Intersections of loci of admissible covers with tautological classes

Abstract

AbstractFor a finite group G, let $$\overline{\mathcal {H}}_{g,G,\xi }$$ H ¯ g , G , ξ be the stack of admissible G-covers $$C\rightarrow D$$ C → D of stable curves with ramification data $$\xi $$ ξ , $$g(C)=g$$ g ( C ) = g and $$g(D)=g'$$ g ( D ) = g ′ . There are source and target morphisms $$\phi :\overline{\mathcal {H}}_{g,G,\xi }\rightarrow \overline{\mathcal {M}}_{g,r}$$ ϕ : H ¯ g , G , ξ → M ¯ g , r and $$\delta :\overline{\mathcal {H}}_{g,G,\xi }\rightarrow \overline{\mathcal {M}}_{g',b}$$ δ : H ¯ g , G , ξ → M ¯ g ′ , 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 $$\phi _* [\overline{\mathcal {H}}_{g,G,\xi }]$$ ϕ ∗ [ H ¯ g , 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 $$\xi _A:\overline{\mathcal {M}}_A\rightarrow \overline{\mathcal {M}}_{g,r}$$ ξ A : M ¯ A → M ¯ g , r associated to a stable graph A we give a combinatorial formula for the pullback $$\xi ^*_A \phi _*[\overline{\mathcal {H}}_{g,G,\xi }]$$ ξ A ∗ ϕ ∗ [ H ¯ g , G , ξ ] in terms of spaces of admissible G-covers and $$\psi $$ ψ classes. This allows us to describe the intersection of the cycles $$\phi _*[\overline{\mathcal {H}}_{g,G,\xi }]$$ ϕ ∗ [ H ¯ g , G , ξ ] with tautological classes. Secondly, the pull–push $$\delta _*\phi ^*$$ 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 $$\phi _* [\overline{\mathcal {H}}_{g,G,\xi }]$$ ϕ ∗ [ H ¯ g , G , ξ ] . In particular, we compute the classes "Equation missing" and "Equation missing" of the hyperelliptic loci in $$\overline{\mathcal {M}}_5$$ M ¯ 5 and $$\overline{\mathcal {M}}_6$$ M ¯ 6 and the class "Equation missing" of the bielliptic locus in $$\overline{\mathcal {M}}_4$$ M ¯ 4 .