For 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 .

The tautological ring of the moduli space of stable curves has been studied extensively in the last decades. We present a SageMath implementation of many core features of this ring. This includes lists of generators and their products, intersection numbers and verification of tautological relations. Maps between tautological rings induced by functoriality, that is pushforwards and pullbacks under gluing and forgetful maps, are implemented. Furthermore, many interesting cycle classes, such as the double ramification cycles, strata of k-differentials and hyperelliptic or bielliptic cycles are available. We show how to apply the package, including concrete example computations. MSC2010: 14H10, 97N80.

Let [B 2,0,20 ] and [B 2,0,20 ] be the classes of the loci of stable resp. smooth bielliptic curves with 20 marked points where the bielliptic involution acts on the marked points as the permutation (1 2)... (19 20). Graber and Pandharipande proved in [GP03] that these classes are nontatoulogical. In this note we show that their result can be extended to prove that [B g ] is nontautological for g ≥ 12 and that [B 12 ] is nontautological. IntroductionThe system of tautological rings {R • (M g,n )} is defined to be the minimal system of Q-subalgebras of the Chow rings A • (M g,n ) closed under pushforward (and hence pullback) along the natural gluing and forgetful morphismsThe tautological ring R • (M g,n ) of the moduli space of smooth curves is the image of Raccordingly. We say a cohomology class is tautological if it lies in the tautological subring of its cohomology ring, otherwise we say it is nontautological. In this note we will work over C and all Chow and cohomology rings are assumed to be taken with rational coefficients.These tautological rings are relatively well understood. An additive set of generators for the groups R • (M g,n ) is given by decorated boundary strata and there exists an algorithm for computing the intersection product (see [GP03]). The class of many "geometrically defined" loci can be shown to be tautological, for example this is the case for the class of the locus H g of hyperelliptic curves in M g (see [FP05, Theorem 1]).Any odd cohomology class of M g,n is nontautological by definition. Deligne proved that H 11 (M 1,11 ) = 0, thus providing a first example of the existence of nontautological classes. In fact it is known that H • (M 0,n ) = RH • (M 0,n ) (see [Kee92]) and that H 2• (M 1,n ) = RH 2• (M 1,n ) for all n (see [Pet14, Corollary 1.2]).Examples of geometrically defined loci which can be proven to be nontautological are still relatively scarce. In [GP03] Graber and Pandharipande hunt for algebraic classes in H 2• (M g,n ) and H 2• (M g,n ) which are nontautological. In particular they show that the classes of the loci B 2,0,20 and B 2,0,20 of stable resp. smooth bielliptic curves of genus 2 with 20 marked points where the bielliptic involution acts on the set of marked points as the permutation (1 2)... (19 20)

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Following Mumford and Chiodo, we compute the Chern character of the derived pushforward ch (• * ()) , for an arbitrary element of the Picard group of the universal curve over the moduli stack of stable marked curves. This allows us to express the pullback of universal Brill-Noether classes via Abel-Jacobi sections to the compactified universal Jacobians, for all compactifications such that the section is a well-defined morphism.