# The $\mathcal{H}$-tautological ring

Carl Lian  Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin,
12489, Germany<EMAIL_ADDRESS>http://sites.google.com/view/carllian/

###### Abstract.

We extend the theory of tautological classes on moduli spaces of stable curves
to the more general setting of moduli spaces of admissible Galois covers of
curves, introducing the so-called $\mathcal{H}$-tautological ring. The main
new feature is the existence of restriction-corestriction morphisms
remembering intermediate quotients of Galois covers, which are a rich source
of new classes. In particular, our new framework includes classes of Harris-
Mumford admissible covers on moduli spaces of curves, which are known in some
(and speculatively many more) examples to lie outside the usual tautological
ring. We give additive generators for the $\mathcal{H}$-tautological ring and
show that their intersections may be algorithmically computed, building on
work of Schmitt-van Zelm. As an application, we give a method for computing
integrals of Harris-Mumford loci against tautological classes of complementary
dimension, recovering and giving a mild generalization of a recent quasi-
modularity result of the author for covers of elliptic curves.

###### Key words and phrases:

moduli of curves, admissible covers, tautological classes, intersection theory

###### 2020 Mathematics Subject Classification:

14H10, 14H30, 14N10, 14C17

###### Contents

  1. 1 Introduction
    1. 1.1 Hurwitz loci on moduli spaces of curves
    2. 1.2 The classical tautological ring
    3. 1.3 Galois covers
    4. 1.4 The $\mathcal{H}$-tautological ring
    5. 1.5 Further directions
    6. 1.6 Conventions
    7. 1.7 Acknowledgments
  2. 2 Tautological classes on moduli spaces of curves
    1. 2.1 The universal family
    2. 2.2 $\psi$ and $\kappa$ classes
    3. 2.3 Stable graphs and boundary strata
    4. 2.4 Intersections of boundary strata
    5. 2.5 Pullbacks of $\psi$ and $\kappa$ classes
    6. 2.6 The tautological ring
    7. 2.7 Chern classes of the tangent bundle
  3. 3 From moduli of Galois covers to the $\mathcal{H}$-tautological ring
    1. 3.1 Admissible $G$-covers
    2. 3.2 The universal family
    3. 3.3 $\psi$ and $\kappa$ classes
    4. 3.4 Admissible $G$-graphs and boundary strata
    5. 3.5 Restriction-corestriction maps
    6. 3.6 Diagonals and non-tautological cycles from $\overline{\mathcal{H}}_{g,G,\xi}$
    7. 3.7 The $\mathcal{H}$-tautological ring
  4. 4 Bivariant pullbacks of $\mathcal{H}$-tautological classes
    1. 4.1 Pullbacks by forgetful maps
      1. 4.1.1 Forgetful maps
      2. 4.1.2 Boundary maps
      3. 4.1.3 Diagonal maps
      4. 4.1.4 Restriction-corestriction maps
    2. 4.2 Intersections of two boundary strata
    3. 4.3 Intersections of restriction-corestriction maps with boundary strata
      1. 4.3.1 Restriction maps
      2. 4.3.2 Corestriction maps
    4. 4.4 Pullbacks of $\psi$ and $\kappa$ classes
      1. 4.4.1 Forgetful maps
      2. 4.4.2 Diagonal maps
      3. 4.4.3 Boundary maps
      4. 4.4.4 Restriction-corestriction maps
    5. 4.5 Interlude: Chern classes of the tangent bundle
    6. 4.6 Intersections of restriction-corestriction maps
      1. 4.6.1 Intersection of two restriction maps
      2. 4.6.2 Intersection of a restriction and corestriction map
      3. 4.6.3 Intersection of two corestriction maps
    7. 4.7 Diagonals
  5. 5 Additive structure
    1. 5.1 Generators for the $\mathcal{H}$-tautological ring
    2. 5.2 Integration of zero-cycles
  6. 6 Classical Hurwitz loci
    1. 6.1 The Harris-Mumford space and its normalization
    2. 6.2 Intersecting Harris-Mumford loci with boundary classes
    3. 6.3 $d$-elliptic loci in genus 2
      1. 6.3.1 Intersection with $\Delta_{01}$
      2. 6.3.2 Intersection with $\Delta_{00}$
    4. 6.4 Proof of Theorem 6.3

##  1\. Introduction

###  1.1. Hurwitz loci on moduli spaces of curves

The cycle theory of the moduli space of stable curves
$\overline{\mathcal{M}}_{g,n}$ is central to understanding its geometry.
Moduli spaces of branched covers of curves provide a particularly rich source
of algebraic cycles on $\overline{\mathcal{M}}_{g,n}$. Let
$\mathcal{H}_{g/h,d}$ be a moduli space of branched covers (a Hurwitz space),
parametrizing finite degree $d$ morphisms $f:X\to Y$, where $X,Y$ are smooth
curves of genus $g,h$, respectively. Suppose further that $f$ is simply
branched over

$b=(2g-2)-d(2h-2)$

points. We then get morphisms

---
$\textstyle{\mathcal{H}_{g/h,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{g/h,d}}$$\scriptstyle{\delta_{g/h,d}}$$\textstyle{\mathcal{M}_{g,b}}$$\textstyle{\mathcal{M}_{h,b}}$

remembering the source and target, marked with the ramification and branch
points, respectively.

Via the Harris-Mumford space of admissible covers, one can compactify
$\mathcal{H}_{g/h,d}$ and extend the morphisms above:

---
$\textstyle{\overline{\mathcal{H}}_{g/h,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{g/h,d}}$$\scriptstyle{\delta_{g/h,d}}$$\textstyle{\overline{\mathcal{M}}_{g,b}}$$\textstyle{\overline{\mathcal{M}}_{h,b}}$

By to the Riemann Existence Theorem, the map $\delta_{g/h,d}$ is flat and
quasi-finite of degree given combinatorially by a Hurwitz number counting
monodromy actions of the fundamental group of the target curve, punctured at
the branch points, on the fibers of $f$.

On the other hand, the map $\phi_{g/h,d}$ gives an interesting cycle of
homological degree $3h-3+b$ on $\overline{\mathcal{M}}_{g,b}$ upon pushing
forward the fundamental class $1\in A_{*}(\overline{\mathcal{H}}_{g/h,d})$; we
refer to such cycles as branched cover loci or Hurwitz loci.

We note three reasons for a detailed study of such cycles:

  1. (i)

Cycles $(\phi_{g/h,d})_{*}(1)$ are interesting from the point of view of
birational geometry. For example, Eisenbud, Harris, and Mumford prove [HM82,
Har84, EH87] that $\overline{\mathcal{M}}_{g}$ is of general type for $g\geq
24$ by producing effective divisors $E$ on $\overline{\mathcal{M}}_{g}$ for
which $K_{\overline{\mathcal{M}}_{g}}=A+E$, for some ample divisor $A$. When
$g$ is odd, they take $E$ to be the branched cover locus
$(\pi_{g/0,(g+1)/2})_{*}(1)$ of curves of sub-generic gonality.

More recently, numerous examples of extremal classes on
$\overline{\mathcal{M}}_{g,n}$ have been constructed via branched cover loci,
see, for instance, [CC14, CC15, CP16, CT16, Bla20].

  2. (ii)

Cycles $(\phi_{g/h,d})_{*}(1)$ satisfy interesting numerological phenomena.
For example, in the case $h=1$, we have the following:

######  Conjecture 1.

[L20, Conjecture 1] Fix an integer $g\geq 2$. We have:

$\sum_{d\geq 1}[(\phi_{g/1,d})_{*}(1)]q^{d}\in
A^{g-1}(\overline{\mathcal{M}}_{g})\otimes\operatorname{Qmod},$

where $\operatorname{Qmod}$ is the ring of quasimodular forms.

Conjecture [L20] is known to be true for $g=2,3$ [L20, Theorem 1.2], with a
number of variants, for example, allowing for marked points to be taken into
account. We give a slight improvement on this result in the case $g=2$ in
Theorem 6.4.

  3. (iii)

Cycles $(\phi_{g/h,d})_{*}(1)$ are known in some cases to be non-tautological,
see, for example, [GP03, vZ18]. We defer a more detailed discussion of this
phenomenon to the next section.

###  1.2. The classical tautological ring

Both the Chow ring $A^{*}(\overline{\mathcal{M}}_{g,n})$ and the cohomology
ring $H^{*}(\overline{\mathcal{M}}_{g.n})$ are extremely complicated and a
full understanding of these objects seems out of reach. However, Mumford
[Mum83] initiated a study of certain tautological classes on
$\overline{\mathcal{M}}_{g,n}$ that appear in many natural geometric
situations and are largely computable in practice.

By definition, the tautological rings
$R^{*}(\overline{\mathcal{M}}_{g,n})\subset
A^{*}(\overline{\mathcal{M}}_{g,n})$ form the smallest system of
$\mathbb{Q}$-subalgebras containing the $\psi$ and $\kappa$ classes and closed
under all pushforwards by forgetful morphisms
$\pi:\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$ and
boundary morphisms
$\xi_{\Gamma}:\overline{\mathcal{M}}_{\Gamma}\to\overline{\mathcal{M}}_{g,n}$,
indexed by stable graphs $\Gamma$ of genus $g$ with $n$ marked points.111In
fact, one can make this definition without mentioning the $\psi$ and $\kappa$
classes, which will automatically be included, but we include them explicitly
to highlight their fundamental importance. Additive generators for the
tautological ring may be given by decorated boundary strata, that is,
pushforwards of polynomials in $\psi$ and $\kappa$ classes by boundary
morphisms, see [GP03, Appendix A], and indeed such classes may be
algorithmically intersected in terms of the combinatorics of dual graphs.

Many cohomology classes on moduli spaces of curves arising in geometry turn
out to be tautological. However, as first shown by Graber-Pandharipande
[GP03], Hurwitz loci, in general, are not. For example, as we review in §3.6,
the $(3h-1)$-dimensional locus of stable curves of genus $2h$ admitting an
admissible double cover of a curve of genus $h$ is non-tautological for $h$
sufficiently large. Indeed, its pullback to the boundary stratum
$\overline{\mathcal{M}}_{h,1}\times\overline{\mathcal{M}}_{h,1}\to\overline{\mathcal{M}}_{2h}$
is a positive multiple of the diagonal class. However, if it were
tautological, it would have a tautological Künneth decomoposition upon
pullback, which is impossible because $\overline{\mathcal{M}}_{h,1}$ is known
to have odd cohomology for $h$ sufficiently large.

This example yields two insights: firstly, one would get a strictly larger
theory by adding Hurwitz loci to the calculus of tautological classes, and
secondly, doing so would require diagonal classes to be added as well. We
will, in fact, go further: we develop this theory in the Chow rings of Hurwitz
spaces themselves.

###  1.3. Galois covers

The main drawback to the Harris-Mumford compactification of Hurwitz spaces by
admissible covers is that in general, spaces of admissible covers fail to be
smooth. In order to carry out intersection theory on them in a robust way,
therefore, we must pass to a resolution of singularities.

We will instead work with Galois covers. Suppose that $f:X\to Y$ is a degree
$d$ étale cover of (possibly non-proper) curves. Then, we have a principal
$S_{d}$ bundle

$\widetilde{f}:\widetilde{X}:=(X\times_{Y}\cdots\times_{Y}X)-\Delta\to Y$

given by taking the $d$-fold product over $X$ and removing all diagonals.
Moreover, the data of $f$ can be recovered from the $S_{d}$-cover
$\widetilde{X}\to Y$ by defining $X=\widetilde{X}/S_{d-1}$.

If $f:X\to Y$ is now assumed to be a possibly branched cover of smooth curves,
one can carry out the same construction on the étale locus, then pass to the
unique extension to proper curves, to produce a map
$\widetilde{f}:\widetilde{X}\to Y$ that is an $S_{d}$-cover, but no longer a
principal $S_{d}$-bundle.

If, however, $f:X\to Y$ is an admissible cover of nodal curves, the
construction breaks down. On the other hand, the inverse construction is still
valid: an admissible $S_{d}$-cover $\widetilde{f}:\widetilde{X}\to Y$ gives
rise to an admissible cover $f:\widetilde{X}/S_{d-1}\to Y$.

Thus, we have a map
$\nu:\widetilde{H}_{g/h,d}\to\overline{\mathcal{H}}_{g/h,d}$, where
$\widetilde{H}_{g/h,d}$ parametrizes $S_{d}$-covers
$\widetilde{f}:\widetilde{X}\to Y$ that, upon taking the intermediate quotient
by $S_{d-1}$, gives an admissible cover of the required type. The work of
Abramovich-Corti-Vistoli in the language of twisted stable maps (where the
scheme-theoretic $S_{d}$-cover $\widetilde{f}:\widetilde{X}\to Y$ is replaced
by a principal $S_{d}$-bundle $\widetilde{f}:\widetilde{X}\to\mathcal{Y}$ over
a stacky curve $\mathcal{Y}$) implies that $\widetilde{H}_{g/h,d}$ is smooth
and $\nu$ is a normalization, see [ACV03, Proposition 4.2.2].

Therefore, we work in the following setting, following [SvZ18]: if $G$ is a
finite group, let $\overline{\mathcal{H}}_{g,G,\xi}$ be the moduli space of
pointed admissible $G$-covers with source genus $g$ and with monodromy data
given by $\xi$, see Definition 3.4. Then, $\overline{\mathcal{H}}_{g,G,\xi}$
is a smooth Deligne-Mumford stack, see Theorem 3.7; it admits source and
target maps

$\displaystyle\phi:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{M}}_{g,r}$
$\displaystyle\delta:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{M}}_{g^{\prime},b}$

as in the Harris-Mumford setting.

###  1.4. The $\mathcal{H}$-tautological ring

We develop the theory of tautological classes on spaces of admissible
$G$-covers $\overline{\mathcal{H}}_{g,G,\xi}$ in parallel with that of the
classical theory.

As in the classical setting, we have natural notions of forgetful morphisms
$\pi:\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\to\overline{\mathcal{H}}_{g,G,\xi}$
and boundary morphisms
$\xi_{(\Gamma,G)}:\overline{\mathcal{H}}_{(\Gamma,G)}\to\overline{\mathcal{H}}_{g,G,\xi}$,
the latter indexed by admissible $G$-graphs. In addition, one can define
$\psi$ and $\kappa$ classes on $\overline{\mathcal{H}}_{g,G,\xi}$ by pullback
under $\phi$.

However, the main new feature of the theory of spaces of admissible $G$-covers
is the existence of restriction morphisms

$\operatorname{res}^{G}_{G_{1}}:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{H}}_{g,G_{1},\xi^{\prime}}$

restricting the $G$-action on a stable curve to that of a subgroup
$G_{1}\subset G$, and corestriction morphisms

$\operatorname{cores}^{G}_{G/G_{1}}:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{H}}_{g_{1},G/G_{1},\xi^{\prime}}$

corestricting the $G$-action on a stable curve $X$ to that of $G/G_{1}$ on the
partial quotient $X/G_{1}$, where $G_{1}\subset G$ is a normal subgroup.

Combining these operations gives a more general notion of restriction-
corestriction morphisms, see Definition 3.22.

For example, restriction maps $\operatorname{res}^{G}_{\\{1\\}}$ recover
source maps $\phi$, and corestriction maps
$\operatorname{cores}^{G}_{\\{1\\}}$ recover target maps $\delta$. The pre-
composition

$\phi_{g/h,d}\circ\nu:\widetilde{H}_{g/h,d}\to\overline{\mathcal{M}}_{g,b}$

of the source map on a Harris-Mumford space with its normalization by
$S_{d}$-covers may also be regarded as the composition

$\operatorname{cores}^{S_{d-1}}_{\\{1\\}}\circ\operatorname{res}^{S_{d}}_{S_{d-1}}.$

Finally, as indicated in §1.2, our theory must also include diagonal morphisms

$\Delta:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{H}}_{g,G,\xi}\times\overline{\mathcal{H}}_{g,G,\xi}$

in the calculus, as pullbacks of restriction maps by boundary morphisms may
include contributions from such classes.

The main definition of the paper (see also Definition 3.25) is then:

######  Definition 1.1.

The $\mathcal{H}$-tautological ring is the smallest system of
$\mathbb{Q}$-subalgebras

$R^{*}_{\mathcal{H}}\left(\prod_{j=1}^{m}\overline{\mathcal{H}}_{g_{j},G_{j},\xi_{j}}\right)\subset
A^{*}\left(\prod_{j=1}^{m}\overline{\mathcal{H}}_{g_{j},G_{j},\xi_{j}}\right),$

ranging over all finite products of moduli spaces of admissible $G$-covers,
that contains all $\psi$ and $\kappa$ classes and is closed under pushfoward
by all (products of) boundary, forgetful, restriction-corestriction, and
diagonal maps.

If $\gamma$ is an arbitrary composition of tautological morphisms and $\theta$
is a polynomial in the $\psi$ and $\kappa$ classes, then certainly
$\gamma_{*}(\theta)$ must be $\mathcal{H}$-tautological. Our main theorem (see
also Theorem 5.1) is that such classes additively generate the
$\mathcal{H}$-tautological ring.

######  Theorem 1.2.

The classes $\gamma_{*}(\theta)$ comprise a set of additive generators for the
$\mathcal{H}$-tautological ring. Moreover, the $\mathcal{H}$-tautological ring
is closed under bivariant pullback by the previously-described tautological
morphisms.

In fact, one can take $\gamma$ to be a composition of diagonal, restriction-
corestriction, forgetful, and boundary morphisms, in that order, see Theorem
5.2.

