Advertisement

Quasisymmetric functions and the Chow ring of the stack of expanded pairs

  • Jakob OesinghausEmail author
Research

Abstract

We show that the Hopf algebra of quasisymmetric functions arises naturally as the integral Chow ring of the algebraic stack of expanded pairs originally described by J. Li, using a more combinatorial description in terms of configurations of line bundles. In particular, we exhibit a gluing map which gives rise to the comultiplication. We then apply the result to calculate the Chow rings of certain stacks of semistable curves.

Keywords

Chow groups and rings Algebraic stacks Moduli problems Quasisymmetric functions 

Mathematics Subject Classification

Primary 14C15 14A20 16W30 Secondary 05E05 14C17 

References

  1. 1.
    Abramovich, D., Cadman, C., Fantechi, B., Wise, J.: Expanded degenerations and pairs. Commun. Algebra 41(6), 2346–2386 (2013).  https://doi.org/10.1080/00927872.2012.658589 MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Borne, N., Vistoli, A.: Parabolic sheaves on logarithmic schemes. Adv. Math. 231(3–4), 1327–1363 (2012).  https://doi.org/10.1016/j.aim.2012.06.015 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Fulghesu, D.: The Chow ring of the stack of rational curves with at most 3 nodes. Commun. Algebra 38(9), 3125–3136 (2010).  https://doi.org/10.1080/00927870903399893 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Fulton, W.: Intersection Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, 2nd edn. Springer, Berlin (1998).  https://doi.org/10.1007/978-1-4612-1700-8 CrossRefGoogle Scholar
  5. 5.
    Gessel, I.M.: Multipartite \(P\)-partitions and inner products of skew Schur functions. In: Combinatorics and Algebra (Boulder, Colo., 1983), Contemp. Math., 34, 289–317., Amer. Math. Soc., Providence, RI (1984).  https://doi.org/10.1090/conm/034/777705
  6. 6.
    Graber, T., Vakil, R.: Relative virtual localization and vanishing of tautological classes on moduli spaces of curves. Duke Math. J. 130(1), 1–37 (2005).  https://doi.org/10.1215/S0012-7094-05-13011-3 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grinberg, D., Reiner, V.: Hopf algebras in combinatorics (2018). arXiv:1409.8356
  8. 8.
    Hazewinkel, M.: The algebra of quasi-symmetric functions is free over the integers. Adv. Math. 164(2), 283–300 (2001).  https://doi.org/10.1006/aima.2001.2017 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hazewinkel, M.: Explicit polynomial generators for the ring of quasisymmetric functions over the integers. Acta Appl. Math. 109(1), 39–44 (2010).  https://doi.org/10.1007/s10440-009-9439-z MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jia, W., Wang, Z., Yu, H.: Rigidity for the hopf algebra of quasi-symmetric functions (2017). arXiv:1712.06499
  11. 11.
    Kresch, A.: Cycle groups for Artin stacks. Invent. Math. 138(3), 495–536 (1999).  https://doi.org/10.1007/s002220050351 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Li, J.: Stable morphisms to singular schemes and relative stable morphisms. J. Differ. Geom. 57(3), 509–578 (2001). http://projecteuclid.org/euclid.jdg/1090348132
  13. 13.
    Li, J.: A degeneration formula of GW-invariants. J. Differ. Geom. 60(2), 199–293 (2002). http://projecteuclid.org/euclid.jdg/1090351102
  14. 14.
    Luoto, K., Mykytiuk, S., van Willigenburg, S.: An Introduction to Quasisymmetric Schur Functions. Springer Briefs in Mathematics. Springer, New York (2013).  https://doi.org/10.1007/978-1-4614-7300-8. (Hopf algebras, quasisymmetric functions, and Young composition tableaux)CrossRefzbMATHGoogle Scholar
  15. 15.
    Olsson, M.C.: Logarithmic geometry and algebraic stacks. Ann. Sci. École Norm. Sup. (4) 36(5), 747–791 (2003).  https://doi.org/10.1016/j.ansens.2002.11.001 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Vezzosi, G., Vistoli, A.: Higher algebraic \(K\)-theory for actions of diagonalizable groups. Invent. Math. 153(1), 1–44 (2003).  https://doi.org/10.1007/s00222-002-0275-2 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© SpringerNature 2018

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZurichSwitzerland

Personalised recommendations