along with discrete relabeling data. The data of the group actions are
equivalent to embeddings of constant group schemes $H_{j}\times
S\hookrightarrow\operatorname{Aut}(\mathcal{X})$ over $S$. We then define $H$
to be the group for which the $H_{j}$ generate the constant group scheme
$H\times S$ inside $\operatorname{Aut}(\mathcal{X})$.

Then, we obtain a non-smoothable augmented admissible $H$-graph
$(\Gamma,H,\\{N_{v}\\})$ as in Lemma 4.25. Any ordered choice of distinguished
marked points yields an $S$-point of the smooth stack
$\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}$, which maps isomorphically
to the original point of
$\overline{\mathcal{H}}_{g,H_{1},\xi_{1}}\times_{\overline{\mathcal{H}}_{g,G,\xi}}\overline{\mathcal{H}}_{g,H_{2},\xi_{2}}$
relative to the unique compatible choice of relabeling data. Moreover, the
genericity conditions of non-smoothability and $H=\langle H_{1},H_{2}\rangle$
guarantee that these are the only points of
$\displaystyle\coprod\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}$ living
over $\mathcal{C}$. ∎

We now compute the degree of
$\displaystyle\coprod\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}$ over the
fiber product in Lemma 4.29.

######  Lemma 4.30.

Given a point
$[\mathcal{C}]\in\overline{\mathcal{H}}_{g,H_{1},\xi_{1}}\times_{\overline{\mathcal{H}}_{g,G,\xi}}\overline{\mathcal{H}}_{g,H_{2},\xi_{2}}(\operatorname{Spec}(\mathbb{C}))$,
the number of isomorphism classes of augmented admissible $H$-graphs
$(\Gamma,H,\\{N_{v}\\})$ for which
$\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}$ contains points mapping to
of $[\mathcal{C}]$ under
$\coprod(\operatorname{res}_{\Gamma,1},\operatorname{res}_{\Gamma,2})$ is

$(\\#L^{\prime}(\Gamma))!\cdot\prod_{\ell\in\\#L^{\prime}(\Gamma)}\frac{\\#H}{\\#\widetilde{H}_{\ell}}=(\\#L^{\prime}(\Gamma))!\cdot\prod_{\ell\in\\#L^{\prime}(\Gamma)}\frac{\\#H}{\operatorname{ord}_{H_{\ell}}(h_{\ell})\cdot\\#N_{v_{\ell}}}.$

where $L^{\prime}(\Gamma)\subset L(\Gamma)$ is any choice of $H$-orbit
representatives. Furthermore, the degree of such a
$\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}$ over
$\overline{\mathcal{H}}_{g,H_{1},\xi_{1}}\times_{\overline{\mathcal{H}}_{g,G,\xi}}\overline{\mathcal{H}}_{g,H_{2},\xi_{2}}$
is

$\\#\operatorname{Aut}((\Gamma,H,\\{N_{v}\\})).$

###### Proof.

As we saw in the proof of Lemma 4.29, the genericity conditions require all
pre-images of $[\mathcal{C}]$ to come from the same $H$-graph with the same
subgroups $N_{v}$, while the only choice available is for the ordered set of
distinguished marked points on $(\Gamma,H,\\{N_{v}\\})$; the relabeling data
for $\operatorname{res}_{\Gamma,j}$ are then determined. In the $H$-orbit of
$\ell\in L(\Gamma)$, there are

$\frac{\\#H}{\\#\widetilde{H}_{\ell}}=\frac{\\#H}{\operatorname{ord}_{H_{\ell}}(h_{\ell})\cdot\\#N_{v_{\ell}}}$

possible choices of a distinguished marked point, where $v_{\ell}$ is the
vertex to which $\ell$ is attached, and $(\\#L^{\prime}(\Gamma))!$ ways to
order these choices.

Finally, similarly as in the proof of Lemma 4.14, each
$\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}$ forms a
$\operatorname{Aut}((\Gamma,H,\\{N_{v}\\}))$-torsor over
$\overline{\mathcal{H}}_{g,H_{1},\xi_{1}}\times_{\overline{\mathcal{H}}_{g,G,\xi}}\overline{\mathcal{H}}_{g,H_{2},\xi_{2}}$,
corresponding to possible choices of distinguished half-edge orbit
representatives. This gives the last claim. ∎

######  Remark 4.31.

The index set for the coproduct can in principle be enumerated by a finite
computation. Indeed, $\\#\operatorname{Aut}(X)$ is uniformly bounded in terms
of $g$ and the number of marked points, so there are only finitely many
possible $H\subset\operatorname{Aut}(X)$, and finitely many choices of the
subsequent data.

It remains to compute the top Chern class of the excess bundle on the fiber
product. We may do so on each component of the finite unramified cover

$\displaystyle\coprod\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}$

and then divide by the degree over the corresponding component of the fiber
product.

We have no combinatorial description of the normal bundles of restriction maps
as in the case of boundary maps, but we may compute explicit formulas for the
Chern classes of the spaces involved, see §4.5.

In the end, we conclude the following.

######  Proposition 4.32.

We have

$\displaystyle(\operatorname{res}_{G}^{H_{2}})^{*}((\operatorname{res}_{G}^{H_{1}})_{*}(1))=\sum_{(\Gamma,H,\\{N_{v}\\})}\left(\\#\operatorname{Aut}((\Gamma,H,\\{N_{v}\\}))\cdot(\\#L^{\prime}(\Gamma))!\cdot\prod_{\ell\in\\#L^{\prime}(\Gamma)}\frac{\\#H}{\operatorname{ord}_{H_{\ell}}(h_{\ell})\cdot\\#N_{v_{\ell}}}\right)^{-1}$
$\displaystyle\cdot\left(\operatorname{res}_{\Gamma,2}\right)_{*}\left(c_{\operatorname{top}}\left(\frac{\mathcal{T}_{\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}}\cdot\mathcal{T}_{\overline{\mathcal{H}}_{g,G,\xi}}}{\mathcal{T}_{\overline{\mathcal{H}}_{g,H_{1},\xi_{1}}}\cdot\mathcal{T}_{\overline{\mathcal{H}}_{g,H_{2},\xi_{2}}}}\right)\right)$

where the sum is over augmented admissible $H$ graphs along with relabeling
data as above, and all tangent bundles are pulled back to
$\overline{\mathcal{H}}_{(\Gamma,H,\\{N_{v}\\})}$.

Recall from §4.5 that the Chern classes of the tangent bundles in question may
be expressed in terms of decorated boundary classes, which, owing to §4.3.1,
themselves pull back to decorated boundary classes by tautological restriction
and boundary maps. However, in pulling back by the maps
$\operatorname{res}_{\Gamma,i}$, we also have to pull such classes back by
diagonals. To do this, we may apply Proposition 4.41 with $\gamma_{i}$ taken
to be boundary maps, which is proven later but independently of this section.
We find, by Proposition 4.10, that we again get decorated boundary classes in
this pullback. In particular, the bivariant pullback of one restriction
morphism by another is $\mathcal{H}$-tautological.

####  4.6.2. Intersection of a restriction and corestriction map

Let $p:H_{1}\to G$ be a surjection and let $i:G\to H_{2}$ be an inclusion of
groups. We consider the intersection of the corestriction
$\operatorname{cores}^{H_{1}}_{G}:\overline{\mathcal{H}}_{g_{1},H_{1},\xi_{1}}\to\overline{\mathcal{H}}_{g,G,\xi}$
and the restriction
$\operatorname{res}^{H_{2}}_{G}:\overline{\mathcal{H}}_{g,H_{2},\xi_{2}}\to\overline{\mathcal{H}}_{g,G,\xi}$.

Consider a point in the fiber product, given by the $H_{1}$-cover $W\to Y$
corestricting to $X\to Y$ and the $H_{2}$-cover $X\to Z$ restricting also to
$X\to Y$. Then, the data of all three of these maps is contained in the
composition

$\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{H_{1}}$$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\scriptstyle{H_{2}}$$\textstyle{Y\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z}$

However, this composition may not be Galois. If we assume that the curves in
question are smooth, we may pass to the Galois closure $\widetilde{W}\to Z$
over $W\to Z$, which is a $\widetilde{H}$-cover for some finite group
$\widetilde{H}$. Let $\widetilde{p}:\widetilde{H}\to H_{2}$ be the natural
surjection. Then, the Galois group of $\widetilde{W}$ over $Y$ is

$\widetilde{H}_{1}:=\widetilde{p}^{-1}(G)\subset\widetilde{H};$

let $\widetilde{i}:\widetilde{H}_{1}\to\widetilde{H}$ be the inclusion and
$\widetilde{q}:\widetilde{H}\to H_{1}$ be the natural surjection. We then have
a commutative diagram

$\textstyle{\widetilde{H}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{i}}$$\scriptstyle{\widetilde{q}}$$\textstyle{\widetilde{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{p}}$$\textstyle{H_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{H_{2}}$

######  Definition 4.33.

Given a surjection $p_{1}:H_{1}\to G$ and injection $i:G\to H_{2}$ of finite
groups, we define a GC-diagram (Galois closure diagram) to consist of the data
of a surjection $\widetilde{p}:\widetilde{H}\to H_{2}$ of finite groups along
with a surjection $\widetilde{q}$ from
$\widetilde{H}_{1}:=\widetilde{p}^{-1}(G)\subset\widetilde{H}$ to $H_{1}$
making the diagram as above commute.

A GC-diagram is minimal if it does not factor through another GC-diagram via a
non-trivial surjection $\widetilde{H}\to\widetilde{H^{\prime}}$, as below:

$\textstyle{\widetilde{H}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{H}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{H}^{\prime}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{i}}$$\scriptstyle{\widetilde{q}}$$\textstyle{\widetilde{H}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widetilde{p}}$$\textstyle{H_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{H_{2}}$

Fix a minimal GC diagram, and monodromy data
$\widetilde{\xi}=(\widetilde{h}_{1},\ldots,\widetilde{h}_{b})\in\widetilde{H}^{b}$
which map to $\xi_{2}=(h^{2}_{1},\ldots,h^{2}_{b})\in H_{2}^{b}$ under the
surjection $\widetilde{p}$. Thus, we have a corestriction map

$\operatorname{cores}^{\widetilde{H}}_{\widetilde{H_{2}}}:\overline{\mathcal{H}}_{\widetilde{g},\widetilde{H},\widetilde{\xi}}\to\overline{\mathcal{H}}_{g,H_{2},\xi_{2}},$

where the genus $\widetilde{g}$ is determined by the Riemann-Hurwitz formula.