The proof of Theorem 1.2 is constructive. We give algorithms for expressing
the bivariant pullback of any $\mathcal{H}$-tautological class by any
tautological morphism. Many of the methods are analogous to those in the
classical situation. For example, one can intersect boundary classes on spaces
of admissible $G$-covers (see §4.2) in an essentially identical fashion to
that of boundary classes on $\overline{\mathcal{M}}_{g,n}$ (see §2.4 or [GP03,
Appendix A]), keeping track of the additional data of a group action.

However, we highlight here two aspects of the theory requiring new ideas.

  1. (i)

Intersections of restriction-corestriction morphisms with boundary strata, see
§4.3. This was initiated in the case of the source map
$\phi=\operatorname{res}^{G}_{\\{1\\}}$ by Schmitt-van Zelm [SvZ18], where
many examples of classes of Galois Hurwitz loci on
$\overline{\mathcal{M}}_{g,n}$ determined by their integrals against classical
tautological classes are computed.

We extend the results of Schmitt-van Zelm to the setting of restriction-
corestriction morphisms, where the functorial intersections are, in general,
non-reduced, owing to the fact that boundary strata in general lie in the
branch locus of corestriction morphisms.

In particular, our methods allow for the integration of classical tautological
classes against arbitrary (non-Galois) Hurwitz loci coming from Harris-Mumford
spaces, which is already new. In §6, we employ our methods to re-compute the
classes of $d$-elliptic loci in genus 2, as originally carried out in [L20],
and state a mild generalization of the quasimodularity observed therein, see
Theorem 6.4.

  2. (ii)

Intersections of two restriction-corestriction morphisms, see §4.6. These
methods are new, but because the intersection-theoretic formulas are quite
complicated, with terms indexed by group-theoretic data that seem difficult to
compute in practice, the results may perhaps be more profitably viewed as
proofs of existence.

###  1.5. Further directions

We briefly propose here four avenues for further investigation on the
$\mathcal{H}$-tautological ring.

  1. (i)

Improvement and implementation of algorithms. SAGE code of Delecroix-Schmitt-
van Zelm is available to perform computations in the classical tautological
ring, see [DSvZ]. Their code has additional functionality for certain Galois
Hurwitz loci on $\overline{\mathcal{M}}_{g,n}$.

A next step would be extend this work to include functionality for the general
$\mathcal{H}$-tautological calculus, particularly that of Harris-Mumford loci,
which, as we explain in §6, can in fact can be dealt with computationally
without making explicit reference to Galois covers.

As we indicated above, the results of §4.6 are, at least as stated, likely too
unwieldy to be implemented computationally; some theoretical speedups would
thus be desirable here.

  2. (ii)

Study of the structure of the $\mathcal{H}$-tautological ring, following, for
instance, the Faber conjectures for the classical tautological ring, see
[Fab99].

Two places to start would be the following questions:

     * •

What is the rank of $(R_{\mathcal{H}})_{0}(\overline{\mathcal{H}}_{g,G,\xi})$?
In the classical situation, the answer is always 1, by a result of Graber-
Vakil [GV01]. When $G=\\{1\\}$, that is,
$\overline{\mathcal{H}}_{g,G,\xi}=\overline{\mathcal{M}}_{g,n}$, the answer in
the $\mathcal{H}$-tautological setting is again 1 assuming that zero-cycles of
$\overline{\mathcal{M}}_{g,n}$ defined over $\overline{\mathbb{Q}}$ are
tautological (see [PS20, Speculation 1.3]), though we do not have an
independent proof. In the general case, the situation is complicated by the
fact that $\overline{\mathcal{H}}_{g,G,\xi}$ may be disconnected, but in
principle the group of zero-dimensional $\mathcal{H}$-tautological classes
could still have rank 1.

     * •

What is the largest integer $k$ for which
$R_{\mathcal{H}}^{k}(\mathcal{M}_{g})=0$? In the classical situation, the
answer is $k=g-1$ due to Looijenga and Faber, see [Loo95, Fab97]. However,
when one allows $\mathcal{H}$-tautological classes, this becomes false when
$g=12$ due to van Zelm’s result that the bielliptic locus on
$\mathcal{M}_{12}$ is non-tautological, see [vZ18, Theorem 2].

  3. (iii)

Construction of non-trivial $\mathcal{H}$-tautological relations. For
classical tautological classes on $\overline{\mathcal{M}}_{g,n}$, the Pixton
relations [Pix12, PPZ15] are conjecturally a complete set of relations; these
can be pulled back to $\overline{\mathcal{H}}_{g,G,\xi}$. However,
establishing $\mathcal{H}$-tautological relations not arising in this way
would require new tools. For example, the full Conjecture 1 is unlikely to be
proven by (classical) “tautological methods,” as the classes appearing are not
all tautological.

As a step in this direction, Clader [Cla17] has constructed relations on the
moduli space of cyclic covers, pushing forward to (classical) tautological
relations on $\overline{\mathcal{M}}_{g,n}$.

  4. (iv)

Study of the size of the $\mathcal{H}$-tautological ring, relative to that
(for example) of the classical tautological ring, or on the other hand the
entire cohomology or Chow ring of $\overline{\mathcal{M}}_{g,n}$. In
particular, it would be interesting now to attempt to construct cycles that
are not $\mathcal{H}$-tautological.

###  1.6. Conventions

We work over $\mathbb{C}$. All curves are assumed projective and connected
with only nodes as singularities, unless otherwise indicated. The genus of a
curve refers to its arithmetic genus. All moduli spaces are understood to be
stacks, rather than coarse moduli spaces.

We will work throughout with rational Chow groups $A^{*}(\mathcal{X})$, where
$\mathcal{X}$ ranges over a system of products of Hurwitz spaces.

All group actions are left group actions, unless otherwise stated.

For sake of clarity of exposition, we will often describe constructions
involving moduli spaces on the level of closed points, rather than functors;
we invite the reader to generalize the discussion appropriately, for example,
by replacing marked points on curves with sections of families of curves.

###  1.7. Acknowledgments

I thank Alessio Cela, Johan de Jong, Johannes Schmitt, Johannes Schwab and
Jason van Zelm for helpful comments and discussions about ideas related to
this paper, and David Yang for providing the proof of Lemma 6.8. I also thank
the anonymous referee for numerous suggestions that improved the exposition
and content of the paper.

This work was completed primarily with the support of a NSF Postdoctoral
Research Fellowship, grant DMS-2001976.

##  2\. Tautological classes on moduli spaces of curves

In this section, we recall the classical intersection theory of tautological
classes on moduli spaces of curves, as initiated by [Mum83]; we follow the
approaches of [AC96] and [GP03].

###  2.1. The universal family

######  Definition 2.1.

We define the forgetful morphism
$\pi:\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$ forgetting
the last marked point and contracting non-stable components, along with its
universal sections
$\sigma_{i}:\overline{\mathcal{M}}_{g,n}\to\overline{\mathcal{M}}_{g,n+1}$
attaching a 2-pointed rational tail to the $i$-th marked point.

Taken together, $\pi$ and the sections $\sigma_{i}$ form the universal family
over $\overline{\mathcal{M}}_{g,n}$.

We have the following easy result:

######  Lemma 2.2.

The commutative diagram

$\textstyle{\overline{\mathcal{M}}_{g,n+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{n+1}}$$\scriptstyle{\pi_{n+2}}$$\textstyle{\overline{\mathcal{M}}_{g,n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\overline{\mathcal{M}}_{g,n+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\overline{\mathcal{M}}_{g,n}}$

is Cartesian up to birational equivalence.222by this we mean:
$\overline{\mathcal{M}}_{g,n+2}$ maps properly and birationally onto the fiber
product. Here, the two maps out of $\overline{\mathcal{M}}_{g,n+2}$ forget the
$(n+1)$-st and $(n+2)$-nd points, and the maps into
$\overline{\mathcal{M}}_{g,n}$ forget the other of these two points.

###  2.2. $\psi$ and $\kappa$ classes

######  Definition 2.3.

Let $\omega_{\pi}$ be the relative dualizing sheaf of the forgetful morphism
$\pi:\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$. The
$\psi$ classes are defined by

$\psi_{i}:=\sigma_{i}^{*}(c_{1}(\omega_{\pi}))\in
A^{1}(\overline{\mathcal{M}}_{g,n}).$

######  Definition 2.4.

Let $K$ be the divisor class of the log-canonical sheaf
$\omega_{\pi}(\sigma_{1}+\cdots+\sigma_{n})$. We define the $\kappa$ classes

$\kappa_{i}:=\pi_{*}(K^{i+1})\in A^{i}(\overline{\mathcal{M}}_{g,n}).$

######  Remark 2.5.

One can also take as a definition

$\widetilde{\kappa}_{i}:=\pi_{*}(c_{1}(\omega_{\pi})^{i+1})\in
A^{i}(\overline{\mathcal{M}}_{g,n}).$

The two possible notions of $\kappa$ classes are related by the formula

$\kappa_{i}=\widetilde{\kappa}_{i}+\sum_{j=1}^{n}\psi_{j}^{i},$

see [AC96, (1.5)].

###  2.3. Stable graphs and boundary strata

The boundary of $\overline{\mathcal{M}}_{g.n}$ admits a stratification by
substacks indexed by stable graphs and parametrized by products of moduli
spaces of curves of smaller dimension.

######  Definition 2.6.

A graph $\Gamma$ consists of the data of a vertex set $V(\Gamma)$, a half-edge
set $H(\Gamma)$, a fixed-point free involution $\iota:H(\Gamma)\to H(\Gamma)$
whose orbits form the edge set $E(\Gamma)$ of $E$, and an assignment
$a:H(\Gamma)\to V(\Gamma)$ of a vertex to each half-edge.

A stable graph includes the additional data of a genus function
$g:V(\Gamma)\to\mathbb{Z}_{\geq 0}$ and an ordered set $L(\Gamma)$ of legs
with an assignment $\zeta:L(\Gamma)\to V(\Gamma)$ of a vertex to each leg. We
denote the valency of a vertex, the sum of the number of half-edges and legs
of $\Gamma$ incident at $v$, by $n(v)$, and require that $\Gamma$ satisfies
the stability condition that $2g(v)-2+n(v)>0$ for any $v\in V(\Gamma)$.

The genus of $\Gamma$ is defined to be

$g(\Gamma):=h^{1}(\Gamma)+\sum_{v\in V(\Gamma)}g(v),$

and the number of elements $n$ of $L(\Gamma)$ is referred to as the number of
marked points of $\Gamma$.

Stable graphs of genus $g$ with $n$ marked points are exactly the dual graphs
of stable curves parametrized by $\overline{\mathcal{M}}_{g,n}$.

######  Definition 2.7.

Given a stable graph $\Gamma$, we define the boundary stratum

$\overline{\mathcal{M}}_{\Gamma}:=\prod_{v\in
V(\Gamma)}\overline{\mathcal{M}}_{g(v),n(v)}$

We then have a natural boundary morphism

$\xi_{\Gamma}:\overline{\mathcal{M}}_{\Gamma}\to\overline{\mathcal{M}}_{g,n}$

gluing together marked points corresponding to paired half-edges, whose image
is the closure of the locus of stable curves with dual graph $\Gamma$.

The map $\xi_{\Gamma}$ is finite and unramified; the degree onto its image is
given by $\\#\operatorname{Aut}(\Gamma)$, where the notion of a morphism (and
hence an automorphism) of a stable graph is given below.

######  Definition 2.8.

Suppose $\Gamma,A$ are stable graphs of genus $g$ with $n$ marked points. A
morphism of stable graphs $f:\Gamma\to A$ (or an $A$-structure on $\Gamma$)
consists of a surjection on vertex sets $f_{V}:V(\Gamma)\to V(A)$ and an
injection (in the opposite direction) on half-edge sets $f_{H}:H(A)\to
H(\Gamma)$, compatible with the half-edge involutions $\iota$, edge attachment
maps $a$, and leg assignment maps $\zeta$, and respecting the genus function
in the sense that the pre-image of a vertex $v\in V(A)$ along with its
incident edges forms a stable graph of genus $g(v)$. (See [SvZ18, Definition
2.5] for a more explicit definition.)

A morphism $f:\Gamma\to A$ captures the degeneration of a curve with dual
graph $A$ into one with dual graph $\Gamma$. Indeed, a map $f:\Gamma\to A$ of
stable graphs induces in a natural way a map $\xi_{\Gamma\to
A}:\overline{\mathcal{M}}_{\Gamma}\to\overline{\mathcal{M}}_{A}$, compatible
with the boundary maps $\xi_{\Gamma},\xi_{A}$.

The moduli space $\overline{\mathcal{M}}_{\Gamma}$ may also be interpreted as
the moduli of stable marked curves $X$ along with a $\Gamma$-structure on its
dual graph, see [GP03, Proposition 8].

The normal bundle of $\xi_{\Gamma}$ is given by a direct sum of line bundle
contributions from the edges of $\Gamma$, or equivalently the nodes of the
general curve parametrized by $\overline{\mathcal{M}}_{\Gamma}$. More
precisely, we have the following well-known fact:

######  Lemma 2.9.

$N_{\xi_{\Gamma}}=\bigoplus_{(\ell,\ell^{\prime})\in E(\Gamma)}T_{\ell}\otimes
T_{\ell^{\prime}},$

where $(\ell,\ell^{\prime})$ denotes the pairs of half-edges comprising edges
of $\Gamma$ and $T_{\ell},T_{\ell^{\prime}}$ are the corresponding tangent
line bundles on the appropriate components
$\overline{\mathcal{M}}_{g(v),n(v)}$. In particular, we have

$c(N_{\xi_{\Gamma}})=\prod_{e\in
E(\Gamma)}(1-\psi_{h_{e}}-\psi_{h^{\prime}_{e}}).$

###  2.4. Intersections of boundary strata

The main reference for this section is [GP03, Appendix A]. Let $A,B$ be stable
graphs of genus $g$ with $n$ marked points. We consider the intersection
$\xi_{A}^{*}(\xi_{B*}(1))\in A^{*}(\overline{\mathcal{M}}_{A})$.

######  Lemma 2.10.

We have a Cartesian diagram

$\textstyle{\displaystyle\coprod\overline{\mathcal{M}}_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{\Gamma\to
A}}$$\scriptstyle{\xi_{\Gamma\to
B}}$$\textstyle{\overline{\mathcal{M}}_{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{B}}$$\textstyle{\overline{\mathcal{M}}_{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{A}}$$\textstyle{\overline{\mathcal{M}}_{g,n}}$

where the disjoint union ranges over generic $(A,B)$-graphs $\Gamma$, that is,
stable graphs $\Gamma$ equipped with morphisms $\Gamma\to A,B$ satisfying the
genericity condition that the map

$E(A)\sqcup E(B)\to E(\Gamma)$

is surjective.

The excess bundle on $\overline{\mathcal{M}}_{\Gamma}$ may be easily
understood in terms of the normal bundles of boundary morphisms as in Lemma
2.9. One finds:

######  Lemma 2.11.

$\xi_{\Gamma\to A}^{*}N_{\xi_{A}}/N_{\xi_{\Gamma\to
B}}\cong\bigoplus_{(\ell,\ell^{\prime})\in E(A)\cap E(B)}T_{\ell}\otimes
T_{\ell^{\prime}},$

where $(\ell,\ell^{\prime})$ denotes the pair of half-edges comprising an
edge.

Therefore, by the excess intersection formula, we conclude:

######  Proposition 2.12.

[GP03, (11)] Let $A,B$ be stable graphs of genus $g$ with $n$ marked points.
We have:

$\xi^{*}_{A}(\xi_{B*}(1))=\sum_{\Gamma}(\xi_{\Gamma\to
A})_{*}\left(\prod_{(\ell,\ell^{\prime})\in E(A)\cap
E(B)}(-\psi_{\ell}-\psi_{\ell^{\prime}})\right),$

where the sum is over $\Gamma$ as above.

###  2.5. Pullbacks of $\psi$ and $\kappa$ classes

Here, we record formulas for the pullbacks of $\psi$ and $\kappa$ classes by
forgetful and boundary morphisms.

######  Proposition 2.13.

Let $\pi:\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$ be the
map forgetting the last marked point. We have:

$\pi^{*}(\psi_{i})=\psi_{i}-{\sigma_{i}}_{*}(1),$

where $\sigma_{i}$ is the $i$-th universal section, and

$\pi^{*}(\kappa_{i})=\kappa_{i}-\psi_{n+1}^{i}$

###### Proof.

The first statement appears in [AC87, §3] and the second is [AC96, (1.10)]. ∎

######  Proposition 2.14.

Let
$\xi_{\Gamma}:\overline{\mathcal{M}}_{\Gamma}\to\overline{\mathcal{M}}_{g,n}$
be the boundary map associated to the stable graph $\Gamma$. We have:

$\xi_{\Gamma}^{*}(\psi_{i})=\operatorname{pr}_{v}^{*}\psi_{i},$

where $\operatorname{pr}$ is the projection to the component
$\overline{\mathcal{M}}_{g(v),n(v)}$ of $\overline{\mathcal{M}}_{\Gamma}$
corresponding to the component containing the $i$-th marked point, and

$\xi_{\Gamma}^{*}(\kappa_{i})=\sum_{v\in
V(\Gamma)}\operatorname{pr}_{v}^{*}(\kappa_{i}).$

###### Proof.

The first statement is trivial, and the second is [AC96, (1.8)]. ∎

###  2.6. The tautological ring

######  Definition 2.15.

The tautological ring is the smallest system of $\mathbb{Q}$-subalgebras
$R^{*}(\overline{\mathcal{M}}_{g,n})\subset
A^{*}(\overline{\mathcal{M}}_{g,n})$ containing all $\psi$ and $\kappa$
classes and closed under pushforwards by all boundary morphisms
$\xi_{\Gamma}:\overline{\mathcal{M}}_{\Gamma}\to\overline{\mathcal{M}}_{g,n}$
and all forgetful morphisms
$\pi:\overline{\mathcal{M}}_{g,n+1}\to\overline{\mathcal{M}}_{g,n}$.

