# On the motive of moduli stack of twisted $G$-covers

Massimo Bagnarol<EMAIL_ADDRESS>and  Fabio Perroni  Dipartimento
di Matematica e Geoscienze, Università degli Studi di Trieste, via Valerio
12/1, 34127 Trieste, Italy<EMAIL_ADDRESS>
(Date: September 3, 2024)

###### Abstract.

We describe the class, in the Grothendieck group of stacks, of the stack of
twisted $G$-covers of genus $0$ curves, in terms of the loci corresponding to
covers over smooth bases.

###### Key words and phrases:

Moduli spaces of covers; Grothendieck group of stacks.

##  1\. Introduction

Moduli stacks of twisted stable maps have been introduced in the framework of
Gromov-Witten theory for orbifolds and for Deligne-Mumford stacks,
independently by Chen-Ruan [CR02] and by Abramovich-Graber-Vistoli [AV02],
[AGV08]. In the case where the target is the classifying stack
${\mathcal{B}G}$ of a finite group $G$, a twisted stable map is called a
twisted $G$-cover [ACV03]. The moduli stack
${\mathcal{B}^{\mathrm{bal}}_{g,n}(G)}$ of balanced twisted stable maps to
${\mathcal{B}G}$ is isomorphic to the stack of admissible $G$-covers, which
have been extensively studied due to its several applications (see e.g.
[HM82], [PdJ95], [ACV03], [FG03], [JKK05]).

In this article we focus on the study of the geometry of these moduli stacks,
restricting our attention to the case where the source curves of the maps
(respectively the bases of the covers) are of genus zero and balanced. Our
main result (Theorem 5.4) gives a description of the motive of
$\mathcal{B}^{\mathrm{bal}}_{0,n}(G)$, i.e. of its class in the Grothendieck
group of stacks. This description is obtained stratifying the moduli stack as
the disjoint union of locally closed substacks $\mathcal{B}^{\rm
bal}(G,\tilde{\tau})$, which correspond to the $G$-covers of a given
combinatorial structure $\tilde{\tau}$, where $\tilde{\tau}$ is a gerby tree
(cf. Definition 4.4). The main technical step is Proposition 5.3, where we
show how these strata are obtained by gluing together the loci in
$\mathcal{B}^{\mathrm{bal}}_{0,n}(G)$ corresponding to covers over smooth
bases. In order to keep track of the actions of the symmetric groups on the
marked points, these gluing constructions correspond to two basic operations,
the Day convolution (Definition 2.2) and the composition (Definition 2.5). As
a result we obtain a formula (12) that computes the motive of
$\mathcal{B}^{\mathrm{bal}}_{0,n}(G)$ recursively in terms of the motives of
the loci $\mathcal{B}_{k}$, for $k\leq n$, corresponding to covers over smooth
bases (cf. Corollary 5.5). Here we follow the strategy of [GP06] (where the
authors compute the Betti numbers of the moduli spaces of stable maps to
projective spaces), further extended in [Bag22] to the case of moduli spaces
of stable maps to Grassmannians. Over the field of complex numbers, by
applying the Poincaré (or Serre) characteristic of [BD07], our formula allows
in principle to compute the Betti numbers of
$\mathcal{B}^{\mathrm{bal}}_{0,n}(G)$ in terms of those of $\mathcal{B}_{k}$,
for $k\leq n$. We will address the problem of computing effectively these
Betti numbers in future works.

The structure of the article is as follows. In Section 2 we define the Day
convolution and the composition, and we prove their main properties that will
be used. In Section 3, following the definition of the Grothendieck group of
stacks from [Eke09], we introduce the Grothendieck group of
$\mathsf{S}$-modules, over which there is an induced composition operation. In
Section 4 we define the notion of gerby tree $\tilde{\tau}$ (following
[AJT16]) and we show that the locus in $\mathcal{B}^{\mathrm{bal}}_{0,n}(G)$
of covers of a fixed combinatorial type $\tilde{\tau}$ is locally closed.
Finally, in Section 5, we prove our main results, Proposition 5.3 and Theorem
5.4.

### Acknowledgments

This article is dedicated to Lothar Göttsche, in occasion of his 60th
birthday, with esteem and admiration.