On the other hand, suppose that we have a choice of relabeling data for the
subgroup $\widetilde{H}_{1}\subset\widetilde{H}$ that the resulting
restriction map

$\operatorname{res}^{\widetilde{H}}_{\widetilde{H}_{1}}:\overline{\mathcal{H}}_{\widetilde{g},\widetilde{H},\widetilde{\xi}}\to\overline{\mathcal{H}}_{g_{1},\widetilde{H}_{1},\widetilde{\xi}_{1}}$

yields monodromy data $\widetilde{\xi}_{1}$ mapping to $\xi_{1}$ under the
quotient map $\widetilde{q}:\widetilde{H}_{1}\to H_{1}$. Then, we have a
restriction-corestriction map

$q_{\widetilde{K}_{1},\widetilde{H}_{1},\widetilde{H}}=\operatorname{cores}^{\widetilde{H}_{1}}_{H_{1}}\circ\operatorname{res}^{\widetilde{H}}_{\widetilde{H}_{1}}:\overline{\mathcal{H}}_{\widetilde{g},\widetilde{H},\widetilde{\xi}}\to\overline{\mathcal{H}}_{g_{1},H_{1},\xi^{\prime}_{1}}$

where $\widetilde{K}_{1}=\ker(\widetilde{q})$

Suppose, in addition, that the relabeling data defining
$\operatorname{res}^{\widetilde{H}}_{\widetilde{H}_{1}}$ is compatible with
that defining $\operatorname{res}^{H_{2}}_{G}$ in the sense that the ordered
set of orbit representatives of $\widetilde{H}_{1}$ acting on the
$\widetilde{H}/\langle\widetilde{h}_{i}\rangle$ maps to that of the orbit
representatives of $G$ acting on $\widetilde{H}_{2}/\langle h^{2}_{i}\rangle$
under $\widetilde{p}$ (with the same indices $i$ everywhere). Then, by
construction, we have a commutative diagram

$\textstyle{\overline{\mathcal{H}}_{\widetilde{g},\widetilde{H},\widetilde{\xi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{\widetilde{K_{1}},\widetilde{H}_{1},\widetilde{H}}}$$\scriptstyle{\operatorname{cores}^{\widetilde{H}}_{H_{2}}}$$\textstyle{\overline{\mathcal{H}}_{g_{1},H_{1},\xi_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{cores}^{H_{1}}_{G}}$$\textstyle{\overline{\mathcal{H}}_{g_{2},H_{2},\xi_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}^{H_{2}}_{G}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}}$

The following is thus immediate:

######  Lemma 4.34.

Consider the commutative diagram

$\textstyle{\coprod\overline{\mathcal{H}}_{\widetilde{g},\widetilde{H},\widetilde{\xi}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{\widetilde{K_{1}},\widetilde{H}_{1},\widetilde{H}}}$$\scriptstyle{\operatorname{cores}^{\widetilde{H}}_{H_{2}}}$$\textstyle{\overline{\mathcal{H}}_{g_{1},H_{1},\xi_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{cores}^{H_{1}}_{G}}$$\textstyle{\overline{\mathcal{H}}_{g_{2},H_{2},\xi_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}^{H_{2}}_{G}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}}$

where the coproduct is taken over all possible minimal $GC$ diagrams with
compatible choices of monodromy data $\widetilde{\xi}$ and relabeling data
defining $q_{\widetilde{K_{1}},\widetilde{H}_{1},\widetilde{H}}$ as detailed
above.

Then, the induced map

$\coprod(q_{\widetilde{K_{1}},\widetilde{H}_{1},\widetilde{H}},\operatorname{cores}^{\widetilde{H}}_{H_{2}}):\coprod\overline{\mathcal{H}}_{\widetilde{g},\widetilde{H},\widetilde{\xi}}\to\overline{\mathcal{H}}_{g_{1},H_{1},\xi_{1}}\times_{\overline{\mathcal{H}}_{g,G,\xi}}\overline{\mathcal{H}}_{g_{2},H_{2},\xi_{2}}$

is proper and quasi-finite.

Indeed, the properness and quasi-finiteness follows from that of all of the
morphisms involved. We also remark that the index set of the coproduct can in
principle be computed, as the order of $\widetilde{H}$ has is bounded below by
$((\\#H_{1}\cdot\\#H_{2})/\\#G)!$.

It remains to compute the degree of
$\coprod(q_{\widetilde{K_{1}},\widetilde{H}_{1},\widetilde{H}},\operatorname{cores}^{\widetilde{H}}_{H_{2}})$.

######  Lemma 4.35.

For any point

$([W\to Y],[X\to
Z])\in\overline{\mathcal{H}}_{g_{1},H_{1},\xi_{1}}\times_{\overline{\mathcal{H}}_{g,G,\xi}}\overline{\mathcal{H}}_{g_{2},H_{2},\xi_{2}}$

where $W,X,Y,Z$ are smooth curves, let $\widetilde{W}\to Z$ be the Galois
closure of the composition $W\to Z$, and let

$\widetilde{K}_{1}\subset\widetilde{K}_{2}\subset\widetilde{H}_{1}\subset\widetilde{H}$

be the Galois groups of $\widetilde{W}$ over $W,X,Y,Z$, respectively. Let
$\widetilde{h}_{i}$, for $i=1,\ldots,b$, be any monodromy elements in the
$\widetilde{H}$-orbits of $\widetilde{W}\to Z$. Then, the degree of
$\coprod(q_{\widetilde{K_{1}},\widetilde{H}_{1},\widetilde{H}},\operatorname{cores}^{\widetilde{H}}_{H_{2}})$
over $([W\to Y],[X\to Z])$ is

$\prod_{i=1}^{b}\\#(\widetilde{K}_{2}/(\langle\widetilde{h}_{i}\rangle\cap\widetilde{K}_{2}))\cdot(\\#\widetilde{K}_{1}/(\langle\widetilde{h}_{i}\rangle\cap\widetilde{K}_{1}))^{\\#H_{2}/\\#G}$

###### Proof.

By the minimality requirement, the GC diagram giving a point in the pre-image
of $([W\to Y],[X\to Z])$ must be that arising from $\widetilde{W}\to Z$. The
points in the fiber above $([W\to Y],[X\to Z])$ then correspond to possible
choices of distinguished marked points of $\widetilde{W}\to Z$ (which
determine $\widetilde{\xi}$), along with possible choices of distinguished
marked points of the restriction $\widetilde{W}\to Y$ (which determine the
relabeling data).

For the cover $\widetilde{W}\to Z$, we must choose distinguished marked points
lying over those of $X\to Z$, and their order is also dictated by that on
$X\to Z$. The number of possible distinguished marked points in the $i$-th
$\widetilde{H}$-orbit of $\widetilde{W}$ is then
$\\#(\widetilde{K}_{2}/(\langle\widetilde{h}_{i}\rangle\cap\widetilde{K}_{2}))$.

Similarly, for the cover $\widetilde{W}\to Y$, we must choose distinguished
marked points lying the given ones of $W\to Y$, in the same order. The $i$-th
$\widetilde{H}$-orbit of $\widetilde{W}$ breaks into $\\#H/\\#H_{1}$
$\widetilde{H}_{1}$-orbits, each of which has
$\\#\widetilde{K}_{1}/(\langle\widetilde{h}_{i}\rangle\cap\widetilde{K}_{1})$
choices for a distinguished marked point over $Y$.

Combining yields the conclusion. ∎

Because a general point of the fiber product parametrizes a pair of covers of
smooth curves, we have proven the following.

######  Proposition 4.36.

With notation as above, we have:

$\displaystyle(\operatorname{res}^{H_{2}}_{G})^{*}$
$\displaystyle((\operatorname{cores}^{H_{1}}_{G})_{*}(1))$ $\displaystyle=$
$\displaystyle\sum\left(\prod_{i=1}^{b}\\#(\widetilde{K}_{2}/(\langle\widetilde{h}_{i}\rangle\cap\widetilde{K}_{2}))\cdot(\\#\widetilde{K}_{1}/(\langle\widetilde{h}_{i}\rangle\cap\widetilde{K}_{1}))^{\\#H_{2}/\\#G}\right)^{-1}(\operatorname{cores}^{\widetilde{H}}_{H_{2}})_{*}(1)$

and

$\displaystyle(\operatorname{cores}^{H_{1}}_{G})^{*}$
$\displaystyle((\operatorname{res}^{H_{2}}_{G})_{*}(1))$ $\displaystyle=$
$\displaystyle\sum\left(\prod_{i=1}^{b}\\#(\widetilde{K}_{2}/(\langle\widetilde{h}_{i}\rangle\cap\widetilde{K}_{2}))\cdot(\\#\widetilde{K}_{1}/(\langle\widetilde{h}_{i}\rangle\cap\widetilde{K}_{1}))^{\\#H_{2}/\\#G}\right)^{-1}(q_{\widetilde{K_{1}},\widetilde{H}_{1},\widetilde{H}})_{*}(1),$

where the sums are over choices of minimal GC-diagrams, monodromy data
$\widetilde{\xi}$, and relabeling data as above. In particular, the
intersections in question are $\mathcal{H}$-tautological.

####  4.6.3. Intersection of two corestriction maps

Let $H_{1}\to G,H_{2}\to G$ be surjections of finite groups with kernels
$K_{1}\subset H_{1},K_{2}\subset H_{2}$. We consider the intersection of the
corestriction maps
$\operatorname{cores}^{H_{1}}_{G}:\overline{\mathcal{H}}_{g_{1},H_{1},\xi_{1}}\to\overline{\mathcal{H}}_{g,G,\xi}$
and
$\operatorname{cores}^{H_{2}}_{G}:\overline{\mathcal{H}}_{g_{2},H_{2},\xi_{2}}\to\overline{\mathcal{H}}_{g,G,\xi}$.

We first explain the idea of the construction. Consider admissible
$H_{j}$-covers $f_{j}:X_{j}\to Y$ equipped with an isomorphism
$X_{1}/K_{1}\cong X_{2}/K_{2}$ over $Y$; we identify both curves with the
$G$-curve $X$. Suppose further that the $X_{j}$ and $Y$ are smooth. Then, away
from the marked points of $Y$, the fiber product
$f^{\prime}_{\bullet}:X^{\prime}_{\bullet}=X_{1}\times_{X}X_{2}\to Y$ is a
principal $H_{\bullet}=H_{1}\times_{G}H_{2}$-bundle over $Y$.

After normalizing $X^{\prime}_{\bullet}$, we then get a $H_{\bullet}$-cover of
smooth curves $f_{\bullet}:X_{\bullet}\to Y$. However, $X_{\bullet}$ may be
disconnected. On the other hand, let $H_{0}$ be the stabilizer of a connected
component $X_{0}\subset X_{\bullet}$; alternatively, $H_{0}$ is the image of
the monodromy representation (obtained after choosing base-points
appropriately) $\pi_{1}(Y,\\{q_{k}\\})\to H_{1}\times_{G}H_{2}$ induced by
$f_{1}$ and $f_{2}$. Then, we can essentially recover the data of the original
$H_{j}$-covers by corestricting from $H_{0}$ to $H_{j}$.

The issue remains that distinguished marked points must be chosen on $X_{0}$,
and that these may not map to those on the $X_{j}$, so we will need to post-
compose with relabeling isomorphisms on
$\overline{\mathcal{H}}_{g_{j},H_{j},\xi_{j}}$.

######  Definition 4.37.

As above, define $H_{\bullet}=H_{1}\times_{G}H_{2}$, and let $H_{0}\subset
H_{\bullet}$ be a subgroup such that the compositions

$\textstyle{\pi_{j}:H_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{\bullet}=H_{1}\times_{G}H_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{pr}_{j}}$$\textstyle{H_{j}}$

are both surjective. Moreover, fix $t_{1},\ldots,t_{b}\in H_{\bullet}$.

