Skip to main content
SpringerLink
Log in

Irrational Toric Varieties and Secondary Polytopes

  • Published:
Discrete & Computational Geometry Aims and scope Submit manuscript

Abstract

The space of torus translations and degenerations of a projective toric variety forms a toric variety associated to the secondary fan of the integer points in the polytope corresponding to the toric variety. This is used to identify a moduli space of real degenerations with the secondary polytope. A configuration \({{\mathcal {A}}}\) of real vectors gives an irrational projective toric variety in a simplex. We identify a space of translations and degenerations of the irrational projective toric variety with the secondary polytope of \({{\mathcal {A}}}\). For this, we develop a theory of irrational toric varieties associated to arbitrary fans. When the fan is rational, the irrational toric variety is the nonnegative part of the corresponding classical toric variety. When the fan is the normal fan of a polytope, the irrational toric variety is homeomorphic to that polytope.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Alexeev, V.: Complete moduli in the presence of semiabelian group action. Ann. Math. 155(3), 611–708 (2002)

    Article  MathSciNet  Google Scholar 

  2. Atiyah, M.F.: Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc. 14(1), 1–15 (1982)

    Article  MathSciNet  Google Scholar 

  3. Battaglia, F., Prato, E.: Generalized toric varieties for simple nonrational convex polytopes. Internat. Math. Res. Notices 2001(24), 1315–1337 (2001)

    Article  MathSciNet  Google Scholar 

  4. Battaglia, F., Zaffran, D.: Foliations modeling nonrational simplicial toric varieties. Internat. Math. Res. Notices 2015(22), 11785–11815 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Birch, M.W.: Maximum likelihood in three-way contingency tables. J. R. Stat. Soc. Ser. B 25, 220–233 (1963)

    MathSciNet  MATH  Google Scholar 

  6. Buchstaber, V.M., Panov, T.E.: Toric Topology. Mathematical Surveys and Monographs, vol. 204. American Mathematical Society, Providence (2015)

  7. Cox, D.A., Little, J.B., Schenck, H.K.: Toric Varieties. Graduate Studies in Mathematics, vol. 124. American Mathematical Society, Providence (2011)

  8. Craciun, G., García-Puente, L.D., Sottile, F.: Some geometrical aspects of control points for toric patches. In: Mathematical Methods for Curves and Surfaces (Tønsberg 2008). Lecture Notes in Comput. Sci., vol. 5862, pp. 111–135. Springer, Berlin (2010)

  9. Darmois, G.: Sur les lois de probabilité à estimation exhaustive. C. R. Acad. Sci. Paris 200, 1265–1266 (1935)

    MATH  Google Scholar 

  10. Davis, M.W., Januszkiewicz, T.: Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J. 62(2), 417–451 (1991)

    Article  MathSciNet  Google Scholar 

  11. De Loera, J.A., Rambau, J., Santos, F.: Triangulations. Algorithms and Computation in Mathematics, vol. 25. Springer, Berlin (2010)

  12. Ewald, G.: Combinatorial Convexity and Algebraic Geometry. Graduate Texts in Mathematics, vol. 168. Springer, New York (1996)

  13. Ford, T.J.: The toroidal embedding arising from an irrational fan. Results Math. 35(1–2), 44–69 (1999)

    Article  MathSciNet  Google Scholar 

  14. Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton (1993)

  15. Garcia-Puente, L.D., Sottile, F.: Linear precision for parametric patches. Adv. Comput. Math. 33(2), 191–214 (2010)

    Article  MathSciNet  Google Scholar 

  16. García-Puente, L.D., Sottile, F., Zhu, Ch.: Toric degenerations of Bézier patches. ACM Trans. Graphics 30(5), # 110 (2011)

  17. Gel’fand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants, and Multidimensional Determinants. Mathematics: Theory & Applications. Birkhäuser, Boston (1994)

  18. Goodman, L.A.: On methods for comparing contingency tables. J. R. Stat. Soc. Ser. A 126, 94–108 (1963)

    Article  MathSciNet  Google Scholar 

  19. Guillemin, V., Sternberg, S.: Convexity properties of the moment mapping. Invent. Math. 67(3), 491–513 (1982)

    Article  MathSciNet  Google Scholar 

  20. Kapranov, M.M., Sturmfels, B., Zelevinsky, A.V.: Quotients of toric varieties. Math. Ann. 290(4), 643–655 (1991)

    Article  MathSciNet  Google Scholar 

  21. Kapranov, M.M., Sturmfels, B., Zelevinsky, A.V.: Chow polytopes and general resultants. Duke Math. J. 67(1), 189–218 (1992)