The second author was partially supported by the national project 2017SSNZAW
005-PE1 “Moduli Theory and Birational Classification”, by the research group
GNSAGA of INDAM and by FRA of the University of Trieste.

### Conventions

The symbol $\mathbb{N}$ denotes the set of nonnegative integers. For each
$n\in\mathbb{N}$, the symmetric group on $n$ letters is denoted by $S_{n}$.

We fix a universe, and we use the term “small” in reference to the chosen
universe. As usual, the category of small categories is denoted by
$\mathsf{Cat}$, the category of small sets is denoted by $\mathsf{Set}$, and
$\bm{1}$ denotes the terminal category. For all (enriched) categories
$\mathsf{C}$ and $\mathsf{D}$, $\mathsf{D}^{\mathsf{C}}$ denotes the
(enriched) category of (enriched) functors from $\mathsf{C}$ to $\mathsf{D}$.
A $2$-category is a $\mathsf{Cat}$-enriched category, which is sometimes
referred to as a strict $2$-category.

In order to avoid set-theoretic issues when considering toposes, by a stack
over a base scheme $T$ we mean a stack in groupoids over a suitable small sub-
site of the big fppf site of $T$ (constructed as in [AGV72, §IV.2.5]).

##  2\. The category of $\mathsf{S}$-modules over a base

Let $\Bbbk$ be a fixed algebraically closed field of characteristic $0$. By an
algebraic $\Bbbk$-stack we mean an algebraic stack of finite type over $\Bbbk$
which has affine stabilizers. The $2$-category of algebraic $\Bbbk$-stacks is
denoted by $\mathsf{AlgSt}_{\Bbbk}$, while $\mathsf{St}_{\Bbbk}$ denotes the
$2$-category of stacks over $\Bbbk$.

Let $G$ be a finite group. Our aim is to study the class of
$\mathcal{B}^{\mathrm{bal}}_{0,n}(G)$, the stack of balanced twisted
$G$-covers of $n$-pointed genus $0$ curves, in the Grothendieck ring of
algebraic $\Bbbk$-stacks $K_{0}(\mathsf{AlgSt}_{\Bbbk})$. For the definition
and further details about the stack $\mathcal{B}^{\mathrm{bal}}_{0,n}(G)$ we
refer to [ACV03].

In order to study the class of $\mathcal{B}^{\mathrm{bal}}_{0,n}(G)$ in
$K_{0}(\mathsf{AlgSt}_{\Bbbk})$ (actually, in a slightly different ring that
we will define later), we need to translate the combinatorics of its boundary
into relations within this ring. To this aim, we shall also take into account
the action of the symmetric group $S_{n}$ on
$\mathcal{B}^{\mathrm{bal}}_{0,n}(G)$.

An approach towards this direction can be recovered from [Pet13] and, more
generally, from the theory of colored operads. Let us recall the main
definitions and results, which we adapt to the setting we are interested in,
namely that of enriched categories. Since these results are effortlessly
reworked versions of well-known facts in the non-enriched case, we do not
present complete proofs, but we rather indicate which classical results they
are adapted from.

We consider categories enriched over a fixed cosmos
$(\mathscr{V},\otimes,\mathds{1})$, i.e., a symmetric monoidal closed category
which is bicomplete. For simplicity, we assume that the unit $\mathds{1}$ is
also the terminal object of $\mathscr{V}$. The initial object of $\mathscr{V}$
is denoted by $\varnothing$. By Mac Lane’s coherence theorem, we may also
assume that $(\mathscr{V},\otimes,\mathds{1})$ is a strict monoidal category,
i.e., the associator and the unitors are identity morphisms. Unless otherwise
stated, all categorical constructions (for instance, coends and Kan
extensions) will be carried out in the $\mathscr{V}$-enriched setting. Such
level of generality is needed because $\mathsf{St}_{\Bbbk}$ and
$\mathsf{AlgSt}_{\Bbbk}$ are $2$-categories, i.e., they are enriched over the
cosmos $(\mathsf{Cat},\times,\bm{1})$ of small categories. A reference for all
the tools of enriched category theory that we will use is [Kel05a].