We define, for $j=1,2$, the map

$\operatorname{cores}^{\prime
H_{0}}_{H_{j}}:\overline{\mathcal{H}}_{h_{0},H_{0},\xi_{0}}\to\overline{\mathcal{H}}_{h_{j},H_{j},\xi_{j}}$

by the composition of the usual corestriction map

$\operatorname{cores}^{H_{0}}_{H_{j}}:\overline{\mathcal{H}}_{h_{0},H_{0},\xi_{0}}\to\overline{\mathcal{H}}_{h_{j},H_{j},\xi^{\prime}_{j}}$

and the relabeling isomorphism

$\operatorname{res}^{H_{j}}_{H_{j}}:\overline{\mathcal{H}}_{h_{0},H_{j},\xi^{\prime}_{j}}\to\overline{\mathcal{H}}_{h_{j},H_{j},\xi_{j}}$

replacing the distinguished marked point $p_{i1}$ with
$\operatorname{pr}_{j}(t_{i})\cdot p_{i1}$.

Write $\xi_{j}=(h^{j}_{1},\ldots,h^{j}_{b})$ for $j=0,1,2$. Then, for the maps
$\operatorname{cores}^{\prime H_{0}}_{H_{j}}$ to make sense, we need

$h^{j}_{i}=\operatorname{pr}_{j}(t_{i})\cdot\pi_{j}(h^{0}_{i})\cdot\operatorname{pr}(t_{i}^{-1})$

for $j=1,2$.

######  Lemma 4.38.

For any choice of $H_{0}\subset H_{\bullet}$ and $t_{1},\ldots,t_{b}\in
H_{\bullet}$ as in Definition 4.37, set, for $i=1,2,\ldots,b$,

$h^{0}_{i}=t_{i}^{-1}h^{\bullet}_{i}t_{i}\in H_{\bullet},$

where $h^{\bullet}_{i}=(h^{1}_{i},h^{2}_{i})$. Then, we have a commutative
diagram

$\textstyle{\coprod\overline{\mathcal{H}}_{h_{0},H_{0},\xi_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{cores}^{\prime
H_{0}}_{H_{1}}}$$\scriptstyle{\operatorname{cores}^{\prime
H_{0}}_{H_{2}}}$$\textstyle{\overline{\mathcal{H}}_{g_{1},H_{1},\xi_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{cores}^{H_{1}}_{G}}$$\textstyle{\overline{\mathcal{H}}_{g_{2},H_{2},\xi_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{cores}^{H_{2}}_{G}}$$\textstyle{\overline{\mathcal{H}}_{g,G,\xi}}$

where the coproduct is taken over all possible choices above.

Moreover, the induced map

$\coprod(\operatorname{cores}^{\prime
H_{0}}_{H_{1}},\operatorname{cores}^{\prime
H_{0}}_{H_{2}}):\coprod\overline{\mathcal{H}}_{h_{0},H_{0},\xi_{0}}\to\overline{\mathcal{H}}_{g_{1},H_{1},\xi_{1}}\times_{\overline{\mathcal{H}}_{g,G,\xi}}\overline{\mathcal{H}}_{g_{2},H_{2},\xi_{2}}$

is proper and quasi-finite.

###### Proof.

The choice of $h^{0}_{i}$ ensures that the maps $\operatorname{cores}^{\prime
H_{0}}_{H_{j}}$ are well-defined. Then, the composition in both directions is
the corestriction from $H_{0}$ to $G$, but where the distinguished marked
points $p^{\prime}_{i1}$ on the resulting $G$-cover are replaced by
$t^{\prime}_{i}\cdot p^{\prime}_{i1}$, where $t^{\prime}_{i}$ is the image of
$t_{i}$ in $G$.

The properness and quasi-finiteness are then immediate from the same
properties for the corestriction maps. ∎

######  Lemma 4.39.

For $j=1,2$, let $f_{j}:X_{j}\to Y$ be an $H_{j}$-cover, agreeing upon
corestriction on the $G$-cover $f:X\to Y$. Assume further that the $X_{j}$ and
$Y$ are smooth. Let $f_{\bullet}:X_{\bullet}\to Y$ be the normalized fiber
product of the $f_{j}$ over $f$, and let $H^{\prime}_{0}\subset H_{\bullet}$
be the stabilizer of any component of $X_{\bullet}$. Then, over
$(f_{1},f_{2})$, the map

$\coprod(\operatorname{cores}^{\prime
H_{0}}_{H_{1}},\operatorname{cores}^{\prime
H_{0}}_{H_{2}}):\coprod\overline{\mathcal{H}}_{h_{0},H_{0},\xi_{0}}\to\overline{\mathcal{H}}_{g_{1},H_{1},\xi_{1}}\times_{\overline{\mathcal{H}}_{g,G,\xi}}\overline{\mathcal{H}}_{g_{2},H_{2},\xi_{2}}$

has degree

$\\#H_{\bullet}\cdot(\\#H^{\prime}_{0})^{b-1}$

###### Proof.

Suppose that we have a point of
$\coprod\overline{\mathcal{H}}_{h_{0},H_{0},\xi_{0}}$ lying over
$(f_{1},f_{2})$, given by a pointed $H_{0}$-cover $f_{0}:X_{0}\to Y$ along
with $t_{1},\ldots,t_{b}\in H_{\bullet}$ defining relabeling isomorphisms,
such that the corestrictions of $f_{0}$ to $H_{j}$ yield the covers $f_{j}$.
Then, by the universal property of fiber product, $f_{0}$ factors through
$X_{\bullet}$; more specifically, it surjects onto a unique component of
$X^{\prime}_{0}\subset X_{\bullet}$.

We claim, however, that the induced map $X_{0}\to X^{\prime}_{0}$ is an
isomorphism. If not, then we have a non-trivial automorphism of $X_{0}$ over
both $X_{1}$ and $X_{2}$, but $H_{0}$ is a subgroup $H_{1}\times_{G}H_{2}$, so
we have a contradiction. Therefore, we may canonically identify $H_{0},X_{0}$
with $H^{\prime}_{0},X^{\prime}_{0}$, respectively.

Now, in identifying points in the fiber over $(f_{1},f_{2})$, we have the
following choices: a component $X^{\prime}_{0}\subset X_{\bullet}$, which
determines $H_{0}$, a collection of distinguished marked points of
$X^{\prime}_{0}$, which determines $\xi_{0}$, and the $t_{i}\in H_{\bullet}$.
The number of choices of a component of $X_{\bullet}$ is
$\\#H/\\#H^{\prime}_{0}$, the number of choices of distinguished marked points
of $X^{\prime}_{0}$ in the $i$-th orbit is
$\\#H^{\prime}_{0}/\operatorname{ord}_{H_{\bullet}}(h^{\bullet}_{i})$, and the
number of possible choices of $t_{i}$ is
$\operatorname{ord}_{H_{\bullet}}(h^{\bullet}_{i})$ (as the set of possible
choices is a left coset of the stabilizer of the distinguished marked point in
$H_{\bullet}$). Combining gives the desired degree. ∎

Because a general point on any component of
$\overline{\mathcal{H}}_{g_{1},H_{1},\xi_{1}}\times_{\overline{\mathcal{H}}_{g,G,\xi}}\overline{\mathcal{H}}_{g_{2},H_{2},\xi_{2}}$
parametrizes a pair of covers of smooth curves, we have proven the following.

######  Proposition 4.40.

We have

$(\operatorname{cores}^{H_{2}}_{G})^{*}((\operatorname{cores}^{H_{1}}_{G})_{*}(1))=\sum_{H_{0},\\{t_{i}\\}}\frac{1}{\\#H_{\bullet}\cdot(\\#H_{0})^{b-1}}\cdot(\operatorname{cores}^{\prime
H_{0}}_{H_{2}})_{*}(1),$

where the sum, as before, is over subgroups $H_{0}\subset
H_{\bullet}=H_{1}\times_{G}H_{2}$ surjecting onto the $H_{i}$ and choices of
$t_{i}\in H_{\bullet}$.

###  4.7. Diagonals

######  Proposition 4.41.

Let $\overline{\mathcal{H}}_{g,G,\xi}$ be any stack of $G$-covers, and let
$\gamma_{i}:\mathcal{H}_{i}\to\overline{\mathcal{H}}_{g,G,\xi}$ for $i=1,2$ be
forgetful, boundary, corestriction, or restriction maps. Then, the
intersections of $\gamma_{1}\times\gamma_{2}$ and the diagonal class $\Delta$
(in either order) are given by $\mathcal{H}$-tautological classes computed
earlier in this section.

###### Proof.

The intersection of $\gamma_{1}\times\gamma_{2}$ and $\Delta$ is given by that
of $\gamma_{1}$ and $\gamma_{2}$, essentially by definition (see [Ful98,
Chapter 8]). We have already computed all of these intersections, in terms of
an $\mathcal{H}$-tautological class on a space $\mathcal{H}_{0}$ mapping (via
a composition of tautological morphisms) to both $\mathcal{H}_{j}$, so in
particular to $\overline{\mathcal{H}}_{g,G,\xi}$ and to
$\mathcal{H}_{1}\times\mathcal{H}_{2}$. ∎

The only remaining case to consider is that of the self-intersection of the
diagonal of $\overline{\mathcal{H}}_{g,G,\xi}$; this is given simply by the
top Chern class of the tangent bundle, which we have already explained how to
compute.

######  Proposition 4.42.

We have
$\Delta^{*}(\Delta_{*}(1))=c_{\operatorname{top}}(\mathcal{T}_{\overline{\mathcal{H}}_{g,G,\xi}})$,
which may be expressed in terms of decorated boundary classes, and is in
particular $\mathcal{H}$-tautological.

###### Proof.

See Proposition 4.22. ∎

##  5\. Additive structure

###  5.1. Generators for the $\mathcal{H}$-tautological ring

Our main theorem is now immediate from the results of the previous section:

######  Theorem 5.1.

Let

$\mathcal{H}=\prod_{i=1}^{m}\overline{\mathcal{H}}_{g_{i},G_{i},\xi_{i}}$

be an arbitrary product of spaces of admissible $G$-covers. Consider all
classes of the form $\gamma_{*}(\theta)\in A^{*}(\mathcal{H})$, where
$\gamma:\mathcal{H}^{\prime}\to\mathcal{H}$ is a composition of forgetful,
boundary, diagonal, and restriction-corestriction morphisms from another
product of spaces of admissible $G$ covers, and $\theta$ is a monomial in the
$\psi$ and $\kappa$ classes on each of the components of
$\mathcal{H}^{\prime}$. Then, the classes $\gamma_{*}(\theta)$ form an
additive set of generators for $R^{*}_{\mathcal{H}}(\mathcal{H})$. Moreover,
the $\mathcal{H}$-tautological ring is closed under bivariant pullback by all
of the aforementionoed tautological morphisms.

###### Proof.

All such classes are $\mathcal{H}$-tautological by definition. The results of
the previous section allow one to compute the bivariant pullback of such a
composition by another one, one tautological morphism at a time. At each step,
we intersect basic $\mathcal{H}$-tautological morphisms
$\gamma_{i}:\mathcal{H}_{i}\to\mathcal{H}$, giving the answer in terms of an
$\mathcal{H}$-tautological class in terms of a space $\mathcal{H}_{0}$ mapping
simultaneously to the $\mathcal{H}_{j}$, so that we can proceed to the next
step. We may also pull $\psi$ and $\kappa$ classes back by any of these
morphisms.

