in Mathematics. Springer, New York (2009)Google Scholar
  8. 8.
    Bruns, W.: The quest for counterexamples in toric geometry. Proc. CAAG 17, 1–17 (2010)Google Scholar
  9. 9.
    Caviglia, Giulio: The pinched Veronese is Koszul. J. Algebraic Comb. 30, 539–548 (2009)MATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Conca, A., De Negri, E., Rossi, M.: Koszul algebras and regularity. In: Peeva, I. (ed.) Commutative Algebra, pp. 285–315. Springer, New York (2013)CrossRefGoogle Scholar
  11. 11.
    Cox, David A., Little, John B., Schenck, Henry K.: Toric Varieties Volume 124 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2011)Google Scholar
  12. 12.
    Dais, D.I., Haase, C., Ziegler, G.M.: All toric local complete intersection singularities admit projective crepant resolutions. Tohoku Math. J. 2(53), 95–107 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)MATHGoogle Scholar
  14. 14.
    Hering, M.: Multigraded regularity and the Koszul property. J. Algebra 323, 1012–1017 (2010)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Higashitani, A.: Non-normal very ample polytopes and their holes. Electron. J. Comb. 21:Paper 1.53, 12, (2014)Google Scholar
  16. 16.
    Katthän, L.: Polytopal affine semigroups with holes deep inside. Discrete Comput. Geom. 50, 503–508 (2013)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Kempf, G., Knudsen, F.F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings. I. Lecture Notes in Mathematics, vol. 339. Springer, New York (1973)Google Scholar
  18. 18.
    Lam, T., Postnikov, A.: Alcoved polytopes. I. Discrete Comput. Geom. 38(3), 453–478 (2007)MATHMathSciNetCrossRefGoogle Scholar
  19. 19.
    Lasoń, M., Michałek, M.: Non-normal, very ample polytopes–constructions and examples. arXiv:1406.4070
  20. 20.
    Mini-Workshop: Projective Normality of Smooth Toric Varieties. Oberwolfach Rep., 4(3):2283–2319, 2007. Abstracts from the mini-workshop held August 12–18, 2007, organized by Christian Haase, Takayuki Hibi and Diane Maclagan, Oberwolfach Reports. Vol. 4, No. 3Google Scholar
  21. 21.
    Nakajima, H.: Affine torus embeddings which are complete intersections. Tohoku Math. J. 2(38), 85–98 (1986)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Oda, T.: Convex bodies and algebraic geometry, volume 15 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Springer, New York (1988)Google Scholar
  23. 23.
    Ohsugi, H., Herzog, J., Hibi, T.: Combinatorial pure subrings. Osaka J. Math. 37, 745–757 (2000)MATHMathSciNetGoogle Scholar
  24. 24.
    Payne, S.: Lattice polytopes cut out by root systems and the Koszul property. Adv. Math. 220, 926–935 (2009)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Peeva, I. (ed.): Infinite free resolutions over toric rings. In: Syzygies and Hilbert functions. Lecture Notes in Pure Applied Mathematics, vol. 254, pp. 233–247. CRC Press, Boca Raton (2007)Google Scholar
  26. 26.
    Priddy, S.B.: Koszul resolutions. Trans. Amer. Math. Soc. 152, 39–60 (1970)MATHMathSciNetCrossRefGoogle Scholar
  27. 27.
    Fröberg, R.: Koszul algebras. In: Dobbs, D.E., Fontana, M., Kabbaj, S.-E. (eds.) Advances in Commutative Ring Theory (Fez, 1997). Lecture Notes in Pure and Applied Mathematics, vol. 205, pp. 337–350. Dekker, New York (1999)Google Scholar
  28. 28.
    Santos F., Ziegler G.M.: Unimodular triangulations of dilated 3-polytopes. Trans. Moscow Math. Soc., pp. 293–311 (2013)Google Scholar
  29. 29.
    Sturmfels, B.: Gröbner bases and convex polytopes. University Lecture Series. vol. 8, American Mathematical Society, Providence (1996)Google Scholar
  30. 30.
    Workshop: Combinatorial Challenges in Toric Varieties. April 27 to May 1, 2009, organized by Joseph Gubeladze, Christian Haase, and Diane Maclagan, American Institute of Mathematics, Palo Alto.

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Matthias Beck
    • 1
  • Jessica Delgado
    • 2
  • Joseph Gubeladze
    • 1
  • Mateusz Michałek
    • 3
  1. 1.Department of MathematicsSan Francisco State UniversitySan FranciscoUSA
  2. 2.Kahikoluamea DepartmentHonoluluUSA
  3. 3.Polish Academy of SciencesWarsawPoland

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