Advertisement

Computing the Chern–Schwartz–MacPherson Class of Complete Simplical Toric Varieties

  • Martin Helmer
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 198)

Abstract

Topological invariants such as characteristic classes are an important tool to aid in understanding and categorizing the structure and properties of algebraic varieties. In this note, we consider the problem of computing a particular characteristic class, the Chern–Schwartz–MacPherson class, of a complete simplicial toric variety \(X_{\Sigma }\) defined by a fan \({\Sigma }\) from the combinatorial data contained in the fan \(\Sigma \). Specifically, we give an effective combinatorial algorithm to compute the Chern–Schwartz–MacPherson class of \(X_{\Sigma }\), in the Chow ring (or rational Chow ring) of \(X_{\Sigma }\). This method is formulated by combining, and when necessary modifying, several known results from the literature and is implemented in Macaulay2 for test purposes.

Keywords

Chern–Schwartz–MacPherson class Chern class Toric varieties Computer algebra Computational intersection theory 

References

  1. 1.
    Aluffi, P.: Computing characteristic classes of projective schemes. J. Symb. Comput. 35(1), 3–19 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barthel, G., Brasselet, J.-P., Fieseler, K.-H.: Classes de Chern de variétés toriques singulières. CR Acad. Sci. Paris Sér. I Math. 315(2), 187–192 (1992)zbMATHGoogle Scholar
  3. 3.
    Brasselet, J.-P., Schwartz, M.-H.: Sur les classes de Chern d’un ensemble analytique complexe. Astérisque 82(83), 93–147 (1981)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cox, D.A., John, B., Schenck, H.K.: Toric varieties. Am. Math. Soc. 124, 575 (2011)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Fulton, W.: Intersection Theory, 2nd edn. Springer, Berlin (1998)CrossRefzbMATHGoogle Scholar
  6. 6.
    Grayson, D.R., Stillman, M.E.: Macaulay2, a software system for research in algebraic geometry. Biometrika 66(2), 339–344 (2013)Google Scholar
  7. 7.
    Helmer, M.: An algorithm to compute the topological Euler characteristic, the Chern–Schwartz–Macpherson class and the Segre class of subschemes of some smooth complete toric varieties. arXiv:1508.03785 (2015)
  8. 8.
    Helmer, M.: Algorithms to compute the topological Euler characteristic, Chern–Schwartz–Macpherson class and Segre class of projective varieties. J. Symb. Comput. 73, 120–138 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Helmer, M.: A direct algorithm to compute the topological Euler characteristic and Chern–Schwartz–Macpherson class of projective complete intersection varieties. Submitted to a Special Issue of the Journal of Theoretical Computer Science for SNC-2014. Available on the, arXiv.org/abs/1410.4113 (2015)
  10. 10.
    Jost, C.: An algorithm for computing the topological Euler characteristic of complex projective varieties. arXiv:1301.4128 (2013)
  11. 11.
    Robert, D.: MacPherson. Chern classes for singular algebraic varieties. Ann. Math. 100(2), 423–432 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Schürmann, J., Yokura, S.: A Survey of Characteristic Classes of Singular Spaces, pp. 865–952. World Scientific, Singapore (2007)zbMATHGoogle Scholar
  13. 13.
    Schwartz, M.-H.: Classes caractéristiques définies par une stratification d’une variété analytique complexe. Comptes Rendus de l’Académie des Sciences Paris 260, 3262–3264 (1965)zbMATHGoogle Scholar
  14. 14.
    Stein, W.A et al.: Sage Mathematics Software (Version 5.11). The Sage Development Team, http://www.sagemath.org (2013)
  15. 15.
    The LinBox Group.: LinBox–Exact Linear Algebra Over the Integers and Finite Rings, Version 1.1.6 (2008)Google Scholar
  16. 16.
    The PARI Group, Bordeaux.: PARI/GP version 2.7.0. Available from http://pari.math.u-bordeaux.fr/ (2014)

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California BerkeleyBerkeleyUSA

Personalised recommendations