Journal of Algebraic Combinatorics

, Volume 35, Issue 3, pp 421–460

Phylogenetic toric varieties on graphs

Open Access
Article

Abstract

We define phylogenetic projective toric model of a trivalent graph as a generalization of a binary symmetric model of a trivalent phylogenetic tree. Generators of the projective coordinate ring of the models of graphs with one cycle are explicitly described. The phylogenetic models of graphs with the same topological invariants are deformation-equivalent and share the same Hilbert function. We also provide an algorithm to compute the Hilbert function.

Keywords

Binary symmetric model GIT quotient Hilbert function 

References

  1. 1.
    Białynicki-Birula, A.: Quotients by actions of groups. In: Algebraic Quotients. Torus Actions and Cohomology. The Adjoint Representation and the Adjoint Action. Encyclopedia Math. Sci., vol. 131, pp. 1–82. Springer, Berlin (2002) Google Scholar
  2. 2.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997). Computational algebra and number theory (London, 1993) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Buczyńska, W., Wiśniewski, J.A.: On geometry of binary symmetric models of phylogenetic trees. J. Eur. Math. Soc. 9(3), 609–635 (2007) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150. Springer, New York (1995) MATHGoogle Scholar
  5. 5.
    Felsenstein, J.: Inferring Phylogenies. Sinauer Press, Sunderland (2004) Google Scholar
  6. 6.
    Fulton, W.: Introduction to Toric Varieties. Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton, NJ (1993). The William H. Roever Lectures in Geometry MATHGoogle Scholar
  7. 7.
    Grothendieck, A.: Éléments de géometrie algébrique (rédigés avec la collaboration de Jean Dieudonné): II. Étude globale élémentaire de quelques classes de mophismes. Graduate Texts in Mathematics, vol. 52. Publications mathématique de l’IHÉS, Bures-sur-Yvette (1961) Google Scholar
  8. 8.
    Haiman, M., Sturmfels, B.: Multigraded Hilbert schemes. J. Algebr. Geom. 13(4), 725–769 (2004) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977). MATHGoogle Scholar
  10. 10.
    Jeffrey, L.C., Weitsman, J.: Bohr-Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula. Commun. Math. Phys. 150(3), 593–630 (1992) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Manon, C.A.: The algebra of conformal blocks. arXiv:0910.0577v3 [math.AG] (2009)
  12. 12.
    Manon, C.A.: Coordinate rings for the moduli of SL 2(ℂ) quasi-parabolic principal bundles on a curve and toric fiber products. arXiv:1105.2045v1 [math.AC]
  13. 13.
    Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, 3rd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34. Springer, Berlin (1994) CrossRefGoogle Scholar
  14. 14.
    Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Graduate Texts in Mathematics, vol. 227. Springer, New York (2005) Google Scholar
  15. 15.
    Oda, T.: Convex Bodies and Algebraic Geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 15. Springer, Berlin, (1988). An introduction to the theory of toric varieties, Translated from the Japanese MATHGoogle Scholar
  16. 16.
    Pachter, L., Sturmfels, B.: Statistics. In: Algebraic Statistics for Computational Biology, pp. 3–42. Cambridge Univ. Press, New York (2005) CrossRefGoogle Scholar
  17. 17.
    Reid, M.: Graded rings and birational geometry. In: Ohno, K. (ed.) Proceedings of Algebraic Geometry Symposium, Kinosaki, Oct. 2000, pp. 1–72 (2000) Google Scholar
  18. 18.
    Semple, C., Steel, M.: Phylogenetics. Oxford Lecture Series in Mathematics and Its Applications, vol. 24. Oxford University Press, Oxford (2003) MATHGoogle Scholar
  19. 19.
    Stanley, R.P.: Hilbert functions of graded algebras. Adv. Math. 28(1), 57–83 (1978) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Sturmfels, B., Xu, Z.: Sagbi basis and Cox-Nagata rings. J. Eur. Math. Soc. 12(2), 429–459 (2010) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA

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