Intersections of loci of admissible covers with tautological classes
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Selecta Mathematica New Series
Intersections of loci of admissible covers with tautological classes Johannes Schmitt1 · Jason van Zelm2 Accepted: 11 October 2020 / Published online: 17 November 2020 © The Author(s) 2020
Abstract For a finite group G, let Hg,G,ξ be the stack of admissible G-covers C → D of stable curves with ramification data ξ , g(C) = g and g(D) = g . There are source and target morphisms φ : Hg,G,ξ → Mg,r and δ : Hg,G,ξ → Mg ,b , remembering the curves C and D together with the ramification or branch points of the cover respectively. In this paper we study admissible cover cycles, i.e. cycles of the form φ∗ [Hg,G,ξ ]. Examples include the fundamental classes of the loci of hyperelliptic or bielliptic curves C with marked ramification points. The two main results of this paper are as follows: firstly, for the gluing morphism ξ A : M A → Mg,r associated to a stable graph A we give a combinatorial formula for the pullback ξ A∗ φ∗ [Hg,G,ξ ] in terms of spaces of admissible G-covers and ψ classes. This allows us to describe the intersection of the cycles φ∗ [Hg,G,ξ ] with tautological classes. Secondly, the pull–push δ∗ φ ∗ sends tautological classes to tautological classes and we give an explicit combinatorial description of this map. We show how to use the pullbacks to algorithmically compute tautological expressions for cycles of the form φ∗ [Hg,G,ξ ]. In particular, we compute the classes [H5 ] and [H6 ] of the hyperelliptic loci in M5 and M6 and the class [B 4 ] of the bielliptic locus in M4 .
The first author was supported by the grants SNF-200020162928, ERC-2017-AdG-786580-MACI and by the SNF early postdoc mobility grant 184245 and thanks the Max Planck Institute for Mathematics in Bonn for its hospitality. The second author was supported by a GTA fellowship from the University of Liverpool and an Einstein fellowship at Humboldt Universität Berlin. The project has received funding from the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (grant agreement No. 786580).
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Johannes Schmitt [email protected] Jason van Zelm [email protected]
1
Mathematical Institute, University of Bonn, Endenicher Allee 60, 53115 Bonn, Germany
2
Department of Mathematics, Humboldt-Universität zu Berlin, Rudower Chaussee 25, Room 1.415, 12489 Berlin, Germany
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J. Schmitt, J. van Zelm
Keywords Admissible covers · Moduli spaces of curves · Tautological classes Mathematics Subject Classification 14H10 · 14H30 · 14C25
Contents 1 Introduction . . . . . . . . . . . . . . . 1.1 Motivation: admissible cover cycles 1.2 The tautological ring . . . . . . . . 1.3 Stacks of admissible G-covers . . . 1.4 Intersection results . . . . . . . . . 1.5 Applications . . . . . . . . . . . . . 1.6 Outlook . . . . . . . . . . . . . . . 1.7 Outline of the paper . . . . . . . . . 2 Intersections in the tautological ring . . . 2.1 Decorated stratum classes . . . . . . 3 Stacks of pointed admissible G-covers . 3.1 Group actions on smooth curves . .
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