Similarly, one can define the tautological
$RH^{*}(\overline{\mathcal{M}}_{g,n})\subset
H^{*}(\overline{\mathcal{M}}_{g,n})$ by the image of the tautological ring in
Chow under the cycle class map.

Additive generators for the tautological ring are given by decorated boundary
classes, that is, pushforwards of monomials in $\psi$ and $\kappa$ classes by
boundary morphisms, see [GP03, Proposition 11]. Indeed, by definition, these
classes must be tautological, and the results of §2.4 and 2.5 show that the
sub-$\mathbb{Q}$-vector spaces $A^{*}(\overline{\mathcal{M}}_{g,n})$ generated
by decorated boundary classes are closed under multiplication. In fact, one
easily sees that the tautological ring is closed under all bivariant pullbacks
by forgetful and boundary morphisms.

Computing intersection numbers of tautological classes amounts to computing to
integrals of polynomials in $\kappa$ and $\psi$ classes in top degree. Such
integrals are given by the KdV hierarchy, see [Kon92].

###  2.7. Chern classes of the tangent bundle

The Chern classes of the tangent bundle of $\overline{\mathcal{M}}_{g,n}$ do
not play a direct role in the calculus of tautological classes as we have
introduced it, but we will need the following result in our extended theory.
Recall from standard deformation theory that the cotangent bundle to
$\overline{\mathcal{M}}_{g,n}$ is given by
$\pi_{*}(\Omega_{\pi}(\sigma_{1}+\cdots+\sigma_{n})\otimes\omega_{\pi})$,
where $\Omega_{\pi}$ denotes the relative sheaf of differentials.

Adapting Mumford’s [Mum83] well-known application of the Grothendieck-Riemann-
Roch theorem to the universal family $\pi$, Bini [Bin05, Theorem 2] gives an
explicit formula for
$\operatorname{ch}(\mathcal{T}^{*}_{\overline{\mathcal{M}}_{g,n}})$. We do not
reproduce the formula here, the following will suffice for us.

######  Theorem 2.16.

[Bin05] We have

$c(\mathcal{T}^{*}_{\overline{\mathcal{M}}_{g,n}})\in
R^{*}(\overline{\mathcal{M}}_{g,n}).$

##  3\. From moduli of Galois covers to the $\mathcal{H}$-tautological ring

###  3.1. Admissible $G$-covers

Stacks of admissible covers of nodal curves were introduced by Harris-Mumford
in [HM82] to compactify Hurwitz spaces of branched covers of smooth curves.
Later work by Mochizuki [Moc95] and Abramovich-Corti-Vistoli [ACV03] clarified
the notion of a family of admissible covers over an arbitrary base scheme. In
addition, the moduli space $\overline{\mathcal{M}}_{g^{\prime},b}(BG,d)$ of
stable maps to $BG$ constructed by Abramovich-Corti-Vistoli gives a
normalization of the Harris-Mumford space by way of twisted $G$-covers.
Incarnations of this space of also appear in [JKK05] and [BR11]; for
enumerative applications, it is most convenient to adopt the conventions of
[SvZ18], which we now recall.

######  Definition 3.1.

Let $G$ be a finite group and let $S$ be an arbitrary base scheme. An
admissible $G$-cover over $S$ is a finite morphism $f:X\to Y$ of connected
nodal curves over $S$ with a fiberwise action of $G$ on $X$ and distinct
marked sections $q_{1},\ldots,q_{b}:S\to Y$ satisfying the following
properties:

  * •

$[D,q_{1},\ldots,q_{b}]$ is a stable curve over $S$,

  * •

if $s\in S$ is a geometric point, then $f(x)\in Y_{s}$ is a node if and only
if $x\in X_{s}$ is,

  * •

$f$, with the given $G$-action on $X$, is a principal $G$-bundle away from
$q_{1},\ldots,q_{b}\in Y$ and the nodes of $Y$

  * •

in analytic local coordinates over a marked point of $D$, the map $X\to Y\to
S$ is isomorphic to

$\operatorname{Spec}(R[x^{\prime}])\to\operatorname{Spec}(R[x])\to\operatorname{Spec}(R),$

where $e\geq 1$ is an integer and $f^{*}(x)=(x^{\prime})^{e}$,

  * •

in analytic local coordinates over a node of $D$, the map $X\to Y\to S$ is
isomorphic to

$\operatorname{Spec}(R[x^{\prime},y^{\prime}]/(x^{\prime}y^{\prime}-r))\to\operatorname{Spec}(R[x,y]/(xy-r^{e}))\to\operatorname{Spec}(R),$

where $e\geq 1$ is an integer and $f^{*}(x)=(x^{\prime})^{e}$ and
$f^{*}(y)=(y^{\prime})^{e}$, and

  * •

in analytic local coordinates at a node $p\in C$ over a geometric point $s\in
S$, the $G$-action on $C_{s}$ has cyclic stabilizer
$G_{p}\cong\mathbb{Z}/e\mathbb{Z}$ and is balanced. That is, if

$\operatorname{Spec}(R[x^{\prime},y^{\prime}]/(x^{\prime}y^{\prime}-r))\to\operatorname{Spec}(R)$

is a local coordinate chart near $p$, then a generator of $g_{p}\in G_{p}$
acts by

$g_{p}\cdot(x^{\prime},y^{\prime})=(\mu x^{\prime},\mu^{-1}y^{\prime}),$

where $\mu$ is a primitive $e$-th root of unity.

Note that, if $f:X\to Y$ is an admissible $G$-cover, then $Y$ is isomorphic to
the scheme-theoretic quotient $X/G$.

By definition, the target of an admissible $G$-cover is a stable marked curve,
allowing us to define later the target morphism in Definition 3.5. In order
for the source to be another such, one needs to mark the points of the marked
fibers of an admissible $G$-cover.

######  Definition 3.2.

A pointed admissible $G$-cover consists of the data of an admissible
$G$-cover, along with, for each $i=1,2,\ldots,b$, a distinguished marked point
$p_{i1}\in f^{-1}(q_{i})$.

Given a choice of $p_{i1}$, the $G$-action on the fibers of $f$ yields a
marking of the entire fiber $f^{-1}(q_{i})$ by the left coset $G/G_{p_{i1}}$.
Indeed, if $t\in G$, then we may label the point $t\cdot p_{i1}$ by the coset
$tG_{p_{i1}}$. The resulting curve with all pre-images of the $q_{i}$ taken as
marked points is then stable, see [SvZ18, Lemma 3.4], allowing us to define
later the source morphism in Definition 3.6.

We also record the monodromy of the $G$-cover $f$ at each distinguished marked
point. The stabilizer of the $G$-action at $p_{i1}$ is a cyclic group
$G_{p_{i1}}$ of order $n_{i}$, see [SvZ18, Lemma 1]. Thus, $G_{p_{i1}}$ acts
on the tangent space $T_{p_{i1}}X\cong\mathbb{C}$ by a character of order
$n_{i}$.

######  Definition 3.3.

Let $f:X\to Y$ be a pointed admissible $G$-cover. For $i=1,2,\ldots,b$, we
define the element $h_{i}\in G_{p_{i1}}\subset G$ to be the unique element
acting on $T_{p_{i1}}X\cong\mathbb{C}$ by multiplication by the root of unity
$e^{2\pi\sqrt{-1}\cdot n_{i}}$. We call the $b$-tuple
$\xi=(h_{1},\ldots,h_{b})\in G^{b}$ the monodromy data of the pointed
admissible cover $f$.

.

The monodromy data at the distinguished marked points of an admissible
$G$-cover $f$ determines the monodromy data at all marked points. Indeed, the
stabilizer of the point $t\cdot p_{i1}$ is the conjugate $tG_{p_{i1}}t^{-1}$
with distinguished generator $th_{i}t^{-1}$.

We are now ready to define the moduli spaces of admissible $G$-covers.

######  Definition 3.4.

Let $b,g\geq 0$ integers, $G$ be a finite group, and let
$\xi=(h_{1},\ldots,h_{b})\in G^{b}$ be a tuple of group elements. We define
$\overline{\mathcal{H}}_{g,G,\xi}$ to be the moduli space parametrizing
admissible $G$-covers $f:X\to Y$, along with the data of marked points
$q_{1},\ldots,q_{b}\in Y$ and distinguished marked points $p_{i1}\in
f^{-1}(q_{i})$, such that $X$ has genus $g$ and $f$ has monodromy data $\xi$
(as measured at the $p_{i1}$).

When $G=\\{1\\}$, we recover the usual moduli spaces of stable pointed curves.
Note that the genus $g^{\prime}$ of $Y$ is determined by the Riemann-Hurwitz
formula:

$2g-2=\\#G\cdot\left[(2g^{\prime}-2)+\sum_{i=1}^{b}\frac{\operatorname{ord}_{G}(h_{i})-1}{\operatorname{ord}_{G}(h_{i})}\right].$

The space $\overline{\mathcal{H}}_{g,G,\xi}$ is equipped with two natural maps
remembering the source and target curve, respectively.

######  Definition 3.5.

The target morphism

$\delta:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{M}}_{g^{\prime},b}$

sends a pointed $G$-cover $f:X\to Y$ to the curve $[Y,\\{y_{i}\\}]$.

######  Definition 3.6.

The source morphism

$\phi:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{M}}_{g,r}$

sends a pointed $G$-cover $f:X\to Y$ to the curve $[X,\\{p_{ia}\\}]$. Here,
the index set for the set of marked points is the disjoint union of the left
coset spaces $G/\langle h_{i}\rangle$.

We point out that there is already a source of ambiguity here, namely that the
target $\overline{\mathcal{M}}_{g,r}$ parametrizes genus $g$ curves with
ordered marked points, whereas, while the marked points on a source curve
parametrized by $\overline{\mathcal{H}}_{g,G,\xi}$ are distinguished, only the
marked points of the target (and hence the $G$-orbits on the source) are
ordered. Thus, the map $\phi$ must be defined relative to an ordering on the
marked points, which we allow to be chosen independently on the ordering of
the $G$. For sake of brevity, we will suppress this choice in the notation. We
will see this ambiguity in the more general definition of restriction
morphisms later.

######  Theorem 3.7.

[SvZ18, Theorem 3.7] The map $\phi$ is representable, finite, unramified, and
a local complete intersection. The map $\delta$ is flat, proper, and quasi-
finite, but not necessarily representable (hence not necessarily finite).

Moreover, the space $\overline{\mathcal{H}}_{g,G,\xi}$ is a smooth, proper,
Deligne-Mumford stack of dimension $3g^{\prime}-3+b$.

The degree of $\delta$ is given by a Hurwitz number, which may be computed
purely group-theoretically, see [SvZ18, Theorem 3.19].

The ramification locus of $\delta$ consists of covers $f:X\to Y$ ramified to
order $e\geq 2$ over at least one node $q\in Y$. Indeed, if $t$ is a smoothing
parameter on the deformation space of $Y$ corresponding to $q$, and $s$ is a
smoothing parameter on the deformation space of $X$ (and thus for the cover
$f$, because $\phi$ is unramified) corresponding to the $G$-orbit above $q$,
then $\delta^{*}(t)=s^{e}$, up to a unit.

However, the deformations of $f$ are controlled by the twisted curve
$\mathcal{Y}$ associated to the cover $f$, see [ACV03, §3]. Here,
$\mathcal{Y}$ is a Deligne-Mumford stack with coarse space $Y$ such that the
natural map $c:\mathcal{Y}\to Y$ is an isomorphism away from the nodes and
marked points, equipped with a principal $G$-bundle
$\widetilde{f}:X\to\mathcal{Y}$ such that $c\circ\widetilde{f}=f$.

###  3.2. The universal family

Given a space of admissible $G$-covers $\overline{\mathcal{H}}_{g,G,\xi}$,
with $\xi=(h_{1},\ldots,h_{b})$, we will denote the monodromy datum
$(h_{1},\ldots,h_{b},1)$ by the somewhat abusive notation $\xi+\\{1\\}$. Then,
the space $\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}$ parametrizes the same
covers as $\overline{\mathcal{H}}_{g,G,\xi}$, along with an additional
$G$-orbit of unramified points.

######  Definition 3.8.

We define the natural forgetful morphism
$\pi:\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\to\overline{\mathcal{H}}_{g,G,\xi}$
forgetting the additional unramified orbit and contracting non-stable
components on the source and target.

The following is straightforward:

######  Lemma 3.9.

The diagram

$\textstyle{\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\scriptstyle{\pi_{G}}$$\textstyle{\overline{\mathcal{M}}_{g,r+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\overline{\mathcal{M}}_{g,r}}$

where $\phi^{\prime}$ is the composition of the source map on
$\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}$ followed by the map forgetting all
marked points in the extra $G$-orbit except the distinguished one, is
Cartesian.

Moreover, the forgetful map
$\pi_{G}:\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\to\overline{\mathcal{H}}_{g,G,\xi}$,
taken with the pullbacks of the universal sections $\sigma_{p_{ia}}$ by
$\phi$, form the universal family of (sources of) admissible $G$-covers over
$\overline{\mathcal{H}}_{g,G,\xi}$.

We also describe a more explicit construction of the universal sections to
$\pi_{G}:\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\to\overline{\mathcal{H}}_{g,G,\xi}$,
also abusively called $\sigma_{p_{ia}}$. One may also re-interpret this
construction in the language of boundary classes, explained in §3.4.

Suppose $f:X\to Y$ is a pointed admissible $G$-cover, and let $p_{ia}$ be a
marked point of $X$ mapping to $q_{i}\in Y$ with ramification index $e_{i}$.
We define the admissible $G$-cover $f^{\prime}:X^{\prime}\to Y^{\prime}$ as
follows. Let $[Y^{\prime}]=\sigma_{i}([Y])$, and let $X^{\prime}$ be the curve
obtained from $X$ by attaching rational tails to the points in the $G$-orbit
of $p_{ia}$. Then, $f^{\prime}$ is is the admissible $G$-cover agreeing with
$f$ after contraction of the new rational components, and totally ramified
over the new node of $Y^{\prime}$ and over $q_{i}$. Finally, any point in the
pre-image of $q_{b+1}$ on the same component of $p_{ia}$ may be taken to be
the distinguished marked point of the new marked fiber, giving $f^{\prime}$
the structure of a pointed admissible $G$-cover. We then have
$\sigma_{p_{ia}}([f])=[f^{\prime}]$.

###  3.3. $\psi$ and $\kappa$ classes

######  Definition 3.10.

We define the $\psi$ and $\kappa$ classes on
$\overline{\mathcal{H}}_{g,G,\xi}$ by:

$\psi_{p_{ia}}=\phi^{*}\psi_{p_{ia}}\in
A^{1}(\overline{\mathcal{H}}_{g,G,\xi})$

and

$\kappa_{i}=\phi^{*}\kappa_{i}\in A^{i}(\overline{\mathcal{H}}_{g,G,\xi}),$

respectively, where as usual
$\phi:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{M}}_{g,r}$ is the
source morphism.

Lemma 3.9 shows that one can equivalently define the $\psi$ and $\kappa$
classes in terms of the relative dualizing sheaf of
$\pi:\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\to\overline{\mathcal{H}}_{g,G,\xi}$,
as in Definitions 2.3 and 2.4.

Up to constant factors, the $\psi$ and $\kappa$ classes on
$\overline{\mathcal{H}}_{g,G,\xi}$ may also be defined by pullback by
$\delta$. Indeed, we have the following:

######  Proposition 3.11.

[SvZ18, Lemma 3.9] We have:

$\psi_{p_{ia}}=\frac{1}{\operatorname{ord}_{G}(h_{i})}\cdot\delta^{*}(\psi_{q_{i}}).$

Moreover,

$\kappa_{i}=(\\#G)\cdot\delta^{*}(\kappa_{i}).$

###  3.4. Admissible $G$-graphs and boundary strata

We review the construction of [SvZ18, §3.4-5]. In analogy with the
stratification of $\overline{\mathcal{M}}_{g,n}$ indexed by stable graphs, we
have a stratification of $\overline{\mathcal{H}}_{g,G,\xi}$ indexed by so-
called admissible $G$-graphs. The dual graph $\Gamma$ of a (source) curve
parametrized by $\overline{\mathcal{H}}_{g,G,\xi}$ naturally acquires a
$G$-action, along with monodromy data at its legs and half-edges.

######  Definition 3.12.

[SvZ18, Definition 3.14] Let $\overline{\mathcal{H}}_{g,G,\xi}$ be a moduli
space of admissible $G$-covers, with $\xi=(h_{1},\ldots,h_{b})$. An admissible
$G$-graph $(\Gamma,G)$ of genus $g$ and monodromy data $\xi$ is a stable graph
of genus $g$ with $G$-action, with the additional data of a distinguished leg
$\ell_{i1}$ in each $G$-orbit of legs and an element $h_{\ell}\in G$ for all
$\ell\in H(\Gamma)\sqcup L(\Gamma)$, satisfying the following conditions:

  * •

(monodromy data generate stabilizers) the stabilizer $G_{\ell}\subset G$ of
$\ell$ is cyclic with generator $h_{\ell}$,

  * •

(monodromy data agrees with $\xi$) $h_{\ell_{i1}}=h_{i}$ for all
$i=1,2,\ldots,b$,

  * •

(monodromy data are compatible in $G$-orbits) $h_{t\cdot\ell}=th_{\ell}t^{-1}$
for all $\ell\in H(\Gamma)\sqcup L(\Gamma)$ and $t\in G$,

  * •

(quotient by $G$ doesn’t collapse edges) if $(\ell,\ell^{\prime})$ is a pair
of half-edges comprising an edge, then $t\cdot\ell\neq\ell^{\prime}$ for all
$t\in G$, and

  * •

(balancing condition) if $(\ell,\ell^{\prime})$ is a pair of half-edges
comprising an edge, then $h_{\ell}=h_{\ell^{\prime}}^{-1}$.