Let $(\mathsf{E},\diamond,I)$ be a bicomplete symmetric monoidal closed
$\mathscr{V}$-category and $B$ an object of $\mathsf{E}$. Our guiding example
is given by

$(\mathscr{V},\otimes,\mathds{1})=(\mathsf{Cat},\times,\bm{1})\,,\quad(\mathsf{E},\diamond,I)=(\mathsf{St}_{\Bbbk},\times_{\Bbbk},\operatorname{Spec}\Bbbk)\,,\quad
B=\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})\,,$

where $\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})$ denotes the stack of
cyclotomic gerbes in ${\mathcal{B}G}$.

In the following $\mathsf{S}$ denotes the free $\mathscr{V}$-category on the
groupoid $\coprod_{n\in\mathbb{N}}S_{n}$ and
${\mathsf{S}}_{B}\colon{\mathsf{S}}\to\mathsf{E}$ is the functor that sends
$n\mapsto B^{n}$ and $\sigma\in S_{n}$ to the morphism that permutes the
factors, where $B^{n}$ denotes $B^{\diamond n}$.

######  Definition 2.1.

An $\mathsf{S}$-module over $B$ is a pair $(X,e)$, where
$X\colon{\mathsf{S}}\to{\mathsf{E}}$ is a functor and $e\colon
X\Rightarrow{\mathsf{S}}_{B}$ is a natural transformation.

The inclusion of $S_{n}\times S_{m}$ into $S_{n+m}$ as
$\operatorname{Aut}\\{1,\dots,n\\}\times\operatorname{Aut}\\{n+1,\dots,n+m\\}$
induces a tensor product $+$ for a $\mathscr{V}$-enriched symmetric monoidal
structure on $\mathsf{S}$ with unit $0$.

######  Definition 2.2 (Day convolution).

For all $\mathsf{S}$-modules $(X,e)$ and $(Y,\varepsilon)$ over $B$, the
convolution $X\ast Y$ is the functor ${\mathsf{S}}\to{\mathsf{E}}$ defined by
the coend

(1) $X\ast Y\colon
k\mapsto\int^{m,n}{\mathsf{S}}(m+n,k)\odot\bigl{(}X_{m}\diamond
Y_{n}\bigr{)}\,,$

where $\odot$ denotes the tensor product (also known as copower) in the
$\mathscr{V}$-enriched category $\mathsf{E}$. The universal property of coends
implies the existence of a natural transformation $e\ast\varepsilon\colon
X\ast Y\Rightarrow{\mathsf{S}}_{B}\ast{\mathsf{S}}_{B}$.

There is a $\mathscr{V}$-natural isomorphism

$X\ast Y\cong\operatorname{Lan}_{+}\bigl{(}\diamond(X\otimes Y)\bigr{)}\,,$

where $\operatorname{Lan}_{+}$ denotes the left Kan extension along
$+\colon{\mathsf{S}}\otimes{\mathsf{S}}\to{\mathsf{S}}$.

By adapting [Day70, §3] to this context, the following proposition is
obtained.

######  Proposition 2.3.

For all functors $X,Y,Z\colon{\mathsf{S}}\to{\mathsf{E}}$, there is a
$\mathscr{V}$-natural isomorphism $(X\ast Y)\ast Z\simeq X\ast(Y\ast Z)$.
Furthermore, if $(X,e),(Y,\varepsilon),(Z,{\mathfrak{e}})$ are
${\mathsf{S}}$-modules over $B$, then the resulting square

${(X\ast Y)\ast Z}$${X\ast(Y\ast
Z)}$${({\mathsf{S}}_{B}\ast{\mathsf{S}}_{B})\ast{\mathsf{S}}_{B}}$${{\mathsf{S}}_{B}\ast({\mathsf{S}}_{B}\ast{\mathsf{S}}_{B})}$$\scriptstyle{(e\ast\varepsilon)\ast{\mathfrak{e}}}$$\scriptstyle{e\ast(\varepsilon\ast{\mathfrak{e}})}$

is commutative.