In particular, the intersection of any two classes of the form
$\gamma_{*}(\theta)$ is a linear combination of such, so the theorem is
proven. ∎

In fact, one can give a smaller set of additive generators for the
$\mathcal{H}$-tautological ring:

######  Proposition 5.2.

The $\mathcal{H}$-tautological ring is generated additively by classes
$\gamma_{*}(\theta)$ as above, where

$\gamma=\xi\circ\pi\circ(\operatorname{cores}\circ\operatorname{res})\circ\Delta$

is taken to be the composition of diagonal, restriction-corestriction,
forgetful, and boundary morphisms, in that order.

###### Proof.

This follows from the following claims, whose verifications we leave to the
reader.

  * •

the composition of a boundary, forgetful, or restriction-corestriction map
followed by a diagonal map is equal to a composition of a diagonal map
followed by maps of the original type,

  * •

the composition of a boundary map followed by a forgetful map is equal to the
composition of forgetful maps followed by a boundary map,

  * •

the composition of a boundary map followed by a restriction-corestriction map
is equal to the composition of restriction-corestriction maps, diagonal maps,
and a boundary map, in that order, and

  * •

the composition of a forgetful map followed by a restriction-corestriction map
is equal to a restriction-corestriction map followed by forgetful maps.

∎

###  5.2. Integration of zero-cycles

We remark that, given our results, one can also compute intersection numbers
between $\mathcal{H}$-tautological classes of complementary dimension. Indeed,
by the results of this section, one only needs to integrate top-degree
monomials in $\psi$ and $\kappa$ classes on
$\overline{\mathcal{H}}_{g,G,\xi}$. However, such classes, up to constant
factors, are pulled back from target maps
$\delta:\overline{\mathcal{H}}_{g,G,\xi}\to\overline{\mathcal{M}}_{g^{\prime},b}$.
Therefore, we need only the information of the degrees of target maps
$\delta$, given by Hurwitz numbers (see [SvZ18, Theorem 3.19]), and integrals
of monomials in $\psi$ and $\kappa$ classes on
$\overline{\mathcal{M}}_{g^{\prime},b}$, given by KdV hierarchy, see [Kon92].

##  6\. Classical Hurwitz loci

###  6.1. The Harris-Mumford space and its normalization

Recall from §1 that our motivation is to study cycles on moduli spaces of
curves coming from classical Hurwitz spaces, that is, cycles
$(\phi_{g/h,d})_{*}(1)$ where
$\phi:\overline{\mathcal{H}}_{g/h,d}\to\overline{\mathcal{M}}_{g,r}$ is the
source map.

The first compactification $\overline{\mathcal{H}}_{g/h,d}$ was given by
Harris-Mumford [HM82] in terms of admissible covers, whose definition we now
recall.

######  Definition 6.1.

Let $X,Y$ be curves. Let $b=(2p_{a}(X)-2)-d(2p_{a}(Y)-2)$, and let
$y_{1},\ldots,y_{b}\in Y$ be such that $(Y,y_{1},\ldots,y_{b})$ is stable.
Then, an admissible cover consists of the data of the stable marked curve
$(Y,y_{1},\ldots,y_{b})$ and a finite morphism $f:X\to Y$ such that:

  * •

$f(x)$ is a smooth point of $Y$ if and only if $x$ is a smooth point of $X$,

  * •

$f$ is simply branched over the $y_{i}$ and étale over the rest of the smooth
locus of $Y$, and

  * •

at each node of $X$, the ramification indices of $f$ restricted to the two
branches are equal.

The definition can also be generalized in an obvious way to compactify spaces
of covers whose ramification is not assumed to be simple. As in the case of
Galois covers, spaces of admissible covers admit source and target maps to
moduli spaces of curves.

In general, the spaces $\overline{\mathcal{H}}_{g/h,d}$ are Cohen-Macaulay,
but fail to be smooth. We recall from [HM82] the explicit description of their
complete local rings. Let $[f:X\to Y]$ be a point of
$\overline{\mathcal{H}}_{g/h,d}$. Let $y^{\prime}_{1},\ldots,y^{\prime}_{n}$
be the nodes of $Y$, and let $y_{1},\ldots,y_{b}\in Y$ be the branch points.
Let $\mathbb{C}[[t_{1},\ldots,t_{3h-3+b}]]$ be the deformation space of
$(Y,y_{1},\ldots,y_{b})$, so that $t_{1},\ldots,t_{n}$ are smoothing
parameters for the nodes $y^{\prime}_{1},\ldots,y^{\prime}_{n}$. Let
$x_{i,1},\ldots,x_{i,r_{i}}$ be the nodes of $X$ mapping to $y^{\prime}_{i}$,
and denote the ramification index of $f$ at $x_{i,j}$ by $a_{i,j}$.

######  Proposition 6.2 ([HM82]).

The complete local ring of $\mathcal{A}dm_{g/h,d}$ at $[f]$ is

$\mathbb{C}\left[\left[t_{1},\ldots,t_{3h-3+b},\\{t_{i,j}\\}^{1\leq i\leq
n}_{1\leq j\leq
r_{i}}\right]\right]/\left(t_{1}=t_{1,1}^{a_{1,1}}=\cdots=t_{1,r_{1}}^{a_{1,r_{1}}},\ldots,t_{n}=t_{n,1}^{a_{n,1}}=\cdots=t_{n,r_{n}}^{a_{n,r_{n}}}\right).$

It follows that $[f:X\to Y]\in\overline{\mathcal{H}}_{g/h,d}$ is a singular
point if and only if there exists a node of $Y$ over which $f$ is ramified at
more than one point.

Abramovich-Corti-Vistoli [ACV03] give a normalization of
$\overline{\mathcal{H}}_{g/h,d}$ by way of twisted $S_{d}$-covers. The precise
definitions will not be important for us, but we recall the idea.

The space $\widetilde{\mathcal{H}}_{g/h,d}^{ACV}$ is defined to be the space
of balanced twisted $S_{d}$-covers $f:\widetilde{X}\to\mathcal{Y}$, where
$\mathcal{Y}$ is a marked nodal curve with certain stacky structure at its
marked and singular points, and $f$ is a principal $S_{d}$-bundle. One can
then recover an admissible cover of degree $d$ by replacing $\widetilde{X}$ by
$X=\widetilde{X}/S_{d-1}$ and $\mathcal{Y}$ by its coarse space.

We then get a map (after restricting to the appropriate components of
$\widetilde{H}_{g/h,d}^{ACV}$)

$\nu:\widetilde{\mathcal{H}}_{g/h,d}^{ACV}\to\overline{\mathcal{H}}_{g/h,d}$

which is a normalization, see [ACV03, Proposition 4.2.2]. If $f:X\to Y$ is a
degree $d$ cover of smooth curves, then the unique balanced twisted
$S_{d}$-cover lying over $[f]$ in the normalization is given over the étale
locus $Y_{0}\subset Y$ of $f$ by

$\widetilde{X}_{0}=(X_{0}\times_{Y_{0}}\cdots\times_{Y_{0}}X_{0})-\Delta.$

where the product is $d$-fold and $\Delta$ is the locus on which the points in
any two factors are equal. The induced map $\widetilde{X}_{0}\to Y_{0}$ then
extends uniquely to a principal $S_{d}$-bundle
$f:\widetilde{X}\to\mathcal{Y}$.

However, one can just as well understand the normalization $\nu$ in terms of
the scheme-theoretic $S_{d}$-covers
$\widetilde{f}^{\prime}:\widetilde{X}\to\widetilde{X}/S_{d-1}=Y$, where the
target is now identified with the coarse space of $\mathcal{Y}$. Now,
$\widetilde{f}^{\prime}$ is no longer a principal $S_{d}$-bundle over the
nodes and marked points of $Y$, but rather is an admissible $S_{d}$-cover in
the sense of this paper. We will therefore work with the normalization

$\nu:\widetilde{\mathcal{H}}_{g/h,d}=\coprod\overline{\mathcal{H}}^{\prime}_{g^{\prime},S_{d},\xi}\to\overline{\mathcal{H}}_{g/h,d}$

which is isomorphic to the normalization by twisted $S_{d}$-covers, see
[SvZ18, Remark 3.6] for a detailed discussion.

Here, $\overline{\mathcal{H}}^{\prime}_{g^{\prime},S_{d},\xi}$ a moduli space
of possibly disconnected admissible $S_{d}$-covers $\widetilde{X}\to Y$,
which, upon applying the restriction-corestriction map remembering
$\widetilde{X}/S_{d-1}$, yield an admissible cover of the form parametrized by
$\overline{\mathcal{H}}_{g/h,d}$. We will see that the intersection-theoretic
tools of §4, where we have dealt with spaces of connected covers, will carry
over to this setting.

###  6.2. Intersecting Harris-Mumford loci with boundary classes

In this section, we will outline an approach to computing intersections of
normalized Harris-Mumford loci

$\textstyle{\widetilde{\mathcal{H}}_{g/h,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{\overline{\mathcal{H}}_{g/h,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\overline{\mathcal{M}}_{g,r}}$

with boundary classes
$\xi_{\Gamma}:\overline{\mathcal{M}}_{\Gamma}\to\overline{\mathcal{M}}_{g,r}$.

For simplicity, we will assume that $\xi_{\Gamma}$ and $\phi\circ\nu$ define
classes of complementary dimension in $\overline{\mathcal{M}}_{g,r}$ and ask
only for the intersection numbers; a more general discussion is developed in
[L21, §3.2].

The computation can be carried out applying the results of §4.3 directly
(after mild changes to allow for disconnected source curves), as we have

$\phi\circ\nu=\delta\circ\operatorname{res}^{S_{d}}_{S_{d-1}}.$

However, we will opt for a more streamlined approach working with the Harris-
Mumford admissible covers directly and invoking the results §4.3 implicitly;
the advantage will be that we will need to compute only a minimal amount of
data for the $S_{d}$-covers parametrized by $\widetilde{\mathcal{H}}_{g/h,d}$,
which are in general much more complicated than the Harris-Mumford covers.

Let

$\textstyle{\widetilde{\mathcal{B}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{\Gamma}}$$\textstyle{\widetilde{\mathcal{H}}_{g/h,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu}$$\textstyle{\mathcal{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{H}}_{g/h,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\overline{\mathcal{M}}_{\Gamma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{\Gamma}}$$\textstyle{\overline{\mathcal{M}}_{g,r}}$

be the fiber diagram resulting from the intersection of the desired classes.

Note, by the results of §4.3, that $\widetilde{B}$ has underlying reduced
space given by the disjoint union of boundary classes indexed by
$S_{d}$-admissible graphs, which is in particular smooth.

We will compute the intersection number of $\xi_{\Gamma}$ and $\phi\circ\nu$
as follows.

  1. (i)

Compute $\mathcal{B}$ set-theoretically as a disjoint union of boundary
classes on $\overline{\mathcal{H}}_{g/h,d}$, expressed in terms of dual graphs
and covers of curves of lower genus.

  2. (ii)

Compute the reduced degree (degree of the induced map of reduced spaces) of
$\nu_{\Gamma}$ over any point of $\mathcal{B}$; this is given by the number of
branches in the normalization of its complete local ring, as given by
Proposition 6.2.

  3. (iii)

Express the normal bundles of the underlying reduced components of the
boundary strata comprising $\widetilde{\mathcal{B}}^{\operatorname{red}}$ in
terms of $\psi$ classes on the dual graphs of the target curves parametrized
by $\widetilde{\mathcal{B}}$, and thus in terms of the target curves
parametrized by $\mathcal{B}$.