As with admissible $G$-covers, the $G$-orbits of the legs of an admissible
$G$-graph are assumed to be ordered. The dual graph of the source curve of any
cover parametrized by $\overline{\mathcal{H}}_{g,G,\xi}$ naturally acquires
the structure of an admissible $G$-graph.

Schmitt-van Zelm distinguish pre-admissible $G$-graphs from admissible
$G$-graphs, where an admissible $G$-graph is required to correspond to a non-
empty boundary stratum; we do not make this distinction here.

We now define the boundary stratum associated to the admissible $G$-graph
$(\Gamma,G)$. We first choose a collection $V^{\prime}(\Gamma)\subset
V(\Gamma)$ of $G$-orbit representatives on the set of vertices of $\Gamma$.
Then, for each $v\in V^{\prime}(\Gamma)$, we choose distinguished orbit
representatives for the $G_{v}$-actions on the sets of legs and half-edges
incident at $v$, where $G_{v}\subset G$ is the stabilizer of $v$. Let
$L^{\prime}_{v},H^{\prime}_{v}$ be the resulting sets of distinguished legs
and half-edges, respectively. Let

$\xi^{\prime}_{v}=\left(h_{\ell}:\ell\in L^{\prime}_{v}\coprod
H^{\prime}_{v}\right).$

######  Definition 3.13.

With notation as above define the boundary stratum

$\overline{\mathcal{H}}_{(\Gamma,G)}=\prod_{v\in
V^{\prime}(\Gamma)}\overline{\mathcal{H}}_{g(v),G_{v},\xi^{\prime}_{v}}$

Suppose $[C_{v}]\in\overline{\mathcal{H}}_{g(v),G_{v},\xi^{\prime}_{v}}$,
where $C_{v}$ is a marked stable curve carrying a $G_{v}$ action. Let

$\widetilde{C}_{v}=\coprod_{g\in G/G_{v}}C_{v}.$

Here, $G/G_{v}$ acts naturally on the components, which are in natural
bijection with the $G$-orbit of $v\in V(\Gamma)$. The component corresponding
to $1\in G/G_{v}$, in addition, carries the given action of $G_{v}$ on
$C_{v}$. We then choose a $G$-action on $\widetilde{C}_{v}$ compatible with
both of these actions; this amounts to the choice of coset representatives of
$G_{v}$ in $G$, and will not affect the definition of boundary morphism.

Given such a $C_{v}$ for each $v\in V^{\prime}(\Gamma)$, we then glue the
curves $\widetilde{C}_{v}$ according to the graph $\Gamma$, to produce a
$G$-curve $C$ with the required genus. Moreover, we also distinguish one point
$G$-orbit of marked points of $C$ according to $\Gamma$.

######  Definition 3.14.

We define the boundary morphism

$\xi_{(\Gamma,G)}:\overline{\mathcal{H}}_{(\Gamma,G)}\to\overline{\mathcal{H}}_{g,G,\xi}.$

by the construction given above, namely:

$\xi_{(\Gamma,G)}\left(\\{[C_{v}]\\}_{v\in
V^{\prime}(\Gamma)(G)}\right)=[f:C\to C/G].$

######  Remark 3.15.

Boundary maps on spaces of admissible $G$-covers are related to the classical
boundary maps by the following commutative diagram, see [SvZ18, (16)]):

$\textstyle{\overline{\mathcal{H}}_{(\Gamma,G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{(\Gamma,G)}}$$\scriptstyle{\phi_{(\Gamma,G)}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\overline{\mathcal{M}}_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{\Gamma}}$$\textstyle{\overline{\mathcal{M}}_{g,r}}$

Here, the map $\phi_{(\Gamma,G)}$ is defined by the composition

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
44.29384pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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The image of $\xi_{(\Gamma,G)}$ is the closure of the locus parametrizing
admissible $G$-covers whose source curve has underlying admissible $G$-graph
$\Gamma$. As in the case of boundary morphisms on
$\overline{\mathcal{M}}_{g,n}$, the morphism $\xi_{(\Gamma,G)}$ is finite and
unramified of degree $\\#\operatorname{Aut}((\Gamma,G))$ onto its image,
according to the following natural notion of morphism of admissible
$G$-graphs:

######  Definition 3.16.

A morphism $f:(\Gamma,G)\to(A,G)$ between admissible $G$-graphs of genus $g$
and monodromy $\xi$ is a $G$-equivariant morphism of stable graphs
$f:\Gamma\to A$ satisfying the obvious notions of compatibility of monodromy
data at half-edges, see Definition [SvZ18, Proposition 3.12].

The morphism $f$ induces a map $\xi_{\Gamma\to
A}:\overline{\mathcal{H}}_{(\Gamma,G)}\to\overline{\mathcal{H}}_{(A,G)}$
compatible with the boundary morphisms to $\overline{\mathcal{H}}_{g,G,\xi}$.
The map $\xi_{\Gamma\to A}$ is itself a product of usual boundary morphisms
into the components of the target.

The moduli space $\overline{\mathcal{H}}_{(\Gamma,G)}$ may also be interpreted
as the moduli of admissible $G$-covers $f:X\to Y$ along with a
$(\Gamma,G)$-structure on the underlying admissible $G$-graph of $X$, see
[SvZ18, Proposition 4.6].

Again in analogy with boundary morphisms on $\overline{\mathcal{M}}_{g,n}$,
the normal bundle of $\xi_{(\Gamma,G)}$ is given by a direct sum of line
bundle contributions from the $G$-orbits of edges of $\Gamma$. More precisely:

######  Lemma 3.17.

We have

$N_{\xi_{(\Gamma,G)}}=\bigoplus_{(\ell,\ell^{\prime})\in
E^{\prime}(\Gamma)}T_{\ell}\otimes T_{\ell^{\prime}}$

and thus

$c(N_{\xi_{(\Gamma,G)}})=\prod_{(\ell,\ell^{\prime})\in
E^{\prime}(\Gamma)}(1-\psi_{\ell}-\psi_{\ell^{\prime}}).$

Here $(\ell,\ell^{\prime})$ denotes a pair of half-edges comprising an edge,
taken over any subset $E^{\prime}(\Gamma)\subset E(\Gamma)$ of orbit
representatives for the $G$-action on $E(\Gamma)$, and $T_{h},T_{h^{\prime}}$
are the corresponding tangent line bundles on
$\overline{\mathcal{H}}_{(\Gamma,G)}$.

Note that the $\psi$ classes in the formula above are independent of the
choice of $E^{\prime}(\Gamma)$, as $\psi$ classes associated to half-edges in
the same $G$-orbit are equal.

###  3.5. Restriction-corestriction maps

This is the main new feature of spaces of admissible $G$-covers not available
in the classical case.

Suppose $X\to Y$ is an admissible $G$-cover, so that $Y$ is isomorphic to the
scheme-theoretic quotient $X/G$. If $G_{1}\subset G$ is a subgroup, one can
restrict to the action of $G_{1}$, that is, remember the admissible
$G_{1}$-cover $X\to X/G_{1}$. If $G_{1}\subset G$ is normal, then we may also
corestrict to the induced action of $G/G_{1}$ on $X/G_{1}$, remembering the
admissible $G/G_{1}$-cover $X/G_{1}\to Y=X/G$.

One must also keep track of the monodromy data in applying these operations.
This is essentially done in [BR11, §2.2.2], but in our setting with
distinguished points in marked $G$-orbits, the definitions require a bit more
care.

First, consider the case of restriction to the action of an arbitrary subgroup
$G_{1}\subset G$. Upon such a restriction, the $G$-orbits of marked points on
$X$ break into $G_{1}$-orbits, and one must make a choice of distinguished
elements $\\{p_{ij}\\}$ of the $G_{1}$-orbits. This amounts, for
$i=1,2,\ldots,b$, to choosing representatives $\langle t_{ij}\rangle$ for the
orbits of the $G_{1}$-action on $G/\langle h_{i}\rangle$.

The stabilizer $G_{ij}$ of the $G$-action at the point $p_{ij}=t_{ij}\cdot
p_{i1}$ is the cyclic group generated by

$t_{ij}h_{i}t_{ij}^{-1}\in G,$

acting on $T_{p_{ij}}X\cong\mathbb{C}$ by multiplication by
$e^{2\pi\sqrt{-1}\cdot\\#G_{ij}}$. (Note that this formula is independent of
the choice of $t_{ij}$ within a left coset of $\langle h_{i}\rangle$.)

On the other hand, the stabilizer at $p_{ij}$ of the $G_{1}$-action is the
cyclic group

$G^{\prime}_{ij}=G_{ij}\cap G_{1}=\langle
t_{ij}h_{i}^{r_{ij}}t_{ij}^{-1}\rangle,$

where

$r_{ij}=\frac{\\#G_{ij}}{\\#G^{\prime}_{ij}}.$

Similarly to before, $t_{ij}h_{i}^{r_{ij}}t_{ij}^{-1}$ acts on
$T_{p_{ij}}X\cong\mathbb{C}$ by multiplication by
$e^{2\pi\sqrt{-1}\cdot\\#G^{\prime}_{p_{ij}}}$.

Once we have chosen distinguished marked points $t_{ij}\cdot p_{i}$ for the
cover $X\to X/G_{1}$, in order to obtain a point of the stack of admissible
$G_{1}$-covers, one needs in addition to order the $G_{1}$-orbits of marked
points, or equivalently, the marked points on the quotient $X/G_{1}$.

Therefore, we make the following definition:

######  Definition 3.18.

Let $G_{1}\subset G$ be a subgroup, and, given a moduli space of admissible
$G$-covers $\overline{\mathcal{H}}_{g,G,\xi}$, fix, for $i=1,2,\ldots,b$, a
choice of representatives $\\{t_{ij}\\}$ of the orbits of the $G_{1}$-action
on $G/\langle h_{i}\rangle$, and an ordering on the $t_{ij}$, ranging over all
$i,j$. The restriction morphism

$\operatorname{res}^{G}_{G_{1}}:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{H}}_{g,G_{1},\xi^{\prime}}$

is defined by

$\operatorname{res}^{G}_{G_{1}}([f:X\to X/G])=[f^{\prime}:X\to X/G_{1}],$

with the new monodromy data $\xi^{\prime}=\\{h^{\prime}_{ij}\\}$ is given by
the formula

$h^{\prime}_{ij}=t_{ij}h_{i}^{r_{ij}}t_{ij}^{-1}$

at the distinguished marked point $p_{ij}=t_{ij}\cdot p_{i1}$, where $r_{ij}$
is defined as above, and the marked points of $X/G_{1}$ (and correspondingly,
the monodromy elements comprising $\xi^{\prime}$) are ordered according to the
ordering on the $t_{ij}$.

We will refer to the ordered choice of $\\{t_{ij}\\}$ as relabeling data. As
in the definition of the source maps $\phi$, we suppress this data from the
notation, but we will need to keep track of it explicitly in many instances.

Taking $G_{1}=\\{1\\}$ recovers the source maps $\phi$. The composition of two
restriction maps is again a restriction map, relative to appropriate
relabeling data. In particular, the restriction maps
$\operatorname{res}^{G}_{G_{1}}$ are compatible with source maps from both
spaces to $\overline{\mathcal{M}}_{g,r}$, and the general restriction maps are
also representable, finite, unramified, and local complete intersections.

Taking $G_{1}=G$, different choices of relabeling data give relabeling
isomorphisms

$\operatorname{res}^{G}_{G}:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{H}}_{g,G,\xi^{\prime}}$

sending an admissible $G$-cover $f:X\to Y$ to itself, but possibly changing
the distinguished marked points in each $G$-orbit, conjugating the monodromy
accordingly, and reordering the $G$-orbits. For general restriction maps,
different relabeling data modify $\operatorname{res}^{G}_{G_{1}}$ by a
relabeling isomorphism on the target.

We also define the operation of restriction on admissible $G$-graphs.

######  Definition 3.19.

Let $G_{1}\subset G$ be a subgroup, and let $(\Gamma,G)$ be an admissible
$G$-graph associated to the space $\overline{\mathcal{H}}_{g,G,\xi}$. Fix
relabeling data associated to $G_{1}\subset G$ and $\xi$. The restriction
$((\Gamma,G),G_{1})$ of $(\Gamma,G)$ to $G_{1}$ is defined by restricting the
action of $G$ on $\Gamma$ to $G_{1}$, and relabeling the distinguished legs
exactly as in Definition §3.18, and recording the new monodromy data
accordingly.

We obtain an induced natural map of boundary strata
$\operatorname{res}_{\Gamma}:\overline{\mathcal{H}}_{(\Gamma,G)}\to\overline{\mathcal{H}}_{((\Gamma,G),G_{1}))}$
defined by the product of the maps

$\textstyle{(\operatorname{res}_{\Gamma})_{v}:\overline{\mathcal{H}}_{g_{v},G_{v},\xi_{v}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}^{G_{v}}_{G_{v}\cap
G_{1}}}$$\textstyle{\overline{\mathcal{H}}_{g_{v},G_{v}\cap
G_{1},\xi^{\prime}_{v}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{\displaystyle\prod_{i=1}^{\frac{\\#G/\\#G_{v}}{\\#G_{1}/(\\#(G_{1}\cap
G_{v}))}}\overline{\mathcal{H}}_{g_{v},G_{v}\cap G_{1},\xi^{\prime}_{v}}}$

where the maps $\operatorname{res}^{G_{v}}_{G_{v}\cap G_{1}}$ are defined
relative to relabeling data with that of $\operatorname{res}^{G}_{G_{1}}$ and
$\Delta$ is the diagonal map. In particular, we get a commutative diagram

$\textstyle{\overline{\mathcal{H}}_{(\Gamma,G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{(\Gamma,G)}}$$\scriptstyle{\operatorname{res}_{\Gamma}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}^{G}_{G_{1}}}$$\textstyle{\overline{\mathcal{H}}_{((\Gamma,G),G_{1})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{((\Gamma,G),G_{1})}}$$\textstyle{\overline{\mathcal{H}}_{g,G_{1},\xi^{\prime}}}$

In the case of corestriction, the situation is more straightforward. Suppose
that $G_{1}\subset G$ is a normal subgroup. The fibers of $f:X\to X/G$
collapse into $G/G_{1}$-orbits upon passing to $f^{\prime}:X/G_{1}\to X/G$,
and distinguished points of the fibers may be chosen to be the images of those
in $X$. The monodromy data is given simply by taking the images of the
$h_{i}\in G$ comprising $\xi$ in $G/G_{1}$. We summarize this in the following
definition.

######  Definition 3.20.

Let $G_{1}\subset G$ be a normal subgroup. The corestriction map

$\operatorname{cores}^{G}_{G/G_{1}}:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{H}}_{g_{1},G/G_{1},\xi^{\prime}}$

is defined by

$\operatorname{cores}^{G}_{G_{1}}([f:X\to X/G])=[f^{\prime}:X/G_{1}\to X/G].$

Here, $g_{1}$ is the genus of $X/G_{1}$ (which may be computed by the Riemann-
Hurwitz formula), the distinguished marked points of the fibers of
$f^{\prime}$ are given by the images of those in $f$, and
$\xi^{\prime}=(h^{\prime}_{1},\ldots,h^{\prime}_{b})$, where $h^{\prime}_{i}$
is the image of $h_{i}$ under the quotient map $G\to G/G_{1}$.

When $G_{1}=G$, we recover the target map $\delta$. The composition of two
corestriction maps (induced by a composition of group surjections) is again a
corestriction map. In particular, the corestriction maps
$\operatorname{cores}^{G}_{G_{1}}$ are compatible with the target maps from
both spaces to $\overline{\mathcal{M}}_{g^{\prime},b}$, and the general
corestriction maps are also flat, proper, and quasi-finite. Their degrees may
be computed in terms of those of the two target maps $\delta$, which are in
turn given by Hurwitz numbers.

As with restriction, we also define the notion of the corestriction of an
admissible $G$-graph $(\Gamma,G)$.

######  Definition 3.21.

Let $G_{1}\subset G$ be a normal subgroup and let $(\Gamma,G)$ be an
admissible $G$-graph. We define the corestriction $(\Gamma/G_{1},X/G_{1})$ of
$(\Gamma,G)$ to $G_{1}$ by the quotient graph $\Gamma/G_{1}$ with the induced
$X/G_{1}$-action. The monodromy elements on the corestricted graph are the
images of those from $\Gamma$ under the quotient quotient map.

