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Mathematics, Algebraic Geometry
Abstract:
For a finite group $G$, let $\H_{g,G,\xi}$ be the stack of admissible
$G$-covers $C\to D$ of stable curves with ramification data $\xi$, $g(C)=g$ and
$g(D)=g'$. There are source and target morphisms $\phi\colon \H_{g,G,\xi}\to
\M_{g,r}$ and $\delta\colon \H_{g,G,\xi}\to \M_{g',b}$, 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 $\phi_* [\H_{g,G,\xi}]$. 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 $\xi_A\colon \M_A\to \M_{g,r}$ associated to to a stable graph $A$ we
give a combinatorial formula for the pullback $\xi^*_A \phi_*[\H_{g,G,\xi}]$ in
terms of spaces of admissible $G$-covers and $\psi$ classes. This allows us to
describe the intersection of the cycles $\phi_*[\H_{g,G,\xi}]$ with
tautological classes. Secondly, the pull-push $\delta_*\phi^*$ sends
tautological classes to tautological classes and we also give a combinatorial
description of this map in terms of standard generators of the tautological
rings.
We show how to use the pullbacks to algorithmically compute tautological
expressions for cycles of the form $\phi_* [\H_{g,G,\xi}]$. In particular, we
compute the classes $[\Hyp_5]$ and $[\Hyp_6]$ of the hyperelliptic loci in
$\M_5$ and $\M_6$ and the class $[\B_4]$ of the bielliptic locus in $\M_4$.