  4. (iv)

Compute the multiplicities of the components appearing in
$\widetilde{\mathcal{B}}$ in the tangent directions parametrized by the edges
of the dual graphs of target curves parametrized by $\mathcal{B}$.

  5. (v)

Combine the previous two steps to compute the Segre class of
$\widetilde{\mathcal{B}}$ in $\widetilde{\mathcal{H}}_{g/h,d}$, and apply the
excess intersection formula to express the desired intersection number in
terms of $\psi$ classes on $\widetilde{\mathcal{B}}^{\operatorname{red}}$.

  6. (vi)

Integrate the resulting classes on
$\widetilde{\mathcal{B}}^{\operatorname{red}}$ by using the fact that they are
pulled back from appropriate target maps
$\delta:\widetilde{\mathcal{B}}\to\overline{\mathcal{M}}_{h^{\prime},b^{\prime}}$,
which factor through $\mathcal{B}$. Because we have already computed the
reduced degree of $\nu_{\Gamma}$, it remains to compute the reduced degrees of
target maps out of $\mathcal{B}$, which are given by Hurwitz numbers.

We emphasize that all of these steps are carried out without computing
$\widetilde{\mathcal{B}}$ explicitly. Rather, we will express all of the
needed data in terms of the combinatorics of the covers appearing in
$\mathcal{B}$. We will see that the only data of the $S_{d}$-covers that we
will need to access is local; more specifically, we will only need to know the
ramification indices that appear.

For zero-dimensional strata of $\mathcal{B}$ and $\widetilde{B}$, the last
four steps simplify, as the intersection appears in the correct dimension.
Once we have the reduced degree of $\nu_{\Gamma}$ over each zero-dimensional
component of $\mathcal{B}$ and the multiplicity of the corresponding points of
$\widetilde{\mathcal{B}}$, we immediately have their contributions to the
intersection.

###  6.3. $d$-elliptic loci in genus 2

We now give an explicit calculation of the classes of Hurwitz loci in the case
$g=2,h=1$ using the method described in the previous section.

Let $\overline{\mathcal{H}}_{2/1,d}$ be the Harris-Mumford space parametrizing
simply ramified admissible covers $f:X\to Y$, where $X,Y$ are stable curves of
genus 2, 1, respectively. By definition, we will mark all $2d-2$ pre-images of
the two branch points of $f$.

Let
$\phi_{d}:\overline{\mathcal{H}}_{2/1,d}\to\overline{\mathcal{M}}_{2,2d-2}$
denote the source map. Let
$\phi^{\prime}_{d}:\overline{\mathcal{H}}_{2/1,d}\to\overline{\mathcal{M}}_{2}$
be the post-composition of $\phi$ with the map forgetting all marked points
(and contracting non-stable components).

The class of $\phi^{\prime}_{d}\in A^{1}(\overline{\mathcal{M}}_{2})$ is
determined by its intersections with the boundary classes
$\Delta_{00},\Delta_{01}\in A^{2}(\overline{\mathcal{M}}_{2})$ parametrizing
curves of the form shown in Figure 1, see [Mum83, Part III] or [L20, §2.2.3].

Figure 1. Boundary classes in $A^{2}(\overline{\mathcal{M}}_{2})$

Let $\Gamma_{00},\Gamma_{01}$ denote the corresponding dual graphs, so that
$\Delta_{0i}=(\xi_{\Gamma_{0i}})_{*}(1)$.

######  Theorem 6.3.

[L20, Theorem 1.3] We have:

$\displaystyle\int_{\overline{\mathcal{M}}_{2}}(\phi^{\prime}_{d})_{*}(1)\cdot\Delta_{00}$
$\displaystyle=4(d-2)!^{2}(d-1)\sigma_{1}(d)$
$\displaystyle\int_{\overline{\mathcal{M}}_{2}}(\phi^{\prime}_{d})_{*}(1)\cdot\Delta_{01}$
$\displaystyle=2(d-2)!^{2}\sum_{d_{1}+d_{2}=d}\sigma_{1}(d_{1})\sigma_{1}(d_{2}),$

where $\sigma_{1}(m)$ denotes the sum of the positive integer divisors of $m$.

In particular,

$\sum_{d\geq 1}\frac{1}{(d-2)!^{2}}\cdot[(\phi^{\prime}_{d})_{*}(1)]q^{d}\in
A^{1}(\overline{\mathcal{M}}_{2})\otimes\mathbb{Q}[[q]]$

is a cycle-valued quasimodular form.

The factors of $(d-2)!^{2}$ appearing here but not in [L20] come from our
marking of the additional $d-2$ points in each of the fibers over the branch
points of a $d$-elliptic cover.

In fact, while we do not give the details, we have been able to check, using
the same computation, the following mild generalization:

######  Theorem 6.4.

Let $\phi_{d,N}:\overline{\mathcal{H}}_{2/1,d}\to\overline{\mathcal{M}}_{2,N}$
be a system of maps obtained by forgetting all but a fixed subset of $N$
marked points for all $d$. Then,

$\sum_{d\geq
1}\frac{1}{\gamma_{d}}\left(\int_{\overline{\mathcal{M}}_{2,N}}(\phi_{d,N})_{*}(1)\cdot\alpha\right)q^{d}\in\mathbb{Q}[[q]]$

is a quasimodular form for any tautological class $\alpha\in
R^{2}(\overline{\mathcal{M}}_{2,N})$, where $\gamma_{d}$ is a combinatorial
factor counting relabelings of the forgotten points.

To prove Theorem 6.3, we follow the method outlined in §6.2, after pulling the
boundary strata of $\overline{\mathcal{M}}_{2}$ back to
$\overline{\mathcal{M}}_{2,2d-2}$. To carry out step (i), we will need a
complete description of all boundary strata of
$\overline{\mathcal{H}}_{2/1,d}$. This can be computed algorithmically, by
considering all possible topological types of the target curve, then all
possible ways in which the components can be covered, then glued.

We do not give full classification here, but rather show in Figures 2-6 the
strata of covers that will appear in our calculations. To label these strata,
we follow the notational convention for admissible covers of [L20], recording
the boundary strata in which the source and target, respectively, lie.

Figure 2. Covers of type $(\Delta_{00},\Delta_{01})$

Figure 3. Cover of type $(\Delta_{001},\Delta_{01})$

Figure 4. Cover of type $(\Delta_{000},\Delta_{00})$

Figure 5. Cover of type $(\Delta_{00},\Delta_{0})$

Figure 6. Cover of type $(\Delta_{000},\Delta_{0})$

####  6.3.1. Intersection with $\Delta_{01}$

See Figure 4. We consider the fiber diagram

$\textstyle{\widetilde{\mathcal{B}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{01}}$$\textstyle{\widetilde{\mathcal{H}}_{2/1,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{H}}_{2/1,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{d}}$$\textstyle{\coprod\overline{M}_{0,3+r}\times\overline{\mathcal{M}}_{1,2d-1-r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{M}}_{2,2d-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{0,3}\times\overline{\mathcal{M}}_{1,1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{01}}$$\textstyle{\overline{\mathcal{M}}_{2}}$

The coproduct is over all possible ways of distributing the $2d-2$ marked
points between the two components.

Following the method of §6.2, we begin by enumerating the points of
$\mathcal{B}$, which are all given by covers of type
$(\Delta_{001},\Delta_{01})$.

Over the nodal component of $Y$, we have an $m$-gon of rational curves whose
components each have degree $a$ over $Y$ and an $n$-gon of rational curves
whose components each have degree $b$ over $Y$; both components are ramified
only at the nodes. (Note that we allow one or both of $m,n$ to be equal to 1,
corresponding to a rational nodal component $Y$.) Over the smooth component of
$Y$, we have a rational bridge of degree 2, connecting the two polygons, along
with $d-2$ components mapping isomorphically.

We distinguish the two cycles of rational curves mapping to $Y$ by declaring
that the non-separating node corresponding to the self-edge of $\Gamma_{01}$
is to come from the $m$-gon. We have $m$ distinct choices for the node of
$x\in X$ corresponding to the self-edge $(h_{0},h^{\prime}_{0})$ of
$\Gamma_{01}$ – these are distinguished by the distance from edge along the
dual graph of $X$ to that connecting to the bridge of $X$, as measured in the
direction of $h$.

We also have 2 choices to identify the other edge $(h_{1},h^{\prime}_{1})$ of
$\Gamma_{01}$ with a node of $X$. We must then label the marked points of $X$;
this can be done in $(d-2)!^{2}$ ways, but we also have actions by $a$-th and
$b$-th roots of unity, respectively, on the non-distinguished components of
the cycles. Therefore, we find

$\\#\mathcal{B}(\mathbb{C})=\sum_{am+bn=d}2m\cdot\frac{(d-2)!^{2}}{a^{m-1}b^{n-1}}$

where we have weighted the count by automorphisms.

Next, we consider the degree of the map $\nu_{01}$ over a point of
$\mathcal{B}$ as above. The complete local ring of $[f:X\to Y]$ is isomorphic
to

$\mathbb{C}[[s_{1},\ldots,s_{m},t_{1},\ldots,t_{n},u]]/(s_{i}^{a}=t_{j}^{b}),$

where the $s_{i},t_{j}$ are smoothing parameters of the vertices of the
$m$-gon and $n$-gon, respectively, and $u$ is a smoothing parameter for the
separating nodes. The normalization will thus have $a^{m-1}b^{n-1}\gcd(a,b)$
branches. It follows that

$\\#\widetilde{\mathcal{B}}(\mathbb{C})=\sum_{am+bn=d}2m(d-2)!^{2}\gcd(a,b).$

Finally, we consider the multiplicities of the points of
$\widetilde{\mathcal{B}}$. Let $\widetilde{f}:\widetilde{X}\to Y$ be any point
of $\widetilde{H}_{2/1,d}$ lying over $[f:X\to
Y]\in\overline{\mathcal{H}}_{2/1,d}$. Then, the common ramification index of
the (possibly disconnected) $S_{d}$-cover $\widetilde{f}$ over the non-
separating node of $Y$ is $\operatorname{lcm}(a,b)$. On the other hand, the
ramification index of $f$ at the chosen node $x\in X$ corresponding to the
self-edge of $\Gamma_{01}$ is $a$. Therefore, the common ramification index
over $x$ in the $S_{d-1}$-cover $\widetilde{X}\to X$ is equal to
$\operatorname{lcm}(a,b)/a$.