We obtain the natural map of boundary strata
$\operatorname{cores}_{\Gamma}:\overline{\mathcal{H}}_{(\Gamma,G)}\to\overline{\mathcal{H}}_{(\Gamma/G_{1},G/G_{1})}$
defined by the product of

$\operatorname{cores}_{G_{v}\cap
G_{1}}^{G_{v}}\overline{\mathcal{H}}_{g_{v},G_{v},\xi_{v}}\to\overline{\mathcal{H}}_{g_{1,v},G_{v}/(G_{v}\cap
G_{1}),\xi^{\prime}_{v}}$

over $G$-orbit representatives $v\in V^{\prime}(\Gamma)$, so that we have the
commutative diagram

$\textstyle{\overline{\mathcal{H}}_{(\Gamma,G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{(\Gamma,G)}}$$\scriptstyle{\operatorname{cores}_{\Gamma}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{cores}^{G}_{G/G_{1}}}$$\textstyle{\overline{\mathcal{H}}_{(\Gamma/G_{1},G/G_{1})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{(\Gamma/G_{1},G/G_{1})}}$$\textstyle{\overline{\mathcal{H}}_{g_{1},G/G_{1},\xi^{\prime}}}$

We may combine restriction and corestriction morphisms in the following
general setting.

######  Definition 3.22.

Suppose that we have a nested pair of subgroups $G_{1}\subset G_{2}\subset G$,
with $G_{1}$ normal in $G_{2}$. We define the restriction-corestriction
morphism

$q_{G_{1},G_{2},G}:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{H}}_{g_{1},G_{2}/G_{1},\xi^{\prime}},$

by

$q_{G_{1},G_{2},G}=\operatorname{cores}^{G_{2}}_{G_{2}/G_{1}}\circ\operatorname{res}^{G}_{G_{2}},$

where the restriction map depends on relabeling data.

On the level of maps of curves, we have

$q_{G_{1},G_{2},G}([X\to X/G])=[X/G_{1}\to X/G_{2}].$

The composition of two restriction-corestriction morphisms is a single such.

An important example for us will be the case $G=S_{d}$, $G_{1}=G_{2}=S_{d-1}$,
which recovers the case of loci of Harris-Mumford admissible covers on
$\overline{\mathcal{M}}_{g,n}$, see §6.

###  3.6. Diagonals and non-tautological cycles from
$\overline{\mathcal{H}}_{g,G,\xi}$

Our goal is to construct a framework including classical tautological classes
that also incorporates the cycles obtained from source maps $\phi$, and from
restriction-corestriction morphisms more generally. As we explain here,
diagonal morphisms also play an essential role in the theory. In fact, we have
already seen diagonals appear naturally in Remark 3.15 and Definition 3.19.

We now recall the example of Graber-Pandharipande [GP03] of a non-tautological
cycle arising from admissible $G$-covers, further illustrating the role of
diagonal maps.

Let $h\geq 1$ be an integer and let
$\overline{\mathcal{H}}=\overline{\mathcal{H}}_{2h,\mathbb{Z}/2\mathbb{Z},(1)^{2}}$
denote the moduli space of admissible $\mathbb{Z}/2\mathbb{Z}$-covers $f:X\to
Y$, where $X,Y$ have genus $2h,h$, respectively, and $f$ is ramified over two
marked points. We have that $\overline{\mathcal{H}}$ is a smooth stack of
dimension $3h-1$. We have, as usual, a source morphism

$\phi:\overline{\mathcal{H}}\to\overline{\mathcal{M}}_{2h,2}.$

We will abusively refer to the map

$\phi:\overline{\mathcal{H}}\to\overline{\mathcal{M}}_{2h}$

obtained by forgetting the two ramification points by the same name.

Let $\Gamma$ be the stable graph of genus $2h$ consisting of two genus $h$
vertices connected by a single edge, with associated boundary map

$\xi_{\Gamma}:\overline{\mathcal{M}}_{h,1}\times\overline{\mathcal{M}}_{h,1}\to\overline{\mathcal{M}}_{2h}.$

######  Lemma 3.23.

[GP03, Lemma 1] The class $\xi_{\Gamma}^{*}(\phi_{*}(1))$ is a non-zero
multiple of the class of the diagonal

$\Delta:\overline{\mathcal{M}}_{h.1}\to\overline{\mathcal{M}}_{h,1}\times\overline{\mathcal{M}}_{h,1}.$

More explicitly, the only admissible $\mathbb{Z}/2$-cover $f:X\to Y$ whose
source lies in the boundary stratum $\overline{\mathcal{M}}_{\Gamma}$ has
$Y=C\cup\mathbb{P}^{1}$, where $C$ is a genus $h$ curve, and
$X=C\cup\mathbb{P}^{1}\cup C$, where the two copies of $C$ in $X$ map
isomorphically to that in $Y$, and the rational component of $X$ has degree 2
over that of $Y$.

The existence of odd singular cohomology in $\overline{\mathcal{M}}_{h,1}$ for
sufficiently large $h$, as shown by Pikaart [Pik95], implies that
$\phi_{*}(1)\in H^{*}(\overline{\mathcal{M}}_{2h})$ cannot have a tautological
Künneth composition upon pullback by $\xi_{\Gamma}$. One may then conclude:

######  Proposition 3.24.

[GP03, §2] For all sufficiently large $h$, we have $\phi_{*}(1)\notin
RH^{*}(\overline{\mathcal{M}}_{2h})$.

Similar constructions of non-tautological cycles, including those which
already fail to be tautological in Chow, use the existence of odd middle
cohomology in $\overline{\mathcal{M}}_{1,11}$, see [GP03, Theorem 2] and
[vZ18].

In our framework, the computation of Graber-Pandharipande shows that, in order
for $\phi_{*}(1)$ to be included in a $\mathbb{Q}$-subalgebra of
$\overline{\mathcal{M}}_{2h}$ containing all tautological classes, and more
generally, in a system of such $\mathbb{Q}$-subalgebras for all spaces of
admissible $G$-covers, this system of $\mathbb{Q}$-subalgebras should also
include classes coming from general diagonal morphisms

$\Delta:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{H}}_{g,G,\xi}\times\overline{\mathcal{H}}_{g,G,\xi}.$

The target of this morphism is a product of isomorphic admissible $G$-cover
spaces, and it will therefore be convenient for us to work with arbitrary
products of such spaces.

###  3.7. The $\mathcal{H}$-tautological ring

We are now ready to make the main definition of this paper.

######  Definition 3.25.

The $\mathcal{H}$-tautological ring is the smallest system of
$\mathbb{Q}$-subalgebras

$R^{*}_{\mathcal{H}}\left(\prod_{j=1}^{m}\overline{\mathcal{H}}_{g_{j},G_{j},\xi_{j}}\right)\subset
A^{*}\left(\prod_{j=1}^{m}\overline{\mathcal{H}}_{g_{j},G_{j},\xi_{j}}\right),$

ranging over all finite products of moduli spaces of admissible $G$-covers,
that contains all $\psi$ and $\kappa$ classes and is closed under pushfoward
by all maps of the form

$\gamma_{1}\times\cdots\times\gamma_{m}:\mathcal{H}_{1}\times\cdots\times\mathcal{H}_{m}\to\mathcal{H}^{\prime}_{1}\times\cdots\times\mathcal{H}^{\prime}_{m}$

where each $\mathcal{H}_{i},\mathcal{H}^{\prime}_{i}$ is itself a product of
moduli spaces of admissible $G$-covers and each map
$\gamma_{i}:\mathcal{H}_{i}\to\mathcal{H}^{\prime}_{i}$ is either a boundary,
forgetful, restriction-corestriction, or diagonal map, as defined earlier.

##  4\. Bivariant pullbacks of $\mathcal{H}$-tautological classes

We now build up the theory of intersections in the $\mathcal{H}$-tautological
ring by explaining the algorithmic computation of bivariant pullbacks of the
basic $\mathcal{H}$-tautological classes.

In all of the following results (except those of §4.4), we will consider the
intersection of two tautological maps
$\gamma_{j}:\mathcal{H}_{j}\to\mathcal{H}$, and give a commutative (not always
Cartesian) diagram

$\textstyle{\mathcal{H}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma^{\prime}_{1}}$$\scriptstyle{\gamma^{\prime}_{2}}$$\textstyle{\mathcal{H}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{1}}$$\textstyle{\mathcal{H}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{2}}$$\textstyle{\mathcal{H}}$

such that the maps $\gamma^{\prime}_{j}$ are compositions of tautological
morphisms and the classes $(\gamma_{2})^{*}(\gamma_{1})_{*}(1)$ and
$(\gamma_{1})^{*}(\gamma_{2})_{*}(1)$ may both be expressed as the pushforward
of an $\mathcal{H}$-tautological class on $\mathcal{H}_{0}$.

###  4.1. Pullbacks by forgetful maps

Here, we record diagrams expressing the pullbacks of certain
$\mathcal{H}$-tautological (bivariant) classes by a forgetful morphism.

####  4.1.1. Forgetful maps

The following is a straightforward analog of Lemma 2.2.

######  Proposition 4.1.

The commutative diagram

$\textstyle{\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}+\\{1^{\prime}\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{1^{\prime}}}$$\scriptstyle{\pi_{1}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi+\\{1^{\prime}\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}}$

is Cartesian up to birational equivalence. In particular, the bivariant
pullback of a forgetful map by itself is again a forgetful map.

####  4.1.2. Boundary maps

Suppose $(A,G)$ is an admissible $G$-graph indexing a boundary stratum of
$\overline{\mathcal{H}}_{g,G,\xi}$ and let $v\in V(A)$ be a vertex. Let
$A+\ell_{v}$ be the admissible $G$-graph obtained from $A$ by adding a
distinguished marked leg with trivial monodromy to $v$, as well as a
corresponding $G$-orbit of legs to the vertices in the $G$-orbit of $v$. This
operation corresponds to adding a distinguished marked orbit to a stable
$G$-curve with underlying admissible $G$-graph $A$.

######  Proposition 4.2.

With notation as above, the commutative diagram

$\textstyle{\displaystyle\coprod_{v\in
V(A)}\overline{\mathcal{H}}_{(A+\ell_{v},G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{(A+\ell_{v},G)}}$$\scriptstyle{\pi_{A}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\overline{\mathcal{H}}_{(A,G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{(A,G)}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}}$

is Cartesian up to birational equivalence, and expresses the bivariant
pullback of a forgetful map by a boundary map, and vice versa.

###### Proof.

The data of an extra orbit of unramified points with a distinguished marked
point on a $G$-curve $X$ is equivalent to the data of the choice of a single
unramified point. When $X$ has an $A$-structure, this amounts to the choice of
a single unramified point on one of the components of $X$, which corresponds
to the addition of a leg at a unique vertex of A if the dual graph of $X$ is
in fact isomorphic to $A$. Thus, we get an inverse to
$(\xi_{(A+\ell_{v},G)},\pi_{A})$ defined on the dense open locus
$\mathcal{H}_{(A,G)}\subset\overline{\mathcal{H}}_{(A,G)}$ of $G$-curves with
dual graph isomorphic to $A$. ∎

####  4.1.3. Diagonal maps

######  Proposition 4.3.

We have the following Cartesian diagram, expressing the bivariant pullback of
a diagonal map by a forgetful map, and vice versa:

$\textstyle{\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\operatorname{id},\pi)}$$\scriptstyle{\pi}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\times\overline{\mathcal{H}}_{g,G,\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi\times\operatorname{id}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}\times\overline{\mathcal{H}}_{g,G,\xi}}$

###### Proof.

Clear. Note in addition that

$(\operatorname{id},\pi)=(\operatorname{id}\times\pi)\circ\Delta,$

so in particular the top arrow is the composition of tautological morphisms. ∎

####  4.1.4. Restriction-corestriction maps

######  Proposition 4.4.

We have the following Cartesian diagram, expressing the bivariant pullback of
a restriction map by a forgetful map, and vice versa:

$\textstyle{\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{G}}$$\scriptstyle{\widetilde{\operatorname{res}}^{G}_{G_{1}}}$$\textstyle{\overline{\mathcal{H}}_{g,G_{1},\xi^{\prime}+\\{1\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{G_{1}}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}^{G}_{G_{1}}}$$\textstyle{\overline{\mathcal{H}}_{g,G_{1},\xi^{\prime}}}$

The map $\widetilde{\operatorname{res}}^{G}_{G_{1}}$ is defined by restricting
to $G_{1}$ (with relabeling data compatible with that of the bottom arrow),
then forgetting all marked points in the new $G$-orbit except those in the
$G_{1}$-orbit of the distinguished one.

###### Proof.

The inverse functor from the fiber product to
$\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}$ may be described as follows. Given
a $G$-cover of $\overline{\mathcal{H}}_{g,G,\xi}$ and an additional
distinguished marked point as dictated by
$\overline{\mathcal{H}}_{g,G_{1},\xi^{\prime}+\\{1\\}}$, one takes the same
distinguished marked point on the $G$-cover to get a point of
$\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}$. ∎

We now consider the fiber product of a corestriction map with a forgetful map.
Here, we find a simple example of a phenomenon that will occur in several
other intersections later: we do not express the fiber product directly in
terms of our spaces of admissible $G$-covers, but only up to a finite cover.

######  Lemma 4.5.

We have a commutative diagram

$\textstyle{\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{G}}$$\scriptstyle{\widetilde{\operatorname{cores}}_{G/G_{1}}^{G}}$$\textstyle{\overline{\mathcal{H}}_{g_{1},G/G_{1},\xi^{\prime}+\\{1\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{G/G_{1}}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{cores}_{G/G_{1}}^{G}}$$\textstyle{\overline{\mathcal{H}}_{g^{\prime},G/G_{1},\xi^{\prime}}}$

for which the induced map

$(\pi_{G},\operatorname{cores}^{G}_{G/G_{1}}):\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\to\overline{\mathcal{H}}_{g,G,\xi}\times_{\overline{\mathcal{H}}_{g^{\prime},G/G_{1},\xi^{\prime}}}\overline{\mathcal{H}}_{g_{1},G/G_{1},\xi^{\prime}+\\{1\\}}$

is proper and quasi-finite of degree $\\#G_{1}$.

###### Proof.

The commutativity is clear. All spaces in question are proper, hence so is the
induced map to the fiber product.

A point of
$\overline{\mathcal{H}}_{g,G,\xi}\times_{\overline{\mathcal{H}}_{g^{\prime},G/G_{1},\xi^{\prime}}}\overline{\mathcal{H}}_{g_{1},G/G_{1},\xi^{\prime}+\\{1\\}}$
consists of a pointed $G$-cover $f:X\to X/G$ along with a distinguished
unramified point of the intermediate quotient $x\in X/G_{1}$. Then, the
$\\#G_{1}$ pre-images of $x$ in $X$ correspond to the $\\#G_{1}$ points of
$\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}$ lying over our target point. ∎

It follows immediately that:

######  Proposition 4.6.

We have:

$(\operatorname{cores}_{G/G_{1}}^{G})^{*}([\pi_{G/G_{1}}])=\frac{1}{\\#G_{1}}\cdot[\pi_{G}]$

and

$\pi_{G/G_{1}}^{*}((\operatorname{cores}_{G/G_{1}}^{G})_{*}(1))=\frac{1}{\\#G_{1}}\cdot(\operatorname{cores}_{G/G_{1}}^{G})_{*}(1)$

where we use brackets to denote bivariant classes.

######  Remark 4.7.

Because the corestriction maps are flat, proper, and quasi-finite, the classes

$(\operatorname{cores}_{G/G_{1}}^{G})_{*}(1),(\operatorname{cores}_{G/G_{1}}^{G})_{*}(1)$

are equal to multiples of the fundamental class, so the second formula can be
viewed simply as expressing an equality of numbers. However, in order later to
be able to carry out intersections of compositions of basic tautological
morphisms, we need to express the intersection of the morphisms
$\mathcal{H}_{1}\to\mathcal{H}$ and $\mathcal{H}_{2}\to\mathcal{H}$ in terms
of a class coming from a space $\mathcal{H}_{0}$ admitting tautological maps
to both $\mathcal{H}_{j}$, as we have here; hence, it is not enough simply to
remember the correct multiple of the fundamental class. This issue will arise
in all of our computations of this section involving corestriction maps.

###  4.2. Intersections of two boundary strata

The intersection of boundary classes on spaces of admissible $G$-covers is
entirely analogous to the classical situation, see §2.4.

######  Lemma 4.8.

Let $(A,G),(B,G)$ be admissible $G$-graphs indexing boundary strata on
$\overline{\mathcal{H}}_{g,G,\xi}$. We have a Cartesian diagram

$\textstyle{\displaystyle\coprod\overline{\mathcal{H}}_{(\Gamma,G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{\Gamma\to
A}}$$\scriptstyle{\xi_{\Gamma\to
B}}$$\textstyle{\overline{\mathcal{H}}_{(B,G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{(B,G)}}$$\textstyle{\overline{\mathcal{H}}_{(A,G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{(A,G)}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}}$

where the disjoint union is over admissible $G$-graphs $(\Gamma,G)$ with maps
$(\Gamma,G)\to(A,G),(B,G)$ with the genericity condition that the induced map
on edges

$E(A)\sqcup E(B)\to E(\Gamma)$

is surjective.

###### Proof.

The commutativity is immediate. We describe an inverse functor

$\overline{\mathcal{H}}_{(A,G)}\times_{\overline{\mathcal{H}}_{g,G,\xi}}\overline{\mathcal{H}}_{(B,G)}\to\coprod\overline{\mathcal{H}}_{(\Gamma,G)}.$

The data of a point on the source is equivalent to that of a pointed $G$-cover
$f:X\to Y$ with the required monodromy data along with maps
$f_{A}:(\Gamma_{f},G)\to(A,G)$ and $f_{B}:(\Gamma_{f},G)\to(B,G)$, where
$(\Gamma_{f},G)$ is the canonical admissible $G$-graph structure on the dual
graph of $X$.