In particular, for a fixed functor $Y\colon{\mathsf{S}}\to{\mathsf{E}}$, Day
convolution determines a $\mathscr{V}$-functor
${\mathsf{S}}\otimes{\mathsf{S}}^{\mathrm{op}}\to\mathsf{E}$ given by
$(m,n)\mapsto Y^{\ast n}(m)$.

######  Definition 2.4.

A rooted ${\mathsf{S}}$-module over $B$ is a triple
$(W,\varepsilon,\tilde{\varepsilon})$ such that $(W,\varepsilon)$ is an
${\mathsf{S}}$-module over $B$ and $\tilde{\varepsilon}\colon
W\Rightarrow\Delta_{B}$ is a natural transformation, where $\Delta_{B}$ is the
constant functor on $B$.

######  Definition 2.5 (Composition).

Given an ${\mathsf{S}}$-module over $B$, $(X,e)$, and a rooted
${\mathsf{S}}$-module over $B$, $(W,\varepsilon,\tilde{\varepsilon})$, the
composition product (or plethysm) $X\circ W$ is the $\mathscr{V}$-functor
$\mathsf{S}\to\mathsf{E}$ defined by

(2) $X\circ W\colon m\mapsto\int^{n}X_{n}\times_{B^{n}}W^{\ast n}(m)\,,$

where the morphism $W^{\ast n}(m)\to B^{n}$ is induced by
$\tilde{\varepsilon}$ as follows

$W^{\ast
n}(m)=\coprod_{\begin{subarray}{c}(m_{1},\ldots,m_{n})\in\mathbb{N}^{n}\\\
m_{1}+\ldots+m_{n}=m\end{subarray}}{\rm Sh}(m_{1},\ldots,m_{n})\odot
W_{m_{1}}\diamond\ldots\diamond W_{m_{n}}\to B^{n}\,,$

here ${\rm Sh}(m_{1},\ldots,m_{n})$ is the set of
$(m_{1},\ldots,m_{n})$-shuffles in $S_{m}$, that is the permutations
$\sigma\in S_{m}$ such that
$\sigma(1)<\ldots<\sigma(m_{1}),\sigma(m_{1}+1)<\ldots<\sigma(m_{1}+m_{2}),\ldots$.

Furthermore there is a morphism $X\circ W(m)\to B^{m}$ induced by
$\varepsilon$ that gives to $X\circ W$ a structure of ${\mathsf{S}}$-module
over $B$, it will be denoted $\varepsilon^{\prime}\colon X\circ
W\Rightarrow\mathsf{S}_{B}$.

######  Definition 2.6.

Let $(Y,e)$ be an ${\mathsf{S}}$-module over $B$ and let us suppose that $B$
has an involution $\iota$. We define $(DY,\varepsilon,\tilde{\varepsilon})$ to
be the rooted ${\mathsf{S}}$-module over $B$ such that $DY_{n}:=Y_{n+1}$ with
the action of $S_{n}$ seen as the permutations of $S_{n+1}$ that fix $n+1$,
$\varepsilon_{n}\colon DY_{n}\to B^{n}$ is given by the first $n$ components
of $e_{n+1}$ and $\tilde{\varepsilon}_{n}\colon DY_{n}\to B$ denotes the last
component of $e_{n+1}$ composed with $\iota$.

Let $B$ as in the previous definition with an involution $\iota$. By $I_{1}$
we denote the rooted ${\mathsf{S}}$-module over $B$ concentrated in arity one

$(I_{1})_{n}=\begin{cases}B&\text{if}\quad n=1\\\
\varnothing&\text{otherwise}\end{cases},$

where $\varepsilon_{1}=\operatorname{Id}_{B}$ and
$\tilde{\varepsilon}_{1}=\iota$.

For later use, we define $I_{2}$ to be the ${\mathsf{S}}$-module over $B$
concentrated in arity two

$(I_{2})_{n}=\begin{cases}B&\textit{if}\quad n=2\\\
\varnothing&\text{otherwise}\end{cases},$

where $e_{2}=(\operatorname{Id}_{B},\iota)$ and the symmetric group $S_{2}$
acts trivially.