The local computation of Lemma 4.15 shows, then, that
$[\widetilde{f}]\in\widetilde{\mathcal{B}}$ appears with this multiplicity.
Putting everything together, we obtain

$\displaystyle\int_{\overline{\mathcal{M}}_{2}}(\phi^{\prime}_{d})_{*}(1)\cdot\Delta_{01}$
$\displaystyle=\sum_{am+bn=d}2m(d-2)!^{2}\gcd(a,b)\cdot\frac{\operatorname{lcm}(a,b)}{a}$
$\displaystyle=2(d-2)!^{2}\sum_{am+bn=d}mb$
$\displaystyle=2(d-2)!^{2}\sum_{d_{1}+d_{2}=d}\sigma_{1}(d_{1})\sigma_{1}(d_{2}).$

####  6.3.2. Intersection with $\Delta_{00}$

We now consider the fiber diagram

$\textstyle{\widetilde{\mathcal{B}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nu_{00}}$$\textstyle{\widetilde{\mathcal{H}}_{2/1,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{\mathcal{H}}_{2/1,d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi_{d}}$$\textstyle{\overline{M}_{0,2d+2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi^{\prime}_{00}}$$\textstyle{\overline{\mathcal{M}}_{2,2d-2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\overline{M}_{0,4}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\xi_{00}}$$\textstyle{\overline{\mathcal{M}}_{2}}$

The space $\mathcal{B}$, and thus also $\widetilde{\mathcal{B}}$, consists of
four families of components, corresponding to the topological types
$(\Delta_{00},\Delta_{01})$, $(\Delta_{000},\Delta_{00})$,
$(\Delta_{00},\Delta_{0})$, and $(\Delta_{000},\Delta_{0})$. Note that, while
the remaining topological type, $(\Delta_{001},\Delta_{01})$, studied in the
previous section, has source curve in the locus $\Delta_{00}$, the chosen non-
separating nodes of $X$ map to the same node of $Y$, so upon pullback to
$\widetilde{\mathcal{B}}$, the genericity condition of Lemma 4.11 will be
violated. More concretely, such covers are specializations of those of type
$(\Delta_{00},\Delta_{0})$, so already appear on another of the enumerated
coomponents.

We will label the families of components of $\mathcal{B}$ and
$\widetilde{\mathcal{B}}$ with subscripts according to their topological type
(for example, $\mathcal{B}_{(\Delta_{00},\Delta_{01})}$ is the union of
components of $\mathcal{B}$ corresponding to covers of type
$(\Delta_{00},\Delta_{01})$). We first consider zero-dimensional components of
the intersection.

Type $(\Delta_{00},\Delta_{01})$. See Figure 2. We have a cycle of $m$
rational curves of degree $a$ over the nodal component of $Y$, a rational
bridge of degree 2 over the smooth component, and all other components mapping
isomorphically to the smooth component.

We first compute the number of points in
$\mathcal{B}_{(\Delta_{00},\Delta_{01})}$. Consider first the identification
of nodes of $X$ with edges of $\Gamma_{00}$. Note that the genericity
condition of Lemma 4.11 requires us to choose nodes over both of those of $Y$,
so one must come from the cycle of rational curves, and the other from one of
two attachment points of the rational bridge. We have two choices for which of
the edges corresponds to which type of node.

We begin with an $m$-gon of rational curves with a distinguished node
corresponding to the prescribed edge of $\Gamma_{00}$, and a fixed
orientation, determining an identification of half-edges with branches.

Next, consider the node of $X$ identified with the other edge of
$\Gamma_{00}$; we have $m$ distinct choices for the component of the $m$-gon
on which it lies, and two choices for the identification of half-edges with
branches. If the rational bridge connects two different components of the
$m$-gon, then there are $m-1$ choices for the other component, while if it
connects the chosen component to itself, there are $a-1$ choices for the
second attachment point.

We next have $(d-2)!^{2}$ labellings of the unramified marked points, but we
must correct for the cyclic actions on the rational components with only two
nodes, of which there are $m-2$ if the rational bridge connects two distinct
components and $m-1$ otherwise. We find:

$\\#\mathcal{B}_{(\Delta_{00},\Delta_{01})}(\mathbb{C})=\sum_{am=d}4(d-2)!^{2}\left[\frac{m(m-1)}{a^{m-2}}+\frac{(a-1)m}{a^{m-1}}\right]$

points, where as usual we weight the count by automorphisms.

A cover $f:X\to Y$ of type $(\Delta_{00},\Delta_{01})$ has complete local ring

$\mathbb{C}[[s_{1},\ldots,s_{m},t]]/(s_{1}^{a}=\cdots s_{m}^{a}),$

so

$\\#\widetilde{\mathcal{B}}_{(\Delta_{00},\Delta_{01})}(\mathbb{C})=\sum_{am=d}4(d-2)!^{2}\left[\frac{m(m-1)}{a^{m-2}}+\frac{(a-1)m}{a^{m-1}}\right]a^{m-1}$

Finally, note that if $[\widetilde{f}:\widetilde{X}\to Y]$ is a point of
$\widetilde{\mathcal{B}}$ over $[f]$, then $f$ and $\widetilde{f}$ will have
the same ramification indices of $a$ and 1 over the nodes of $Y$. Thus,
$\widetilde{X}\to X$ is unramified over the nodes, and in particular,
$\widetilde{\mathcal{B}}_{(\Delta_{00},\Delta_{01})}$ is reduced. We therefore
get a contribution from
$\\#\widetilde{\mathcal{B}}_{(\Delta_{00},\Delta_{01})}$ to the desired
intersection number of

$\displaystyle\quad\sum_{am=d}4(d-2)!^{2}\left[\frac{m(m-1)}{a^{m-2}}+\frac{(a-1)m}{a^{m-1}}\right]a^{m-1}$
$\displaystyle=\sum_{am=d}4(d-2)!^{2}[am(m-1)+(a-1)m]$
$\displaystyle=4(d-2)!^{2}\sum_{am=d}m(am-1)$
$\displaystyle=4(d-2)!^{2}(d-1)\sigma_{1}(d).$

Type $(\Delta_{000},\Delta_{00})$. See Figure 4.

######  Lemma 6.5.

Given $a,b\geq 1$, there is a unique degree $a+b$ cover
$g:\mathbb{P}^{1}\to\mathbb{P}^{1}$ that is totally ramified over 0, simply
ramified over $\infty$, and has ramification profile $(a,b)$ over 1. Here, we
consider the two points in $f^{-1}(1)$ to be distinguished if $a=b$.

###### Proof.

The number of such covers is given by the number of pairs
$(\sigma_{0},\sigma_{\infty})\in(S_{a+b})^{2}$ such that $\sigma_{0}$ is an
$(a+b)$-cycle, $\sigma_{\infty}$ is a transposition, and
$\sigma_{\infty}\sigma_{0}$ has cycle type $(a,b)$, up to simultaneous
conjugation. It is straightforward to check that the unique such choice is

$(\sigma_{0},\sigma_{\infty})=((12\cdots(a+b)),(1(a+1)),$

∎

We consider pointed admissible covers $f:X\to Y$, where $Y$ is a union of
rational curves $Y_{1},Y_{2}$ intersecting at two points $y_{+},y_{-}$, and
$X$ contains two rational curves $X_{1},X_{2}$ mapping to $Y_{1},Y_{2}$ in a
unique way as in Lemma 6.5. We take $y_{+}$ to be the node of $Y$ over which
the $X_{j}$ are totally ramified, and $Y_{j}$ to be the component containing
the $j$-th branch point of $f$. The components $X_{1}$ and $X_{2}$ are then
joined by three chains of rational curves totally ramified over the nodes of
$Y$. Note that intersecting components of $X$ must map to opposite components
of $Y$, and different nodes on components other than $X_{1},X_{2}$ must map to
different nodes of $Y$.

Suppose that the chains of rational curves with degrees $a+b,a,b$ over one of
the $Y_{j}$ contains $k,m,n$ pre-images, respectively, of $y_{+}$. Then, these
chains contain $k-1,m+1,n+1$ pre-images, respectively, of $y_{-}$. Here, we
must have $k\geq 1$ and $m,n\geq 0$. In order for the degree of $f$ to be
equal to $d$, we require $(a+b)k+am+bn=d$.

We first choose nodes of $X$ which will correspond to the edges of
$\Gamma_{00}$. By the usual genericity condition, one node each must be chosen
above the nodes $y_{\pm}$. First, suppose that the two nodes chosen have
ramification indices $a$ and $b$ over $Y$; we will distinguish the chains of
these ramification indices by identifying them with the two edges of
$\Gamma_{00}$.

Suppose first that the node of ramification index $a$ is chosen over $y_{+}$,
and that of ramification index $b$ is chosen over $y_{-}$. Then, there are
$m(n+1)$ ways to choose the two nodes and $2\cdot 2=4$ ways to identify their
branches with half-edges. There are $(d-2)!^{2}$ ways to label the unramified
marked points, and $f$ has an automorphism group of order
$(a+b)^{2k-2}a^{2m}b^{2n}$.

We compute the reduced degree of $\nu_{00}$ over $[f]$ to be

$\gcd(a,b)^{4}(a+b)^{2k-3}a^{2m-1}b^{2n-1}.$

and the intersection multiplicity at any point $[\widetilde{f}]$ over $[f]$ to
be

$\frac{\operatorname{lcm}(a+b,a,b)}{a}\cdot\frac{\operatorname{lcm}(a+b,a,b)}{b}=\frac{(a+b)^{2}a^{2}b^{2}}{ab\gcd(a,b)^{4}}.$

Combining, we get a total contribution of

$\displaystyle(d-2)!^{2}\sum_{k(a+b)+ma+nb=d}4m(n+1)\frac{\gcd(a,b)^{4}(a+b)^{2k-3}a^{2m-1}b^{2n-1}}{(a+b)^{2k-2}a^{2m}b^{2n}}\frac{(a+b)^{2}a^{2}b^{2}}{ab\gcd(a,b)^{4}}$
$\displaystyle=$ $\displaystyle(d-2)!^{2}\sum_{(a+b)k+am+bn=d}4m(n+1)(a+b).$

to the desired intersection number.

Similarly, if the nodes with ramification indices $a,b$ live over
$y_{-},y_{+}$, respectively, we get a contribution of

$\sum_{(a+b)k+am+bn=d}4(m+1)n(a+b).$

On the other hand, we may also choose one of our nodes to have ramification
index $a+b$. In this case, to distinguish $a$ and $b$, we declare that the
other chosen node has ramification index $a$. We then have two ways to
identify these nodes with the edges of $\Gamma_{00}$. Analogous computations
to the above show that we get a total contribution from these pointed covers
of

$(d-2)!^{2}\sum_{(a+b)k+am+bn=d}8[k(m+1)+(k-1)m]b.$

Combining everything, covers of type $(\Delta_{000},\Delta_{00})$ contribute

$(d-2)!^{2}\sum_{\begin{subarray}{c}(a+b)k+am+bn=d\\\ k\geq 1,m,n\geq
0\end{subarray}}[(8mn+4m+4n)(a+b)+(16km+8k-8m)b]$

to the intersection of $\phi_{d}$ and $\xi_{00}$.

Now, we consider one-dimensional components of the intersection; here will
need to implement the full program of §6.2.

Type $(\Delta_{00},\Delta_{0})$. See Figure 6.

######  Lemma 6.6.

Given $a,b\geq 1$, the number of degree $a+b$ covers
$g:\mathbb{P}^{1}\to\mathbb{P}^{1}$ simply ramified over two fixed points and
with ramification profile $(a,b)$ over another two fixed points is equal to
$2\max(a,b)$. Here, we consider the points in the latter two fibers
distinguished if $a=b$.

Equivalently, if $\overline{\mathcal{H}}_{(a,b)}$ denotes the Harris-Mumford
space of admissible covers of this type, the degree of the target map
$\delta_{(a,b)}:\overline{\mathcal{H}}_{(a,b)}\to\overline{M}_{0,4}$ is
$2\max(a,b)$.

###### Proof.

We degenerate the target to a union of two copies of $\mathbb{P}^{1}$ attached
at a node, so that each component contains one branch point of each type.
Without loss of generality, assume that $a\geq b$. There are two admissible
covers of the desired type over this target:

  * •

two covers of the type in Lemma 6.5, glued together at the ramification and
branch points of order $a+b$,

  * •

two covers of the type in Lemma 6.5, where the pair $(a,b)$ is replaced by
$(a-b,b)$, glued together at the ramification and branch points of order
$a-b$, along with one additional component covering each component of the
target via the map $x\mapsto x^{b}$.