We define $(\Gamma,G)$ to be the admissible $G$-graph obtained from
$\Gamma_{f}$ contracting all edges not coming from $E(A)$ or $E(B)$, and
modifying the genus function accordingly. Then, the maps $f_{A},f_{B}$ factor
respectively as

$\displaystyle(\Gamma_{f},G)$ $\displaystyle\to(\Gamma,G)\to(A,G),$
$\displaystyle(\Gamma_{f},G)$ $\displaystyle\to(\Gamma,G)\to(B,G)$

where the natural map $(\Gamma_{f},G)\to(\Gamma,G)$ defines, along with the
data of $f:X\to Y$, a point of $\overline{\mathcal{H}}_{(\Gamma,G)}$. This
furnishes the desired inverse. ∎

######  Lemma 4.9.

We have the following formula for the excess bundle on
$\overline{\mathcal{H}}_{(H,G)}$:

$\xi^{*}_{\Gamma\to
A}N_{\overline{\mathcal{H}}_{(A,G)}/\overline{\mathcal{H}}_{g,G,\xi}}/N_{\overline{\mathcal{H}}_{(\Gamma,G)}/\overline{\mathcal{H}}_{(B,G)}}\cong\bigoplus_{(\ell,\ell^{\prime})\in(E(A)\cap
E(B))^{\prime}}T_{\ell}\otimes T_{\ell^{\prime}},$

where $(\ell,\ell^{\prime})$ denotes the pair of half-edges comprising an edge
and $(E(A)\cap E(B))^{\prime}$ is any sets of $G$-orbit representatives in
$E(A)\cap E(B)$

###### Proof.

This follows from Lemma 3.17. ∎

Applying the excess intersection formula as in the classical case yields:

######  Proposition 4.10.

Let $(A,G),(B,G)$ be stable graphs associated to
$\overline{\mathcal{H}}_{g,G,\xi}$. We have:

$\xi^{*}_{(A,G)}(\xi_{(B,G)*}(1))=\sum_{(\Gamma,G)}(\xi_{(\Gamma,G)\to(A,G)})_{*}\left(\prod_{(\ell,\ell^{\prime})\in(E(A)\cap
E(B))^{\prime}}(-\psi_{h}-\psi_{h^{\prime}})\right),$

where the sum is over $(\Gamma,G)$ as above.

###  4.3. Intersections of restriction-corestriction maps with boundary strata

####  4.3.1. Restriction maps

We consider the intersection of a restriction map
$\operatorname{res}^{G}_{G_{1}}:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{H}}_{g,G_{1},\xi^{\prime}}$
with a boundary class on $\overline{\mathcal{H}}_{g,G_{1},\xi^{\prime}}$. The
case of $\operatorname{res}^{G}_{\\{1\\}}=\phi$ is the main technical result
of Schmitt-van Zelm, [SvZ18, Theorem 4.9]; the situation here is similar.

######  Lemma 4.11.

Suppose that $(A,G_{1})$ is an admissible $G_{1}$-graph. Then, we have a
Cartesian diagram

$\textstyle{\displaystyle\coprod\overline{\mathcal{H}}_{(\Gamma,G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}_{\alpha}}$$\scriptstyle{\xi_{(\Gamma,G)}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}_{G_{1}}^{G}}$$\textstyle{\overline{\mathcal{H}}_{(A,G_{1})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{(A,G_{1})}}$$\textstyle{\overline{\mathcal{H}}_{g,G_{1},\xi^{\prime}}}$

Here, the coproduct is over admissible $G$-graphs $(\Gamma,G)$ equipped with a
map of admissible $G_{1}$-graphs $\alpha:((\Gamma,G),G_{1})\to(A,G_{1})$,
satisfying the genericity condition that the map

$\alpha_{E}:E(A)\to E(\Gamma)$

is surjective.

The map $\operatorname{res}_{\alpha}$ is defined by the composition

$\textstyle{\overline{\mathcal{H}}_{(\Gamma,G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}_{\Gamma}}$$\textstyle{\overline{\mathcal{H}}_{(\Gamma,G),G_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{\Gamma\to
A}}$$\textstyle{\overline{\mathcal{H}}_{(A,G_{1})}.}$

where $((\Gamma,G),G_{1})$ is the admissible $G_{1}$-graph obtained by
restricting the $G$-action to $G_{1}$, as in Definition 3.19, and the
relabeling data defining the restriction map is the same as that of the given
map $\operatorname{res}^{G}_{G_{1}}$.

###### Proof.

We describe the inverse functor

$\overline{\mathcal{H}}_{(A,G_{1})}\times_{\overline{\mathcal{H}}_{g,G_{1},\xi^{\prime}}}\overline{\mathcal{H}}_{g,G,\xi}\to\coprod\overline{\mathcal{H}}_{(\Gamma,G)}.$

The construction is exactly analogous to that in the proof of [SvZ18,
Proposition 4.3]. For brevity’s sake, we will describe the functor on the
level of points without making explicit reference to families of objects, but
the entire construction works in this setting.

On the source, we are given a $G$-cover $f:X\to X/G$ along with an
$(A,G_{1})$-structure on $X$, with appropriate marked points distinguished.

We construct an admissible $G$-graph $(\Gamma,G)$ as follows. We define
$E(\Gamma)$ to be the set of nodes of $X$ in the $G$-orbit of some node
corresponding to an edge of $A$. Similarly, $H(\Gamma)$ is defined to be the
set of pre-images of such nodes in the normalization $\widetilde{X}$ of $X$.
The vertex set $V(\Gamma)$ then corresponds to the connected components of
$X-E(\Gamma)$. The set of legs $L(A)$ naturally has a $G$-action coming from
$X$.

It is now straightforward to check that $V(\Gamma),H(\Gamma),L(\Gamma)$ taken
together with the data $g,\iota,a,\zeta$ induced in the obvious way defines
the desired graph $(\Gamma,G)$ with a map
$\alpha:((\Gamma,G),G_{1})\to(A,G_{1})$ satisfying the needed genericity
condition. Moreover, $f:X\to X/G$ with the $(\Gamma,G)$-structure on its dual
graph defines a point of $\overline{\mathcal{H}}_{(\Gamma,G)}$, furnishing the
needed inverse functor. ∎

The excess bundle on each $\overline{\mathcal{H}}_{(\Gamma,G)}$ also admits a
similar description as in [SvZ18, §4.2].

######  Lemma 4.12.

The excess bundle on $\overline{\mathcal{H}}_{(\Gamma,G)}$ is

$\operatorname{res}_{\alpha}^{*}N_{\overline{\mathcal{H}}_{(A,G_{1})}/\overline{\mathcal{H}}_{g,G_{1},\xi^{\prime}}}/N_{\overline{\mathcal{H}}_{(\Gamma,G)}/\overline{\mathcal{H}}_{g,G,\xi}}\cong\bigoplus_{(\ell,\ell^{\prime})}T_{\ell}\otimes
T_{\ell^{\prime}}$

where the sum is over edges $(\ell,\ell^{\prime})$ ranging over a set of
$G_{1}$-orbit representatives in the image of $\alpha_{E}:E(A)\to E(\Gamma)$,
excluding a choice of $G$-orbit representatives among the chosen edges.

###### Proof.

The normal bundle of $\overline{\mathcal{H}}_{(A,G_{1})}$ in
$\overline{\mathcal{H}}_{g,G_{1},\xi^{\prime}}$ is the direct sum of line
bundle contributions from the $G_{1}$-orbits of edges of $A$, and the normal
bundle of $\overline{\mathcal{H}}_{(\Gamma,G)}$ in
$\overline{\mathcal{H}}_{g,G,\xi}$ is the direct sum of line bundle
contributions of $G$-orbits of edges of $\Gamma$. Removing a choice of the
latter set from the former yields the desired formula. ∎

Because the maps $\xi_{(A,G_{1})}$ and $\operatorname{res}_{G_{1}}^{G}$ are
both unramified, we get the same excess bundle upon pushing forward and
pulling back in either order. We conclude the following.

######  Proposition 4.13.

We have:

$\xi^{*}_{(A,G_{1})}((\operatorname{res}_{G_{1}}^{G})_{*}(1))=\sum_{(\Gamma,G),\alpha}(\operatorname{res}_{\alpha})_{*}\left(\prod_{(\ell,\ell^{\prime})}(-\psi_{\ell}-\psi_{\ell^{\prime}})\right),$

where the product is over a set of edges $(\ell,\ell^{\prime})$ of $\Gamma$
chosen as described above, as well as

$(\operatorname{res}_{G_{1}}^{G})^{*}((\xi_{(A,G_{2})})_{*}(1))=\sum_{(\Gamma,G),\alpha}(\xi_{(\Gamma,G)})_{*}\left(\prod_{(\ell,\ell^{\prime})}(-\psi_{\ell}-\psi_{\ell^{\prime}})\right).$

Taking $G_{1}=1$ recovers [SvZ18, Theorem 4.9]. The $\psi$ classes are
invariant under the $G$-action on the marked points of
$\overline{\mathcal{H}}_{g,G,\xi}$, so the formulas above are independent of
the choices of orbit representatives. In fact, we may write

$\prod_{(\ell,\ell^{\prime})}(-\psi_{\ell}-\psi_{\ell^{\prime}})=\prod_{(\ell,\ell^{\prime})\in
E^{\prime}(\Gamma)}(-\psi_{\ell}-\psi_{\ell^{\prime}})^{k_{(\ell,\ell^{\prime})}-1},$

where the product on right hand side is over any choice of $G$-orbit
representatives of $E(\Gamma)$, and

$k_{(\ell,\ell^{\prime})}=\frac{\\#[\\{G\cdot{(\ell,\ell^{\prime})}\\}\cap\operatorname{im}(\alpha_{E})]}{\\#G_{1}}.$

The surjectivity of $\alpha_{E}$ onto $G$-orbits guarantees that
$k_{(\ell,\ell^{\prime})}\geq 1$.

####  4.3.2. Corestriction maps

We now consider the case of corestriction maps
$\operatorname{cores}_{G/G_{1}}^{G}$, where $G_{1}\subset G$ is a normal
subgroup. Suppose that $(A,G/G_{1})$ is an admissible $G/G_{1}$-graph.
Consider the set of admissible $G$-graphs $(\Gamma,G)$ together with an
isomorphism of admissible $G/G_{1}$-graphs
$\alpha:(\Gamma/G_{1},G/G_{1})\to(A,G/G_{1})$. For such an $\alpha$, let
$\operatorname{Aut}(\alpha)$ be the group of automorphisms of $(\Gamma,G)$
inducing the identity automorphism on $\Gamma/G_{1}\cong A$, that is, the
kernel of map
$\operatorname{Aut}((\Gamma,G))\to\operatorname{Aut}((A,G/G_{1}))$.

######  Lemma 4.14.

We have a commutative diagram

$\textstyle{\displaystyle\coprod\overline{\mathcal{H}}_{(\Gamma,G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{cores}_{\alpha}}$$\scriptstyle{\xi_{(\Gamma,G)}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{cores}_{G/G_{1}}^{G}}$$\textstyle{\overline{\mathcal{H}}_{(A,G/G_{1})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{A}}$$\textstyle{\overline{\mathcal{H}}_{g_{1},G/G_{1},\xi^{\prime}}}$

where the disjoint union is over admissible $G$-graphs $(\Gamma,G)$ together
with isomorphisms $\alpha:(\Gamma/G_{1},G/G_{1})\to(A,G/G_{1})$ as above.

The induced map of groupoids (on the level of closed points)

$\displaystyle\coprod\overline{\mathcal{H}}_{(\Gamma,G)}(\operatorname{Spec}(\mathbb{C}))\to\overline{\mathcal{H}}_{g,G,\xi}\times_{\overline{\mathcal{H}}_{g_{1},G/G_{1},\xi^{\prime}}}\overline{\mathcal{H}}_{(A,G/G_{1})}(\operatorname{Spec}(\mathbb{C}))$

is surjective, and the fiber over any point of the target consists of
$\\#\operatorname{Aut}(\alpha)$ points of a unique component
$\overline{\mathcal{H}}_{(\Gamma,G)}$ coming from a pair
$((\Gamma,G),\alpha)$.

Recall from Definition 3.21 that $(\Gamma/G_{1},G/G_{1})$ denotes the
corestriction of $\Gamma$ to $G/G_{1}$. The map
$\operatorname{cores}_{\alpha}$ is induced as the composition

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
12.09595pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\overline{\mathcal{H}}_{(\Gamma,G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
9.85493pt\raise 5.19028pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.82361pt\hbox{$\scriptstyle{\operatorname{cores}_{\Gamma}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 36.09595pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
36.09595pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\overline{\mathcal{H}}_{(\Gamma/G_{1},G/G_{1})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
82.4638pt\raise 6.145pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.71611pt\hbox{$\scriptstyle{\xi_{\Gamma\to
A}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 102.20493pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
102.20493pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\overline{\mathcal{H}}_{(A,G/G_{1})}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Above, we refer to $\overline{\mathcal{H}}_{(\Gamma,G)}$ as a “component”
without claiming its connectedness, but rather to specify a single boundary
stratum equipped with an isomorphism of $G/G_{1}$-graphs $\alpha$.

We remark that the map
$\operatorname{Aut}((\Gamma,G))\to\operatorname{Aut}((A,G/G_{1}))$ induced by
$\alpha$ is not always surjective. For example, suppose
$G_{1}=G=\mathbb{Z}/6\mathbb{Z}$, the graph $A$ has a single edge connecting
two vertices of genus 1, and $(\Gamma,G)$ is the bipartite graph $K_{2,3}$.
The $G$-action on $\Gamma$ is such that $E(\Gamma)$ forms a $G$-torsor, and a
vertex of valence $r$ is stabilized by $r\mathbb{Z}/6\mathbb{Z}$. Then,
$\mathcal{H}_{(\Gamma,G)}$ is a boundary divisor of
$\overline{\mathcal{H}}_{7,\mathbb{Z}/6\mathbb{Z},\emptyset}$ dominating the
boundary divisor $\overline{\mathcal{M}}_{A}$ of $\overline{\mathcal{M}}_{2}$
via the target map. Now, we have
$\operatorname{Aut}((A,G/G_{1}))\cong\mathbb{Z}/2\mathbb{Z}$, but the map
$\operatorname{Aut}((\Gamma,G))\to\operatorname{Aut}((A,G/G_{1}))$ induced by
$\alpha$ is trivial.

###### Proof.

The commutativity of the diagram is clear. Consider a fiber of the map

$\displaystyle\coprod\overline{\mathcal{H}}_{(\Gamma,G)}(\operatorname{Spec}(\mathbb{C}))\to\overline{\mathcal{H}}_{g,G,\xi}\times_{\overline{\mathcal{H}}_{g_{1},G/G_{1},\xi^{\prime}}}\overline{\mathcal{H}}_{(A,G/G_{1})}(\operatorname{Spec}(\mathbb{C})).$

A point of the target consists of a stable $G$-curve $X$ along with an
$A$-marking on its partial quotient $X/G_{1}$. One then constructs the graph
$\Gamma$ by taking $E(\Gamma)$ to be the set of pre-images of the nodes
corresponding to $E(A)$ under the map $X\to X/G_{1}$, with half-edges given by
the pre-images of the individual branches at such nodes. The vertex set
$V(\Gamma)$ is then the set of connected components of $X-E(\Gamma)$, and the
legs of $\Gamma$ corresponding to the marked points on $X$ are attached
accordingly. One easily checks that $\Gamma$ has the natural structure of an
admissible $G$-graph along with an isomorphism
$\alpha:(\Gamma/G_{1},G/G_{1})\to(A,G/G_{1})$.

On the other hand, we have multiple choices of a point of
$\overline{\mathcal{H}}_{(\Gamma,G)}(\operatorname{Spec}(\mathbb{C}))$ lying
over our chosen point of
$\overline{\mathcal{H}}_{g,G,\xi}\times_{\overline{\mathcal{H}}_{g_{1},G/G_{1},\xi^{\prime}}}\overline{\mathcal{H}}_{(A,G/G_{1})}(\operatorname{Spec}(\mathbb{C}))$,
owing to different choices of distinguished marked points in the half-edge
orbits. Such choices naturally form a torsor under the action of
$\operatorname{Aut}(\alpha)$, yielding the last claim. ∎

However, we note that the above local inverse construction cannot work in
families as it does, for instance, in Lemmas 4.8 and 4.11, where the morphisms
being intersected are unramified.

Indeed, suppose for example that $A$ contains exactly one orbit of edges, and
that the monodromy is non-trivial, so that
$\overline{\mathcal{H}}_{(A,G/G_{1})}$ parametrizes $G/G_{1}$-covers whose
target generically has one node, and with non-trivial ramification over that
node. Then, the map $\operatorname{cores}^{G}_{G/G_{1}}$ is ramified over the
image of $\overline{\mathcal{H}}_{(A,G/G_{1})}$, so the fiber product must be
non-reduced.

In particular, let $S=\operatorname{Spec}(\mathbb{C}[\epsilon]/\epsilon^{2})$.
There exists a family of admissible $G$-covers $f:X\to Y$ over $S$ smoothing
to first order, such that the induced deformation of $X/G_{1}$ has an
$A$-structure over all of $S$ (that is, the nodes of $X/G_{1}$ corresponding
to the edge of $A$ are not smoothed over $S$). Correspondingly, we have a non-
constant map $S\to\overline{\mathcal{H}}_{g,G,\xi}$ along with a constant map
$S\to\overline{\mathcal{H}}_{(A,G/G_{1})}$ agreeing on the constant map
$S\to\overline{\mathcal{H}}_{g_{1},G/G_{1},\xi^{\prime}}$. However, because
$A$ has only one edge, we have that $\operatorname{cores}_{\alpha}$ is
unramified, so we see that there will be no simultaneous lift
$S\to\overline{\mathcal{H}}_{(\Gamma,G)}$.