    Article  MathSciNet  Google Scholar 

  22. Katzarkov, L., Lupercio, E., Meersseman, L., Verjovsky, A.: The definition of a non-commutative toric variety. In: Algebraic Topology: Applications and New Directions (Stanford 2012). Contemp. Math., vol. 620, pp. 223–250. American Mathematical Society, Providence (2014)

  23. Koopman, B.O.: On distributions admitting a sufficient statistic. Trans. Am. Math. Soc. 39(3), 399–409 (1936)

    Article  MathSciNet  Google Scholar 

  24. Krasauskas, R.: Toric surface patches. Adv. Comput. Math. 17(1–2), 89–113 (2002)

    Article  MathSciNet  Google Scholar 

  25. López de Medrano, S., Verjovsky, A.: A new family of complex, compact, non-symplectic manifolds. Bol. Soc. Brasil. Mater. (N.S.) 28(2), 253–269 (1997)

  26. Meersseman, L.: A new geometric construction of compact complex manifolds in any dimension. Math. Ann. 317(1), 79–115 (2000)

    Article  MathSciNet  Google Scholar 

  27. Miller, E.: Essential graded algebra over polynomial rings with real exponents (2020). arXiv:2008.03819

  28. Miller, E.: Primary decomposition over partially ordered groups (2020). arXiv:2008.00093

  29. Miller, E.: Stratifications of real vector spaces from constructible sheaves with conical microsupport (2020). arXiv:2008.00091

  30. Munkres, J.R.: Topology: A First Course. Prentice-Hall, Englewood Cliffs (1975)

    MATH  Google Scholar 

  31. Pachter, L., Sturmfels, B. (eds.): Algebraic Statistics for Computational Biology. Cambridge University Press, New York (2005)

    MATH  Google Scholar 

  32. Pir, A.F.: Irrational Toric Varieties. PhD thesis, Texas A &M University (2018). https://core.ac.uk/download/pdf/187124453.pdf

  33. Pitman, E.J.G.: Sufficient statistics and intrinsic accuracy. Math. Proc. Cambr. Phil. Soc. 32(4), 567–579 (1936)

    Article  Google Scholar 

  34. Postinghel, E., Sottile, F., Villamizar, N.: Degenerations of real irrational toric varieties. J. Lond. Math. Soc. 92(2), 223–241 (2015)

    Article  MathSciNet  Google Scholar 

  35. Prato, E.: Simple non-rational convex polytopes via symplectic geometry. Topology 40(5), 961–975 (2001)

    Article  MathSciNet  Google Scholar 

  36. Putcha, M.S.: Linear Algebraic Monoids. London Mathematical Society Lecture Note Series, vol. 133. Cambridge University Press, Cambridge (1988)

  37. Renner, L.E.: Linear Algebraic Monoids. Encyclopaedia of Mathematical Sciences, vol. 134. Invariant Theory and Algebraic Transformation Groups, vol. 5. Springer, Berlin (2005)

  38. Sottile, F.: Toric ideals, real toric varieties, and the moment map. In: Topics in Algebraic Geometry and Geometric Modeling (Vilnius 2002). Contemp. Math., vol. 334, pp. 225–240. American Mathematical Society, Providence (2003)

  39. Sturmfels, B.: Gröbner Bases and Convex Polytopes. University Lecture Series, vol. 8. American Mathematical Society, Providence (1996)

  40. Verjovsky, A.: Intersection of quadrics in \(\mathbb{C}^n\), moment-angle manifolds, complex manifolds and convex manifolds. In: Complex non-Kähler Geometry (Cetraro 2018). Lecture Notes in Math., vol. 2246. Fond. CIME/CIME Found. Subser., pp. 163–240. Springer, Cham (2019)

  41. Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995)

Download references

Author information

Authors and Affiliations

  1. San Jose, USA

    Ata Firat Pir

  2. Department of Mathematics, Texas A &M University, College Station, TX, 77843, USA

    Frank Sottile

Authors
  1. Ata Firat Pir
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Frank Sottile
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Frank Sottile.

Additional information

Editor in Charge: János Pach

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Research of Pir and Sottile supported in part by NSF Grant DMS-1501370.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pir, A.F., Sottile, F. Irrational Toric Varieties and Secondary Polytopes. Discrete Comput Geom 67, 1053–1079 (2022). https://doi.org/10.1007/s00454-021-00334-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00454-021-00334-z

Keywords

Mathematics Subject Classification

Search

Navigation