On the Harris-Mumford stack $\overline{\mathcal{H}}_{(a,b)}$, the first cover
appears with multiplicity $a+b$ over the desired target. The second appears
with multiplicity $b^{2}(a-b)$, but the two components of the target with the
action by a cyclic group of order $b$ introduce a correction factor of
$1/b^{2}$. Summing these contributions yields the claim. ∎

  1. (i)

Suppose we have integers $a,m,b,n\geq 1$ with $am+bn=d$. Then,
$\overline{\mathcal{H}}_{(a,b)}$ gives rise to components of
$\mathcal{B}_{(\Delta_{00},\Delta_{0})}$ in the following way. Let $g:X_{0}\to
Y_{0}$ be a cover as in Lemma 6.6, where $X_{0}$ and $Y_{0}$ are allowed to be
singular. We attach chains of rational curves of lengths $m-1$ and $n-1$,
where $m,n\geq 1$, totally ramified over $Y_{0}$ of degrees $a$ and $b$, to
the two points each of $X_{0}$. Then, we glue the images of these nodes on
$Y_{0}$ to get the curve $Y$.

Suppose that we have an identification of the half-edges of $\Gamma_{00}$ with
branches of nodes of $X_{0}$; the pairs $(a,m)$ and $(b,n)$ may be
distinguished by this identification, and we get orientations on both chains.
The $\Gamma_{00}$-structure on $X$ gives us a map
$\overline{\mathcal{H}}_{(a,b)}\to\overline{M}_{0,2d+2}$ and identifies
$\overline{\mathcal{H}}_{(a,b)}$ with a component of
$\mathcal{B}^{\operatorname{red}}$.

For a given choice of $a,m,b,n$, the number of components of
$\mathcal{B}_{(\Delta_{00},\Delta_{0})}$ associated to
$\overline{\mathcal{H}}_{(a,b)}$ in the way we have described is

$2mn\cdot\frac{(d-2)!^{2}}{a^{m-1}b^{m-1}}.$

Here, the factor of 2 comes from a choice of identification of half-edges and
branches at one of the two edges of $\Gamma_{00}$; the two choices give
distinct components, because both nodes map to the same node of $Y$, so their
branches may be distinguished in pairs. The factor of $mn$ comes from choices
of nodes along each chain of rational curves, the factor of $(d-2)!^{2}$ comes
from choices of labels of unramified points, and the denominator comes from
automorphisms of $X$.

  2. (ii)

The reduced degree of $\nu_{00}$ is the number of branches in the
normalization over a general cover $[f:X\to Y]$ of type
$(\Delta_{00},\Delta_{0})$. The complete local ring of such a cover is of the
form

$\mathbb{C}[[s_{1},\ldots,s_{m},t_{1},\ldots,t_{n},v]]/(s_{i}^{a}=t_{j}^{b}),$

where the $s_{i}$ and $t_{j}$ are deformation parameters for the nodes of
ramification indices $a,b$, respectively, and the deformation parameter $v$
comes from that on the target $\overline{M}_{0,4}$. Therefore, the number of
branches upon normalization is $a^{m-1}b^{n-1}\gcd(a,b)$.

  3. (iii)

Fix $a,m,b,n$ as before, and let
$\widetilde{\mathcal{B}}_{(a,b)}\subset\widetilde{\mathcal{B}}_{(\Delta_{00},\Delta_{0})}$
be the union of the components lying over $\widetilde{B}$ identified with
$\overline{\mathcal{H}}_{(a,b)}$ as above. Then, a point of
$\widetilde{\mathcal{B}}_{(a,b)}$ is an $S_{d}$-cover constructed from Galois
covers $f_{0}:\widetilde{X}_{0}\to Y_{0}\cong\mathbb{P}^{1}$ ramified over
four points with common ramification indices
$2,2,\operatorname{lcm}(a,b),\operatorname{lcm}(a,b)$, by gluing together
copies of $\widetilde{X}_{0}$ at ramification points of order
$\operatorname{lcm}(a,b)$.

The Chern class of the normal bundle of
$\widetilde{\mathcal{B}}_{(a,b)}^{\operatorname{red}}$ in
$\widetilde{\mathcal{H}}_{2/1,d}$ is then given by $1-\psi$, where
$\psi=\psi_{h}+\psi_{h^{\prime}}$ is the sum of the $\psi$ classes on
$\widetilde{\mathcal{B}}_{(a,b)}^{\operatorname{red}}$ corresponding to the
$S_{d}$-orbits of branches with ramification order $\operatorname{lcm}(a,b)$.

Let
$\widetilde{\delta}_{(a,b)}:\widetilde{\mathcal{B}}_{(a,b)}^{\operatorname{red}}\to\overline{M}_{0,4}$
be the target map. Then, by Lemma 3.11, we have

$\psi=\frac{2}{\operatorname{lcm}(a,b)}\cdot\widetilde{\delta}_{(a,b)}^{*}\psi_{t},$

where $\psi_{t}$ is any of the rationally equivalent $\psi$ classes on
$\overline{M}_{0,4}$.

  4. (iv)

We claim that the multiplicity of $\widetilde{\mathcal{B}}_{(a,b)}$ is given
by $\operatorname{lcm}(a,b)/\max(a,b)$. Indeed, the map on complete local
rings at $[f:X\to Y]$ induced by
$\widetilde{\phi}^{\prime}:\widetilde{H}_{2/1,d}\to\overline{\mathcal{M}}_{2,2d-2}$
takes the deformation parameters corresponding to the two edges of
$\Gamma_{00}$ to $t^{\operatorname{lcm}(a,b)/a}$ and
$t^{\operatorname{lcm}(a,b)/b}$, where $t$ is the common deformation parameter
of the nodes on $\widetilde{\mathcal{H}}_{2/1,d}$. Thus, we get the claimed
multiplicity in the direction corresponding to the smoothing of the nodes.

  5. (v)

Analytically locally, $\widetilde{\mathcal{B}}_{(a,b)}$ is a Cartier divisor
in $\widetilde{\mathcal{H}}_{2/1,d}$ cut out by the
$\operatorname{lcm}(a,b)/\max(a,b)$-th power of an equation in the normal
direction corresponding to the smoothing of the nodes. It follows that

$s(\widetilde{\mathcal{B}}_{(a,b)},\widetilde{\mathcal{H}}_{2/1,d})|_{\widetilde{\mathcal{B}}_{(a,b)}^{\operatorname{red}}}=c(N_{\widetilde{\mathcal{B}}_{(a,b)}/\widetilde{\mathcal{H}}_{2/1,d}})^{-1}|_{\widetilde{\mathcal{B}}_{(a,b)}^{\operatorname{red}}}=\frac{\operatorname{lcm}(a,b)}{\max(a,b)}+\left(\frac{\operatorname{lcm}(a,b)}{\max(a,b)}\right)^{2}\psi.$

######  Remark 6.7.

In the general setting, a component of $\widetilde{\mathcal{B}}$ mapping to a
normalized Harris-Mumford space $\widetilde{\mathcal{H}}$ will always have the
local picture of the closed embedding

$\operatorname{Spec}\mathbb{C}[[t_{1},\ldots,t_{r}]]/(t_{1}^{c_{1}},\ldots,t_{r}^{c_{r}})\to\mathbb{C}[[t_{1},\ldots,t_{n}]],$

where $t_{1},\ldots,t_{r}$ correspond to deformation parameters at nodes of
the target curve $\widetilde{X}$. The Segre class of such an embedding in this
general situation may then be computed using the formula of [Alu13, §3.2].

By the excess intersection formula, the contribution of
$\widetilde{\mathcal{B}}_{(a,b)}$ to the desired intersection number is

$\int_{\mathcal{B}_{(a,b)}^{\operatorname{red}}}c(N_{\overline{M}_{0,2d+2}/\overline{\mathcal{M}}_{2,2d-2}})|_{\mathcal{B}_{\operatorname{red}}}\cdot
s(\mathcal{B},\mathcal{H}_{2/1,d})|_{\mathcal{B}_{(a,b)}^{\operatorname{red}}}.$

We have that $c(N_{\overline{M}_{0,2d+2}/\overline{\mathcal{M}}_{2,2d-2}})$ is
given by $\psi$ class contributions corresponding to the two chosen nodes of
$X$, with ramification indices $a$ and $b$. More precisely, by Lemma 3.11, we
have

$c(N_{\overline{M}_{0,2d+2}/\overline{\mathcal{M}}_{2,2d-2}})|_{\mathcal{B}_{(a,b)}^{\operatorname{red}}}=\left(1-\frac{\operatorname{lcm}(a,b)}{a}\psi\right)\left(1-\frac{\operatorname{lcm}(a,b)}{b}\psi\right)\\\
$

  6. (vi)

We are now ready to put everything together. Continuing from above, the
desired contribution is

$\displaystyle\int_{\mathcal{B}_{(a,b)}^{\operatorname{red}}}\left(1-\frac{\operatorname{lcm}(a,b)}{a}\psi\right)\left(1-\frac{\operatorname{lcm}(a,b)}{b}\psi\right)\left(\frac{\operatorname{lcm}(a,b)}{\max(a,b)}+\left(\frac{\operatorname{lcm}(a,b)}{\max(a,b)}\right)^{2}\psi\right)$
$\displaystyle=$
$\displaystyle\frac{\operatorname{lcm}(a,b)^{2}}{\max(a,b)}\cdot\left(-\frac{1}{a}-\frac{1}{b}+\frac{1}{\max(a,b)}\right)\int_{\mathcal{B}_{(a,b)}^{\operatorname{red}}}\psi$
$\displaystyle=-\frac{\operatorname{lcm}(a,b)^{2}}{\min(a,b)\max(a,b)}\cdot\int_{\mathcal{B}_{(a,b)}^{\operatorname{red}}}\psi$
$\displaystyle=-\frac{\operatorname{lcm}(a,b)}{\gcd(a,b)}\int_{\mathcal{B}_{(a,b)}^{\operatorname{red}}}\psi.$

On the other hand, we found in step (iii) that

$\displaystyle\int_{\mathcal{B}_{(a,b)}^{\operatorname{red}}}\psi$
$\displaystyle=\frac{2}{\operatorname{lcm}(a,b)}\int_{\mathcal{B}_{(a,b)}^{\operatorname{red}}}\widetilde{\delta}_{(a,b)}^{*}\psi_{t}$
$\displaystyle=\frac{2}{\operatorname{lcm}(a,b)}\cdot\deg(\widetilde{\delta}_{(a,b)})\int_{\overline{M}_{0,4}}\psi_{t}$
$\displaystyle=\frac{2}{\operatorname{lcm}(a,b)}\cdot\deg(\delta_{(a,b)})\cdot\deg(\nu_{00})\int_{\overline{M}_{0,4}}\psi_{t}$
$\displaystyle=\frac{2}{\operatorname{lcm}(a,b)}\cdot 2\max(a,b)\cdot
a^{m-1}b^{n-1}\gcd(a,b),$

where we by $\deg(\nu_{00})$ we mean the reduced degree we computed in step
(ii), that is, the degree of the induced map on reduced spaces, measured by
the number of branches of the normalization. Therefore, the total contribution
from $\mathcal{B}_{(a,b)}$ is

$-4\max(a,b)\cdot a^{m-1}b^{n-1}.$

Finally, we must multiply by the number of components of
$\mathcal{B}_{(\Delta_{00},\Delta_{0})}$ associated to $a,m,b,n$ as computed
in step (i), and sum over all $a,m,b,n$. This gives the final excess
contribution from covers of type $(\Delta_{00},\Delta_{0})$ of

$-(d-2)!^{2}\sum_{\begin{subarray}{c}am+bn=d\\\ m,n\geq
0\end{subarray}}8\max(a,b)mn$

Type $(\Delta_{000},\Delta_{0})$. See Figure 6.