On the other hand, we have the following:

######  Lemma 4.15.

The multiplicity of a component in the image of the map

$(\xi_{(\Gamma,G)},\operatorname{cores}_{\alpha}):\overline{\mathcal{H}}_{(\Gamma,G)}\to\overline{\mathcal{H}}_{g,G,\xi}\times_{\overline{\mathcal{H}}_{g_{1},G/G_{1},\xi^{\prime}}}\overline{\mathcal{H}}_{(A,G/G_{1})}$

is

$\prod_{(\ell,\ell^{\prime})\in
E^{\prime}(A)}\frac{\operatorname{ord}_{G}(h_{\widetilde{\ell}})}{\operatorname{ord}_{G/G_{1}}(h_{\ell})}.$

where $E^{\prime}(A)\subset E(A)$ is a set of $G/G_{1}$-orbit representatives,
and $\widetilde{\ell}$ is any half-edge of $\Gamma$ lying over
$\alpha_{L}(\ell)\in L(\Gamma/G_{1})$.

We remark also that the terms in the denominators depend only on
$(A,\Gamma/G_{1})$, but the terms in the numerator depend on $(\Gamma,G)$.

In fact, we will obtain more precise information, that the multiplicity in the
tangent direction on $\overline{\mathcal{H}}_{(\Gamma,G)}$ in the direction
corresponding to the edge $(\ell,\ell^{\prime})$ is

$\frac{\operatorname{ord}_{G}(h_{\widetilde{\ell}})}{\operatorname{ord}_{G/G_{1}}(h_{\ell})}=\frac{\operatorname{ord}_{G}(h_{\widetilde{\ell^{\prime}}})}{\operatorname{ord}_{G/G_{1}}(h_{\ell^{\prime}})}.$

###### Proof.

We compute the local picture of the intersection of $\xi_{(\Gamma,G)}$ and
$\operatorname{cores}_{\alpha}$.

The complete local ring of $\overline{\mathcal{H}}_{g,G,\xi}$ at $[X\to X/G]$
is isomorphic to

$\mathbb{C}[[t_{1},\ldots,t_{m},t_{m+1},\ldots,t_{3g^{\prime}-3+b}]],$

where we may take $t_{1},\ldots,t_{m}$ to be smoothing parameters for the
$G$-orbits of nodes of $X$ coming from edges of $A$. The map on deformation
spaces induced by $\xi_{(\Gamma,G)}$ is then the quotient by the ideal
$(t_{1},\ldots,t_{m})$, by the surjectivity of $\alpha_{E}$.

Similarly, the complete local ring of
$\overline{\mathcal{H}}_{g_{1},G/G_{1},\xi^{\prime}}$ at $[X/G_{1}\to X/G]$ is
isomorphic to

$\mathbb{C}[[t^{\prime}_{1},\ldots,t^{\prime}_{m},t^{\prime}_{m+1},\ldots,t^{\prime}_{3g^{\prime}-3+b}]],$

where $t^{\prime}_{1},\ldots,t^{\prime}_{m}$ are smoothing parameters for the
$G/G_{1}$-orbits of nodes of $X/G_{1}$, and the map on deformation spaces
induced by $\xi_{(A,G)}$ is given by the quotient by the ideal
$(t^{\prime}_{1},\ldots,t^{\prime}_{m})$.

For $j=1,2,\ldots,m$, we have that $t_{j}^{e_{j}}$ is a smoothing parameter
for the corresponding node $y_{j}\in X/G$, where $e_{j}$ is the common
ramification index of $X\to X/G$ above $y_{j}$. On the other hand,
${t^{\prime}_{j}}^{e^{\prime}_{j}}$ is also a smoothing parameter at $y_{j}$,
where $e^{\prime}_{j}$ is the common ramification index of $X/G_{1}\to X/G$
above $y_{j}$. Therefore, up to units, we have

${t^{\prime}_{j}}^{e^{\prime}_{j}}={t_{j}}^{e_{j}}.$

Thus, the map

$\mathbb{C}[[t^{\prime}_{1},\ldots,t^{\prime}_{m},t_{m+1},\ldots,t_{3g^{\prime}-3+b}]]\to\mathbb{C}[[t_{1},\ldots,t_{m},t^{\prime}_{m+1},\ldots,t^{\prime}_{3g^{\prime}-3+b}]]$

induced by $\operatorname{cores}_{G/G_{1}}^{G}$ decomposes into the map

$\mathbb{C}[[t^{\prime}_{1},\ldots,t^{\prime}_{m}]]\to\mathbb{C}[[t_{1},\ldots,t_{m}]]$

taking (up to units) ${t^{\prime}_{j}}\mapsto{t_{j}}^{e_{j}/e^{\prime}_{j}}$,
and the map

$\mathbb{C}[[t_{m+1},\ldots,t_{3g^{\prime}-3+b}]]\to\mathbb{C}[[t^{\prime}_{m+1},\ldots,t^{\prime}_{3g^{\prime}-3+b}]]$

induced on complete local rings by
$\operatorname{cores}_{\alpha}:\overline{\mathcal{H}}_{(\Gamma,G)}\to\overline{\mathcal{H}}_{(A,G/G_{1})}$.

It follows that, on the level of complete local rings, the map
$(\xi_{(\Gamma,G)},\operatorname{cores}_{\alpha})$ is of the form

$\mathbb{C}[[t_{1},\ldots,t_{m}]]/(t_{1}^{e_{1}/e^{\prime}_{1}},\ldots,t_{m}^{e_{m}/e^{\prime}_{m}})\otimes_{\mathbb{C}}\mathbb{C}[[t_{m+1},\ldots,t_{3g^{\prime}-3+b}]]\to\mathbb{C}[[t_{m+1},\ldots,t_{3g^{\prime}-3+b}]],$

where the variables $t_{1},\ldots,t_{m}$ map to 0 and
$t_{m+1},\ldots,t_{3g^{\prime}-3+b}$ to themselves. The desired conclusion
follows. ∎

We have therefore proven the following:

######  Proposition 4.16.

With notation as above, we have

$\xi_{A}^{*}((\operatorname{cores}_{G/G_{1}}^{G})_{*}(1))=\sum_{(\Gamma,G),\alpha}\\#\operatorname{Aut}(\alpha)^{-1}\cdot\left(\prod_{(\ell,\ell^{\prime})\in
E^{\prime}(A)}\frac{\operatorname{ord}_{G}(h_{\widetilde{\ell}})}{\operatorname{ord}_{G/G_{1}}(h_{\ell})}\cdot(\operatorname{cores}_{\alpha})_{*}(1)\right)$

and

$(\operatorname{cores}_{G/G_{1}}^{G})^{*}((\xi_{A})_{*}(1))=\sum_{(\Gamma,G),\alpha}\\#\operatorname{Aut}(\alpha)^{-1}\cdot\left(\prod_{(\ell,\ell^{\prime})\in
E^{\prime}(A)}\frac{\operatorname{ord}_{G}(h_{\widetilde{\ell}})}{\operatorname{ord}_{G/G_{1}}(h_{\ell})}.\cdot(\xi_{(\Gamma,G)})_{*}(1)\right)$

###  4.4. Pullbacks of $\psi$ and $\kappa$ classes

We consider the pullbacks of $\psi$ and $\kappa$ classes by the four types of
tautological morphisms.

####  4.4.1. Forgetful maps

######  Proposition 4.17.

Let
$\pi_{G}:\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\to\overline{\mathcal{H}}_{g,G,\xi}$
be a forgetful map. We have:

  1. (a)

$\pi_{G}^{*}(\psi_{p_{ia}})=\psi_{p_{ia}}-\sum_{g\in G/\langle
h_{ia}\rangle}(r_{g}\circ\sigma_{p_{ia}})_{*}(1),$

for any marked point $p_{ia}$ parametrized by
$\overline{\mathcal{H}}_{g,G,\xi}$, where $\sigma_{p_{ia}}$ denotes the
universal section described in §3.2, the sum is over coset representatives
$g\in G/\langle h_{ia}\rangle$, where $\langle h_{ia}\rangle\subset G$ is the
stabilizer of $p_{ia}$, and
$r_{g}:\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\to\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}$
denotes the relabeling isomorphism sending the distinguished additional
ramification point to its image under multiplication by $g$.

  2. (b)

$\pi^{*}(\kappa_{i})=\kappa_{i}-\\#G\cdot\psi_{b+1}^{i},$

where $\psi_{b+1}$ denotes the $\psi$ class on
$\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}$ associated to any point in the
additional unramified $G$-orbit.

In particular, the pullbacks of $\psi$ and $\kappa$ classes by $\pi_{G}$ are
$\mathcal{H}$-tautological.

###### Proof.

We have a commutative diagram

$\textstyle{\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\scriptstyle{\pi_{G}}$$\textstyle{\overline{\mathcal{M}}_{g,r+\\#G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\overline{\mathcal{M}}_{g,r}}$

where we recall that the source map $\phi^{\prime}$ remembers all of the
marked points in the additional unramified orbit, and $\pi$ denotes the
forgetful map forgetting all $|G|$ of these points.

By definition, the $\psi$ and $\kappa$ classes on
$\overline{\mathcal{H}}_{g,G,\xi}$ are pulled back from
$\overline{\mathcal{M}}_{g,r}$, so we may compute the desired pullbacks by
pulling back by $\pi$, then $\phi^{\prime}$. Let $B$ be the subset of marked
points forgotten under the map $\pi$. Propositions 2.13 and 4.2 imply

$\pi^{*}(\psi_{p_{ia}})=\psi_{p_{ia}}-\sum_{B^{\prime}\subset
B}\Delta_{0,p_{ia}\cup B^{\prime}},$

where the sum is over non-empty subsets $B^{\prime}\subset B$, and
$\Delta_{0,p_{ia}\cup B^{\prime}}$ denotes the boundary divisor of curves with
a rational tail containing the marked points of $p_{ia}\cup B^{\prime}$.

By Lemma 4.11, the pullback of $\Delta_{0,p_{ia}\cup B^{\prime}}$ by
$\phi^{\prime}$ is equal to the union of boundary divisors corresponding to
admissible $G$-graphs $(\Gamma,G)$ for which the legs associated to
$p_{ia}\cup B^{\prime}$ are attached to a common genus 0 vertex, each
appearing with multiplicity 1. This is possible exactly when $B^{\prime}$ is
an orbit of $B$ under $\langle h_{ia}\rangle\subset G$, in which case the
boundary divisor $\overline{\mathcal{H}}_{(\Gamma,G)}$ is of the form
$(r_{g}\circ\sigma_{p_{ia}})_{*}(1)$. The union of such divisors is obtained
by varying over a set of coset representatives $g\in G/\langle h_{ia}\rangle$,
yielding (a).

For (b), Propositions 2.13 and 4.2 show that

$\pi^{*}(\kappa_{i})=\kappa_{i}-\left(\sum_{1}^{\\#G}\psi_{r+j}^{i}\right)+\gamma,$

where $\psi_{r+j}$ denote the $\psi$ classes at the marked points forgotten
under $\pi$, and $\gamma\in A^{i}(\overline{\mathcal{M}}_{g,r+\\#G})$ is
supported on the union of boundary divisors where at least two of these marked
points are equal to each other. Note that $\phi^{\prime*}(\gamma)=0$, because
these points are constrained to be pairwise distinct on
$\overline{\mathcal{H}}_{g,G,\xi+\\{1\\}}$. Thus, pulling back by
$\phi^{\prime}$ yields (b), as the $\psi_{r+j}^{i}$ all become equal to
$\psi_{b+i}^{i}$.

∎

####  4.4.2. Diagonal maps

######  Proposition 4.18.

Let
$\Delta:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{H}}_{g,G,\xi}\times\overline{\mathcal{H}}_{g,G,\xi}$
be a diagonal map. Then,

$\Delta^{*}(\theta,1)=\theta$

for any class $\theta$; in particular, $\theta$ can be taken to be a $\psi$ or
$\kappa$ class.

###### Proof.

Trivial. ∎

####  4.4.3. Boundary maps

######  Proposition 4.19.

Let
$\xi_{(\Gamma,G)}:\overline{\mathcal{H}}_{(\Gamma,G)}\to\overline{\mathcal{H}}_{g,G,\xi}$
be a boundary class. We have:

$\xi_{(\Gamma,G)}^{*}\psi_{p_{ia}}=\operatorname{pr}_{v}^{*}\psi_{p_{ia^{\prime}}},$

where $\operatorname{pr}_{v}$ is the projection to the component
$\overline{\mathcal{H}}_{g_{v},G_{v},\xi_{v}}$ of
$\overline{\mathcal{H}}_{(\Gamma,G)}$ corresponding to the component
containing the $i$-th marked orbit and $p_{ia^{\prime}}$ is any point
belonging to this orbit. Moreover,

$\xi_{(\Gamma,G)}^{*}\kappa_{i}=\sum_{v\in
V^{\prime}(\Gamma)}\frac{\\#G}{\\#G_{v}}\operatorname{pr}_{v}^{*}(\kappa_{i})$

###### Proof.

Recall from Remark §3.15 that we have the commutative diagram

$\textstyle{\overline{\mathcal{H}}_{(\Gamma,G)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{(\Gamma,G)}}$$\scriptstyle{\phi_{(\Gamma,G)}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\overline{\mathcal{M}}_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{\Gamma}}$$\textstyle{\overline{\mathcal{M}}_{g,r}}$

where $\phi_{(\Gamma,G)}$ is a composition of the map $\phi$ on individual
components followed by a product of diagonal maps. The claims now follow from
Propositions 2.14 and 4.18. ∎

####  4.4.4. Restriction-corestriction maps

######  Proposition 4.20.

Let
$\operatorname{res}_{G_{1}}^{G}:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{H}}_{g,G_{1},\xi^{\prime}}$
be a restriction map. Then,

$(\operatorname{res}_{G_{1}}^{G})^{*}(\psi_{p})=\psi_{p}$

for any marked point $p$ on the source, and

$(\operatorname{res}_{G_{1}}^{G})^{*}(\kappa_{i})=\kappa_{i}$

for all $i$.

###### Proof.

Immediate from the compatibility of restriction maps with the source maps
$\phi$. ∎

######  Proposition 4.21.

Let
$\operatorname{cores}_{G/G_{1}}^{G}:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{H}}_{g_{1},G/G_{1},\xi^{\prime}}$
be a corestriction map. Then,

$(\operatorname{cores}_{G/G_{1}}^{G})^{*}(\psi_{p})=\frac{\operatorname{ord}_{G/G_{1}}(h)}{\operatorname{ord}_{G}(h)}\cdot\psi_{\widetilde{p}},$

where $p$ is a marked point among the data of
$\overline{\mathcal{H}}_{g_{1},G/G_{1},\xi^{\prime}}$, $\widetilde{p}$ is any
marked point among the data of $\overline{\mathcal{H}}_{g,G,\xi}$ living over
$p$, and $h$ abusively denotes the corresponding monodromy element on both
spaces. Moreover,

$(\operatorname{res}_{G_{1}}^{G})^{*}(\kappa_{i})=\frac{1}{\\#G_{1}}\cdot\kappa_{i}$

for all $i$.

###### Proof.

Immediate from the compatibility of corestriction maps with the target maps
$\delta$, along with Proposition 3.11. ∎

###  4.5. Interlude: Chern classes of the tangent bundle

In order to be able (in principle) to carry out further excess intersection
calculations in the $\mathcal{H}$-tautological ring, we will need to be able
to compute the Chern classes of the tangent bundles of the spaces
$\overline{\mathcal{H}}_{g,G,\xi}$. We do this by pullback from the proper and
quasi-finite morphism
$\delta:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{M}}_{g^{\prime},b}$,
applying Bini’s formula for the Chern classes of
$\overline{\mathcal{M}}_{g^{\prime},b}$.

Let $R$ be the ramification divisor of $\delta$. We have

$\mathcal{T}_{\overline{\mathcal{H}}_{g,G,\xi}}\cong\delta^{*}(\mathcal{T}_{\overline{\mathcal{M}}_{g^{\prime},b}})\otimes\mathcal{O}_{\overline{\mathcal{H}}_{g,G,\xi}}(-R).$

Recall from §3.1 that $\delta$ is ramified at the loci parametrizing
admissible $G$-covers with ramification at nodes. More precisely, $R$ is a sum
of divisors generically parametrizing admissible $G$-covers $f:X\to Y$ where
$X$ has exactly one $G$-orbit of nodes with ramification index $e_{i}$ over
the unique node of $Y$; such a divisor appears in $R$ with multiplicity
$e_{i}-1$.

In the language of admissible $G$-graphs, we consider $(\Gamma,G)$ for which
$G$ acts transitively on the non-empty set $E(\Gamma)$. Let
$e_{\Gamma}=\operatorname{ord}_{G}(h_{\ell})$ for any half-edge $\ell\in
H(\Gamma)$. Let $\overline{\mathcal{H}}^{0}_{(\Gamma,G)}$ be the image
substack of $\overline{\mathcal{H}}_{(\Gamma,G)}$ in
$\overline{\mathcal{H}}_{g,G,\xi}$, which has class

$[\overline{\mathcal{H}}^{0}_{(\Gamma,G)}]=\frac{1}{\\#\operatorname{Aut}(\Gamma,G)}\cdot(\xi_{\Gamma})_{*}(1)\in
A^{1}(\overline{\mathcal{H}}_{g,G,\xi}).$

Putting everything together, we conclude

######  Proposition 4.22.

We have

$\mathcal{T}_{\overline{\mathcal{H}}_{g,G,\xi}}\cong\delta^{*}(\mathcal{T}_{\overline{\mathcal{M}}_{g^{\prime},b}})\left(-\sum_{(\Gamma,G)}(e_{\Gamma}-1)\cdot\overline{\mathcal{H}}^{0}_{(\Gamma,G)}\right),$

where the sum is over admissible $G$-graphs $(\Gamma,G)$ with exactly one
$G$-orbit of edges.

