For a finite group G, let \(\overline{\mathcal {H}}_{g,G,\xi }\) be the stack of admissible G-covers \(C\rightarrow D\) of stable curves with ramification data \(\xi \), \(g(C)=g\) and \(g(D)=g'\). There are source and target morphisms \(\phi :\overline{\mathcal {H}}_{g,G,\xi }\rightarrow \overline{\mathcal {M}}_{g,r}\) and \(\delta :\overline{\mathcal {H}}_{g,G,\xi }\rightarrow \overline{\mathcal {M}}_{g',b}\), 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 \(\phi _* [\overline{\mathcal {H}}_{g,G,\xi }]\). 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 \(\xi _A:\overline{\mathcal {M}}_A\rightarrow \overline{\mathcal {M}}_{g,r}\) associated to a stable graph A we give a combinatorial formula for the pullback \(\xi ^*_A \phi _*[\overline{\mathcal {H}}_{g,G,\xi }]\) in terms of spaces of admissible G-covers and \(\psi \) classes. This allows us to describe the intersection of the cycles \(\phi _*[\overline{\mathcal {H}}_{g,G,\xi }]\) with tautological classes. Secondly, the pull–push \(\delta _*\phi ^*\) 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 \(\phi _* [\overline{\mathcal {H}}_{g,G,\xi }]\). In particular, we compute the classes

对于有限群G，令\（\ overline {\ mathcal {H}} _ {g，G，\ xi} \）是可允许G的堆栈-覆盖具有分支的稳定曲线\（C \ rightarrow D \）数据\（\ xi \），\（g（C）= g \）和\（g（D）= g'\）。有源和目标射态\（\ phi：\ overline {\ mathcal {H}} _ {g，G，\ xi} \ rightarrow \ overline {\ mathcal {M}} __ {g，r} \）和\ （\ delta：\ overline {\ mathcal {H}} _ {g，G，\ xi} \ rightarrow \ overline {\ mathcal {M}} _ {g'，b} \），记住曲线C和D以及覆盖物的分叉点或分支点。在本文中，我们研究了可允许的覆盖周期，即\（\ phi _ * [\ overline {\ mathcal {H}} _ {g，G，\ xi}] \\}形式的周期。示例包括具有明显分枝点的超椭圆曲线或双椭圆曲线C的轨迹的基本类别。本文的两个主要结果如下：首先，对于胶合射态\（\ xi _A：\ overline {\ mathcal {M}} _ A \ rightarrow \ overline {\ mathcal {M}} _ {g，r} \）相关联，以稳定的图表甲我们给出用于拉回一个组合式\（\ XI ^ * _ A \披_ * [\划线{\ mathcal {H}} _ {克，G，\ XI}] \）在允许的G覆盖和\（\ psi \）的空间项类。这允许我们用重言式类描述循环\（\ phi _ * [\ overline {\ mathcal {H}} _ {g，G，\ xi}] \）的交集。其次，push-push \（\ delta _ * \ phi ^ * \）将重言式类别发送到重言式类别，我们对该地图进行了明确的组合描述。我们展示了如何使用回撤算法以算法计算形式为\（\ phi _ * [\ overline {\ mathcal {H}} _ {g，G，\ xi}] \）的周期重言式。特别是，我们计算类