We follow the same method as before, so we omit the details in this case. We
first assume that the $\Gamma_{00}$-structure on $X$ identifies two nodes of
ramification indices $a,b$ with the edges of $\Gamma_{00}$.

Fix integers $a,m,b,n,k\geq 1$.

  1. (i)

We form a cover $f:X\to Y$ of type $(\Delta_{000},\Delta_{0})$ by starting
with two (distinguishable) components $X_{1},X_{2}\subset X$ mapping to
$\mathbb{P}^{1}$ via a cover as in Lemma 6.5. Then, we connect these
components by three chains of lengths $k-1,m-1,n-1$, respectively, totally
ramified over the same $\mathbb{P}^{1}$. Finally, we glue the two images of
the nodes of $X$ on the target $\mathbb{P}^{1}$ to get $Y$.

In this way, we get a total of

$4mn\cdot\frac{(d-2)!^{2}}{a^{m-1}b^{n-1}(a+b)^{k-1}}$

components of $\mathcal{B}_{(\Delta_{000},\Delta_{0})}$.

  2. (ii)

Normalizing the complete local ring of $[f]$ as above shows that the reduced
degree of $\nu_{00}$ above such a cover is

$\gcd(a,b)^{2}a^{m-1}b^{n-1}(a+b)^{k-1}.$

  3. (iii)

If $\widetilde{\mathcal{B}}_{(a,b)}$ is the union of components of
$\widetilde{\mathcal{B}}_{(\Delta_{000},\Delta_{0})}$ lying over one of
$\mathcal{B}_{(\Delta_{000},\Delta_{0})}$ as above, then the covers
parametrized by $\widetilde{\mathcal{B}}_{(a,b)}$ are obtained by gluing
Galois covers of ramification indices
$2,2,\operatorname{lcm}(a,b,a+b),\operatorname{lcm}(a,b,a+b)$ in a similar way
to the case of covers of type $(\Delta_{00},\Delta_{0})$.

The Chern class of the normal bundle of
$\widetilde{\mathcal{B}}_{(a,b)}^{\operatorname{red}}$ is given by $1-\psi$,
where $\psi=\psi_{h}+\psi_{h^{\prime}}$ is the sum of $\psi$ classes on
$\widetilde{\mathcal{B}}_{(a,b)}^{\operatorname{red}}$ corresponding to the
$S_{d}$-orbits of branches with ramification order
$\operatorname{lcm}(a,b,a+b)=ab(a+b)/\gcd(a,b)$.

Let
$\widetilde{\delta}_{(a,b)}:\widetilde{\mathcal{B}}^{\operatorname{red}}_{(a,b)}\to\overline{M}_{0,4}$
be the target map. Then, we have

$\psi=\frac{2\gcd(a,b)}{ab(a+b)}\widetilde{\delta}_{(a,b)}^{*}\psi_{t}.$

  4. (iv)

The multiplicity of $\widetilde{\mathcal{B}}_{(a,b)}$ is given by

$\frac{\operatorname{lcm}(a,b)\cdot(a+b)}{\gcd(a,b)\cdot\max(a,b)},$

and occurs in the direction of smoothing of the nodes of an $S_{d}$-cover
$\widetilde{X}\to Y$ parametrized by $\widetilde{\mathcal{B}}_{(a,b)}$

  5. (v)

We have

$s(\widetilde{\mathcal{B}}_{(a,b)},\widetilde{\mathcal{H}}_{2/1,d})|_{\widetilde{\mathcal{B}}_{(a,b)}^{\operatorname{red}}}=\frac{\operatorname{lcm}(a,b)(a+b)}{\gcd(a,b)\max(a,b)}+\left(\frac{\operatorname{lcm}(a,b)(a+b)}{\gcd(a,b)\max(a,b)}\right)\psi$

and

$c(N_{\overline{M}_{0,2d+2}/\overline{\mathcal{M}}_{2,2d-2}})|_{\mathcal{B}_{(a,b)}^{\operatorname{red}}}=\left(1-\frac{b(a+b)}{\gcd(a,b)}\psi\right)\left(1-\frac{a(a+b)}{\gcd(a,b)}\psi\right)\\\
$

  6. (vi)

Applying the excess intersection formula as before and incorporating the above
data yields a total contribution of

$-8(d-2)!^{2}\cdot\sum_{\begin{subarray}{c}(a+b)k+am+bn\\\ k,m,n\geq
1\end{subarray}}(a+b)mn.$

Finally, we must also consider pointed curves where the
$\Gamma_{00}$-structure on $X$ identifies one of the nodes of ramification
index $a+b$ with an edge of $\Gamma_{00}$. Following the steps above, this
yields an additional contribution of

$-16(d-2)!^{2}\cdot\sum_{\begin{subarray}{c}(a+b)k+am+bn\\\ k,m,n\geq
1\end{subarray}}bkm.$

###  6.4. Proof of Theorem 6.3

We will need the following lemma. We thank David Yang for providing the proof.

######  Lemma 6.8.

We have

$\sum_{\begin{subarray}{c}am+bn=d\\\ a,b,m,n>0\end{subarray}}(mn-
am)\min(a,b)=0.$

and

$\sum_{\begin{subarray}{c}am+bn=d\\\ a,b,m,n>0\end{subarray}}(mn-
bn)\min(a,b)=0.$

###### Proof.

The two statements are clearly equivalent. We prove the first. We have

$\displaystyle\sum_{\begin{subarray}{c}am+bn=d\\\
a,b,m,n>0\end{subarray}}m(n-1)\min(a,b)$
$\displaystyle=\sum_{\begin{subarray}{c}am+bn=d\\\
a,b,m,n>0\end{subarray}}m(n-a)(\min(a,b)-b)$
$\displaystyle=\sum_{\begin{subarray}{c}am+bn=d\\\ a,b,m,n>0\\\
a<b\end{subarray}}m(n-a)(a-b).$

On the index set in the last sum, we have an involution

$(a,b,m,n)\mapsto(n,n+m,b-a,a)$

on which the terms being summed become $(b-a)(a-n)(-m)$, that is, they change
sign. The conclusion is now immediate. ∎

###### Proof of Theorem 6.3.

We have already obtained the intersection number with $\Delta_{01}$ in §6.3.1.
It remains to combine the above contributions to the intersection number with
$\Delta_{00}$. We have the desired intersection number coming from covers of
type $(\Delta_{00},\Delta_{01})$; it suffices to show now that the other three
contributions cancel. Ignoring the common factors of $(d-2)!^{2}$, we have

$\displaystyle-$ $\displaystyle 8\left(\sum_{\begin{subarray}{c}am+bn=d\\\
m,n\geq
1\end{subarray}}\max(a,b)mn\right)-8\left(\sum_{\begin{subarray}{c}(a+b)k+am+bn=d\\\
k,m,n\geq
1\end{subarray}}(a+b)mn\right)-16\left(\sum_{\begin{subarray}{c}(a+b)k+am+bn\\\
k,m,n\geq 1\end{subarray}}bkm\right)$
$\displaystyle+\sum_{\begin{subarray}{c}(a+b)k+am+bn=d\\\ k\geq 1,m,n\geq
0\end{subarray}}[(8mn+4m+4n)(a+b)+(16km+8k-8m)b]$ $\displaystyle=-$
$\displaystyle 8\left(\sum_{\begin{subarray}{c}am+bn=d\\\ m,n\geq
1\end{subarray}}\max(a,b)mn\right)+4\left(\sum_{\begin{subarray}{c}(a+b)k+am+bn=d\\\
k,m,n\geq 1\end{subarray}}[(m+n)(a+b)+2(k-m)b]\right)$
$\displaystyle+16\left(\sum_{\begin{subarray}{c}(a+b)k+am=d\\\ k\geq 1,m\geq
0\end{subarray}}bkm\right)$

where we have cancelled the second and third sums into the fourth, but with
extra terms left over remaining from allowing $n=0$ in the sum over $16bkm$.

Now, note that we can replace the sum over $2(k-m)b$ with the sum over
$(k-m)b+(k-n)a$, by symmetry. This yields simply

$-8\left(\sum_{\begin{subarray}{c}am+bn=d\\\ m,n\geq
1\end{subarray}}\max(a,b)mn\right)+4d\left(\sum_{\begin{subarray}{c}(a+b)k+am+bn=d\\\
k,m,n\geq
1\end{subarray}}1\right)+16\left(\sum_{\begin{subarray}{c}(a+b)k+am=d\\\ k\geq
1,m\geq 0\end{subarray}}bkm\right)$

We will now combine these sums by changing the index conditions on the second
and third terms also to be $am+bn=d$. We rewrite the condition in the second
sum as $a(k+m)+b(k+n)=d$, and change coordinates to get:

$-8\left(\sum_{\begin{subarray}{c}am+bn=d\\\ m,n\geq
1\end{subarray}}\max(a,b)mn\right)+4d\left(\sum_{\begin{subarray}{c}am+bn=d\\\
m,n\geq
1\end{subarray}}\min(m,n)\right)+16\left(\sum_{\begin{subarray}{c}(a+b)k+am=d\\\
k\geq 1,m\geq 0\end{subarray}}bkm\right)$

We rewrite the third sum as

$\displaystyle 16\left(\sum_{\begin{subarray}{c}(a+b)k+am=d\\\ k\geq 1,m\geq
0\end{subarray}}bkm\right)$
$\displaystyle=8\left(\sum_{\begin{subarray}{c}a(k+m)+bk=d\\\ k\geq 1,m\geq
0\end{subarray}}bkm\right)+8\left(\sum_{\begin{subarray}{c}ak+b(k+n)=d\\\
k\geq 1,n\geq 0\end{subarray}}akn\right)$
$\displaystyle=8\sum_{\begin{subarray}{c}am+bn=d\\\ m,n\geq
1\end{subarray}}\theta(\min(m,n))\min(m,n)|m-n|,$

where $\theta(m)=a$ and $\theta(n)=b$. On the other hand, combining back with
the second sum, note that

$4d\min(m,n)+8\theta(\min(m,n))\min(m,n)|m-n|=8mn(a+b)-4d\min(m,n).$

Therefore, putting everything together, we are left with

$\displaystyle\quad\sum_{\begin{subarray}{c}am+bn=d\\\ m,n\geq
1\end{subarray}}[-8\max(a,b)mn+8mn(a+b)-4d\min(m,n)]$
$\displaystyle=\sum_{\begin{subarray}{c}am+bn=d\\\ m,n\geq
1\end{subarray}}[8mn\min(a,b)-4(am+bn)\min(m,n)]$ $\displaystyle=0,$

by Lemma 6.8. The proof is complete. ∎

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