In particular, the Chern classes of
$\mathcal{T}_{\overline{\mathcal{H}}_{g,G,\xi}}$ may be computed explicitly by
combining the formula of Bini [Bin05, Theorem 2] with the earlier results of
this section.

Bini’s formula is expressed in terms of classical tautological classes, so it
follows from §4.2, §4.3.2, and §4.4 that
$\mathcal{T}_{\overline{\mathcal{H}}_{g,G,\xi}}$ may be expressed in terms of
decorated boundary strata, that is, pushforwards of polynomials in $\psi$ and
$\kappa$ classes by boundary morphisms.

###  4.6. Intersections of restriction-corestriction maps

In this section, we show that the intersections of restriction-corestriction
maps $\gamma_{i}:\mathcal{H}_{i}\to\mathcal{H}$ may be expressed in terms of
$\mathcal{H}$-tautological classes on a space mapping simultaneously to both
$\mathcal{H}_{i}$. These intersections may be computed in principle, in terms
of a finite amount of data, but, as will be clear, the algorithmic complexity
seems prohibitive in practice.

As usual, it suffices to consider restriction and corestriction maps
separately; there are three cases.

####  4.6.1. Intersection of two restriction maps

Consider two inclusions of groups $G\subset H_{1}$ and $G\subset H_{2}$, with
restriction maps
$\operatorname{res}_{G}^{H_{1}}:\overline{\mathcal{H}}_{g,H_{1},\xi_{1}}\to\overline{\mathcal{H}}_{g,G,\xi}$
and
$\operatorname{res}_{G}^{H_{2}}:\overline{\mathcal{H}}_{g,H_{2},\xi_{2}}\to\overline{\mathcal{H}}_{g,G,\xi}$.
We first consider the fiber product of the $\operatorname{res}_{G}^{H_{j}}$.

Let us first explain the main idea of the construction. Suppose that $X$ is a
stable curve admitting simultaneous actions by the two groups $H_{1},H_{2}$,
both of which are free away from the marked points of $X$, and which agree
when restricted to the common subgroup $G$. Let $H$ be the subgroup of
$\operatorname{Aut}(X)$ generated by $H_{1}$ and $H_{2}$. Then, $H$ acts on
$X$, the actions of $H_{1},H_{2}$ are both induced by that of $H$, and
$G\subset H_{1}\cap H_{2}$.

However, the action of $H$ on $X$ may not be generically free; rather, for
individual components $X_{0}\subset X$, the action of
$H_{0}=\operatorname{Stab}_{H}(X_{0})$ on $X_{0}$ may have a normal subgroup
$N_{0}\subset H_{0}$ acting trivially.

######  Example 4.23.

[SvZ18, Proposition 5.15] Consider the locus of stable genus 4 curves $X$
admitting both a hyperelliptic and bielliptic involution, that is, the
intersection of the morphisms below:

$\textstyle{\overline{\mathcal{H}}_{4,\mathbb{Z}/2\mathbb{Z},(1)^{6}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{0}}$$\textstyle{\overline{\mathcal{H}}_{4,\mathbb{Z}/2\mathbb{Z},(1)^{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{1}}$$\textstyle{\overline{\mathcal{M}}_{4}}$

For simplicity, we have forgotten the marked points on
$\overline{\mathcal{M}}_{4}$. We have also labelled the maps $\phi_{i}$ with a
subscript keeping track of the target genus. Hyperelliptic and bielliptic
involutions necessarily commute, so the two actions combine to give an action
of $H=\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ on $X$.

The intersection of these two morphisms consists of two strata:

  * •

The locus $\mathcal{A}$ of curves $X=X_{1}\cup X_{2}$, where $X_{k}$ is a
curve of genus $k$ and the two components are attached at two nodes, such that
$X_{2}$ admits a bielliptic involution interchanging the nodes, and

  * •

The locus $\mathcal{B}$ of curves $X=X_{1}\cup X_{2}$, where $X_{k}$ is a
curve of genus $k$ and the two components are attached at two nodes, such that
$X_{1}$ admits a bielliptic involution interchanging the nodes.

In both cases, the hyperelliptic involution $(1,0)\in H$ acts by a
hyperelliptic involution on both components, interchanging the nodes. The
bielliptic involution $(0,1)\in H$ acts by the bielliptic involution on one of
the two components, but the hyperelliptic one on the other. Thus, the diagonal
element $(1,1)$ acts trivially on the non-bielliptic component.

In particular, a point in either locus cannot smooth to a non-singular genus 4
curve admitting both a hyperelliptic and bielliptic involution; in fact, no
such non-singular curves exist for $g\geq 4$.

We thus make the following definition, generalizing the notion of an
admissible $H$-graph to keep track of the possibility that the action of $H$
on a stable curve $X$ may have kernels when restricted to individual
components, but that upon restriction to the subgroups $H_{j}\subset H$, gives
an admissible $H_{j}$-action in the usual sense.

######  Definition 4.24.

Let $H$ be a finite group and let $H_{1},H_{2}$ be subgroups with $\langle
H_{1},H_{2}\rangle=H$. An augmented admissible $H$-graph relative to
$H_{1},H_{2}$

$(\Gamma,H,\\{N_{v}\\}_{v\in V(\Gamma)})$

consists of the following data:

  * •

a stable graph $\Gamma$ with an $H$-action,

  * •

an ordering on the set of $H$-orbits of $L(\Gamma)$, and a distinguished
element $\ell_{i1}$ of each such orbit, so that all elements of $L(\Gamma)$
may be labelled in the usual way by left cosets of the stabilizer
$H_{\ell_{i1}}$ of $\ell_{i1}$,

  * •

for each $v\in V(\Gamma)$, a normal subgroup $N_{v}$ of the stabilizer $H_{v}$
of $v$, that additionally acts trivially on the set $L_{v}\subset
H(\Gamma)\cup L(\Gamma)$ of half-edges and legs incident at $v$, and

  * •

for all $v$ and $\ell\in L_{v}$, a monodromy datum $h_{\ell}\in H_{v}/N_{v}$.

We require the following conditions:

  * •

if $(\ell,\ell^{\prime})$ is a pair of half-edges comprising an edge, then
$t\cdot\ell\neq\ell^{\prime}$ for all $t\in H$,

  * •

$N_{t\cdot v}=tN_{v}t^{-1}$ for any $v\in V(\Gamma)$ and $t\in H$,

  * •

the stabilizer $H_{\ell}\subset H_{v}/N_{v}$ of $\ell$ is a cyclic group
generated by $h_{\ell}$ for all $v\in V(\Gamma)$ and $\ell\in L_{v}$,

  * •

$h_{t\cdot\ell}=th_{\ell}t^{-1}$ in $H_{t\cdot v}/N_{t\cdot
v}=tH_{v}t^{-1}/tN_{v}t^{-1}$ for all $\ell\in L_{v}$ and $t\in H$,

  * •

$\bigcap_{v\in V}N_{v}=\\{1\\}$,

  * •

$N_{v}\cap H_{j}=\\{1\\}$ for any $v\in V(\Gamma)$ and $j=1,2$, and

  * •

the $h_{\ell}$ satisfy the balancing condition when restricted, in the sense
of Definition 3.19, to $H_{1}$ and $H_{2}$.

Let us spell out the last condition in more detail. Suppose that
$(\ell,\ell^{\prime})$ is a pair of half-edges comprising an edge and let
$v,v^{\prime}$ be the vertices to which $\ell,\ell^{\prime}$ are attached,
respectively. Let $\Gamma_{v}$ be the (unpointed) admissible
$H_{v}/N_{v}$-graph with the single vertex $v$ and leg set $L_{v}$ with
monodromy data $\ell_{v}$.

Then, under the restriction of the $H_{v}/N_{v}$-action on $\Gamma_{v}$ to
$H_{j}\cap H_{v}$ (which is a subgroup of $H_{v}/N_{v}$ because $N_{v}\cap
H_{j}=\\{1\\}$), the monodromy datum $h_{\ell}\in H_{\ell}\subset H_{v}/N_{v}$
becomes $h_{\ell,j}=h_{\ell}^{r_{\ell,j}}$, where

$r_{\ell,j}=\frac{\\#H_{\ell}}{\\#H_{\ell}\cap H_{j}}.$

Similarly, we define $h_{\ell^{\prime},j}$. The balancing condition then
requires that $h_{\ell,j}=h_{\ell^{\prime},j}^{-1}\in H_{j}$ for $j=1,2$.

We may define a morphism (resp. automorphism) of augmented admissible
$H$-graphs to be an $H$-equivariant morphism (resp. automorphism) of graphs
compatible with the data of the leg orbit orderings, normal subgroups $N_{v}$,
and monodromy data $h_{\ell}$.

The following explains how augmented admissible $H$-graphs arise when
intersecting restriction morphisms.

######  Lemma 4.25.

Suppose that $X$ is a closed point of the functor
$\overline{\mathcal{H}}_{g,H_{1},\xi_{1}}\times_{\overline{\mathcal{H}}_{g,G,\xi}}\overline{\mathcal{H}}_{g,H_{2},\xi_{2}}$.
Let $H=\langle H_{1},H_{2}\rangle\subset\operatorname{Aut}(X)$, where by
$\operatorname{Aut}(X)$ we mean the finite group of automorphisms of $X$ that
permute the marked points. Then, we have an augmented admissible $H$-graph

$(\Gamma_{X},H,\\{N_{v}\\}),$

where $\Gamma_{X}$ is the dual graph of $X$, $N_{v}$ is the kernel of the
$H$-action on the component of $X$ corresponding to $v$, and we make arbitrary
choices of distinguished legs $\ell_{i1}$ in the $H$-orbits of $L(\Gamma)$ and
of an ordering of these $H$-orbits.

###### Proof.

Each component $X_{v}\in X$ has the structure of an $H_{v}/N_{v}$-curve, and
we obtain monodromy data $h_{\ell}\in H_{v}/N_{v}$ satisfying the needed
compatibilities. Because $H\subset\operatorname{Aut}(X)$, the action $H$ on
$X$ has no global kernel, so the $N_{v}$ have no common intersection. Finally,
restricting to the action on $X$ recovers the underlying admissible
$H_{j}$-structures, so we get the last two conditions. ∎

Note that the construction of $(\Gamma_{X},H,\\{N_{v}\\})$ also makes sense
for a family of curves, as we may replace the nodes of a single curve with the
nodal sections of a family. Moreover, the restriction maps are unramified, so
nodes are smoothed to any order on $\overline{\mathcal{H}}_{g,H_{j},\xi_{j}}$
if and only if they are on $\overline{\mathcal{H}}_{g,G,\xi}$.

We have the following space of curves with a given underlying augmented
admissible $H$-graph:

######  Definition 4.26.

Given any augmented admissible $H$-graph $(\Gamma,H,\\{N_{v}\\})$, we define

$\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}=\prod_{v\in
V(\Gamma)/H}\overline{\mathcal{H}}_{g_{v},H_{v}/N_{v},\xi_{v}}.$

where in the factors on the right hand side, we make arbitrary choices of
distinguished marked points, as in the definition of boundary strata,
Definition 3.13.

Using an analogous gluing procedure to Definition 3.14, a point of
$\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}$ may be interpreted as a
stable curve on $X$ with $H$-action. The condition $N_{v}\cap H_{j}=\\{1\\}$
and the balancing condition upon restriction to $H_{j}$ ensure that this
$H$-curve may be restricted, relative to a choice of relabeling data for the
subgroup $H_{j}\subset H$, to an admissible $H_{j}$-curve. Then, the data of a
point of $\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}$ is equivalent to
that of a stable curve on $X$ with $H$-action and a marking by
$(\Gamma,H,\\{N_{v}\\})$, that is, a morphism of augmented admissible
$H$-graphs from the dual graph of $X$ to
$\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}$.

In the augmented setting, the monodromy data are elements of $H_{v}/N_{v}$,
rather than $H$, but the new distinguished marked points are still chosen
among $H_{j}$-orbits of the marked points of the $H$-curve; the relabeling
data, as before, amounts to an ordered choice of the corresponding
distinguished points.

######  Definition 4.27.

Following the constructions of Definitions 3.14 and 3.19, we define the
proper, quasi-finite, and unramified restriction maps

$\operatorname{res}_{\Gamma,j}:\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}\to\overline{\mathcal{H}}_{g,H_{j},\xi_{j}}.$

given by sending a point in $\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}$
first to the corresponding stable curve equipped with a $\Gamma$-structure and
$H$-action, then restricting to $H_{j}$.

The details of the construction are left to the reader. Note that the maps
$\operatorname{res}_{\Gamma,j}$ are compositions of (products of) diagonal,
boundary, and restriction maps.

In order to compute the fiber product of two restriction morphisms, we need
one more notion that will provide the correct genericity condition.

######  Definition 4.28.

Let $(\Gamma,H,\\{N_{v}\\})$ be an augmented admissible $H$-graph, and let
$(\ell,\ell^{\prime})$ be a pair of half-edges comprising an edge of $\Gamma$,
attached to the (possibly equal) vertices $v,v^{\prime}$, respectively. Let
$\widetilde{H}_{\ell}=\widetilde{H}_{\ell^{\prime}}\subset H$ denote the
stabilizer of the edge $(\ell,\ell^{\prime})$, or equivalently the stabilizer
of either half-edge individually, so that
$\widetilde{H}_{\ell}/N_{v}=H_{\ell}$ and
$\widetilde{H}_{\ell^{\prime}}/N_{v^{\prime}}=H_{\ell^{\prime}}$.

We say that the edge $(\ell,\ell^{\prime})$ is smoothable if
$N_{v}=N_{v^{\prime}}$ and we have the balancing condition
$h_{\ell}=h_{\ell^{\prime}}^{-1}$ in
$\widetilde{H}_{\ell}/N_{v}=\widetilde{H}_{\ell^{\prime}}/N_{v^{\prime}}$.

We call the augmented admissible $H$-graph non-smoothable if none of its edges
are smoothable.

The smoothability condition on the edges of $\Gamma$ corresponds to the
smoothability of the nodes as a point of
$\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}$. Namely, $X$ is a
specialization of another closed point $X_{\eta}$ in the fiber product, where
a given node of $X$ is smoothed into $X_{\eta}$, if and only if the
corresponding edge of $\Gamma$ is smoothable. Correspondingly, if $\Gamma$
fails to be smoothable, we obtain a non-smoothable
$(\Gamma^{\prime},H,\\{N_{v}\\})$ by applying the $H$-equivariant operation of
contracting all smoothable edges and modifying the genus function accordingly.

######  Lemma 4.29.

We have a commutative diagram

$\textstyle{\displaystyle\coprod\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}_{\Gamma,1}}$$\scriptstyle{\operatorname{res}_{\Gamma,2}}$$\textstyle{\overline{\mathcal{H}}_{g,H_{1},\xi_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}^{H_{1}}_{G}}$$\textstyle{\overline{\mathcal{H}}_{g,H_{2},\xi_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}^{H_{2}}_{G}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}}$

where the coproduct is over all non-smoothable augmented admissible $H$-graphs

$(\Gamma,H,\\{N_{v}\\})$

relative to $H_{1},H_{2}$, along with all possible relabeling data defining
the maps $\operatorname{res}_{\Gamma_{j}}$ that are compatible with the
monodromy data $\xi_{j}$ and the maps $\operatorname{res}^{H_{j}}_{G}$.

Moreover, the induced map

$\coprod(\operatorname{res}_{\Gamma,1},\operatorname{res}_{\Gamma,2}):\displaystyle\coprod\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}\to\overline{\mathcal{H}}_{g,H_{1},\xi_{1}}\times_{\overline{\mathcal{H}}_{g,G,\xi}}\overline{\mathcal{H}}_{g,H_{2},\xi_{2}}$

is proper, quasi-finite, étale, and surjective. In particular, the fiber
product is smooth.

More precisely, the compatibility of the relabeling data entails the following
two conditions:

  * •

The resulting restriction maps $\operatorname{res}_{\Gamma,j}$ produce the
correct monodromy data $\xi_{j}$ from that of $(\Gamma,H,\\{N_{v}\\})$, and

  * •

The relabeling data defining the composite morphisms
$\operatorname{res}^{H_{j}}_{G}\circ\operatorname{res}^{\Gamma,j}$ agree, that
is, produce the same ordered collection of distinguished marked points.

###### Proof.

The commutativity is immediate from the requirements on compatibility; the
common composition is simply the restriction to the subgroup $G\subset H$,
with the induced relabeling data. All of the restriction maps are proper and
quasi-finite so
$(\operatorname{res}_{\Gamma,1},\operatorname{res}_{\Gamma,2})$ is as well.

We now show that
$(\operatorname{res}_{\Gamma,1},\operatorname{res}_{\Gamma,2})$ is an analytic
local isomorphism. Consider a point of
$\overline{\mathcal{H}}_{g,H_{1},\xi_{1}}\times_{\overline{\mathcal{H}}_{g,G,\xi}}\overline{\mathcal{H}}_{g,H_{2},\xi_{2}}$
over a local base scheme $S$. This consists of the data of a family
$\mathcal{C}\to S$ of $G$-curves equipped with fiberwise $H_{j}$-actions,