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

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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.

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Notes

  1. I.e., isomorphic to a polynomial algebra.

  2. All \(M_I\) for \(\ell (I) > \vert \mathfrak {I}\vert \) are identically 0.

  3. To see how an integral basis can be constructed with the same index set, see [9], or [7, 6.5] for a more detailed explanation.

  4. The proof given in [4] is for schemes, but will work unchanged for stacks as a base if the vector bundle is trivial.

  5. In the topological setting, we have \(BU(1) = {{\mathbb {C}}}{{\mathbb {P}}}^\infty \). In this spirit, one could regard \(B{{\mathbb {G}}}_m\) as an algebraic version of \({{\mathbb {P}}}^\infty \).

  6. Except over the empty set.

  7. The symbol “nr” stands for “non-rigid.”

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  1. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057, Zurich, Switzerland

    Jakob Oesinghaus

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  1. Jakob Oesinghaus
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Oesinghaus, J. Quasisymmetric functions and the Chow ring of the stack of expanded pairs. Res Math Sci 6, 5 (2019). https://doi.org/10.1007/s40687-018-0168-7

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