From now on we restrict our attention to the case where
$(\mathscr{V},\otimes,\mathds{1})=(\mathsf{Cat},\times,\bm{1})$ and
$(\mathsf{E},\diamond,I)=(\mathsf{St}_{\Bbbk},\times_{\Bbbk},\operatorname{Spec}\Bbbk)$.

######  Proposition 2.7.

The $\mathscr{V}$-category of rooted ${\mathsf{S}}$-modules over $B$, together
with the composition product $\circ$ and the unit $I_{1}$, forms a monoidal
$\mathscr{V}$-category.

To prove the Proposition we follow Kelly’s arguments in [Kel05b, §3]. The next
Lemma is needed.

######  Lemma 2.8.

Let $(V,\delta,\tilde{\delta}),(W,\varepsilon,\tilde{\varepsilon})$ be rooted
${\mathsf{S}}$-modules over $B$. Then $(V\circ W)^{*m}$ is naturally
isomorphic to $V^{*m}\circ W$.

###### Proof.

We have

$\begin{split}(V\circ
W)^{*m}&=\int^{n_{1},\ldots,n_{m}}{\mathsf{S}}(n_{1}+\ldots+n_{m},\\_)\times(V\circ
W)_{n_{1}}\times\ldots\times(V\circ W)_{n_{m}}\\\
&\cong\int^{n_{i},k_{i}}{\mathsf{S}}(n_{1}+\ldots+n_{m},\\_)\times(V_{k_{1}}\times_{B^{k_{1}}}(W^{*k_{1}})_{n_{1}})\times\ldots\times(V_{k_{m}}\times_{B^{k_{m}}}(W^{*k_{m}})_{n_{m}})\\\
&\cong\int^{n_{i},k_{i}}(V_{k_{1}}\times\ldots\times
V_{k_{m}})\times_{B^{k_{1}+\ldots+k_{m}}}{\mathsf{S}}(n_{1}+\ldots+n_{m},\\_)\times(W^{*k_{1}})_{n_{1}}\times\ldots(W^{*k_{m}})_{n_{m}}\\\
&\cong\int^{k_{i}}(V_{k_{1}}\times\ldots\times
V_{k_{m}})\times_{B^{k_{1}+\ldots+k_{m}}}(W^{*k_{1}}*\ldots*W^{*k_{m}})\\\
&\cong\int^{k_{i}}(V_{k_{1}}\times\ldots\times
V_{k_{m}})\times_{B^{k_{1}+\ldots+k_{m}}}(W^{*k_{1}+\ldots+k_{m}})\\\
&\cong\int^{k_{i}}(V_{k_{1}}\times\ldots\times
V_{k_{m}})\times_{B^{k_{1}+\ldots+k_{m}}}\int^{t}{\mathsf{S}}(k_{1}+\ldots+k_{m},t)\times
W^{*t}\\\
&\cong\int^{k_{i},t}{\mathsf{S}}(k_{1}+\ldots+k_{m},t)\times(V_{k_{1}}\times\ldots\times
V_{k_{m}})\times_{B^{k_{1}+\ldots+k_{m}}}W^{*t}\\\
&\cong\int^{t}(V^{*m})_{t}\times_{B^{t}}W^{*t}\\\ \end{split}$

where in the second and sixth isomorphisms we have used the fact that

$(S_{1}\times_{B^{k_{1}}}R_{1})\times(S_{2}\times_{B^{k_{2}}}R_{2})\cong(S_{1}\times
S_{2})\times_{B^{k_{1}+k_{2}}}(R_{1}\times R_{2})\,,$

for $R_{i},S_{i}$ be $B^{k_{i}}$-modules; in the third isomorphism we have
used the definition of $*$ and the property that coends commute with fiber
products in the case of stacks; the fifth isomorphism follows from Yoneda’s
Lemma and the last one follows from the definition of Day convolution. ∎

###### Proof.

(Of Proposition 2.7) Let
$(X,e,\tilde{e}),(Y,\varepsilon,\tilde{\varepsilon}),(Z,{\mathfrak{e}},\tilde{\mathfrak{e}})$
be rooted ${\mathsf{S}}$-modules over $B$. Then

$\begin{split}X\circ(Y\circ Z)&=\int^{m}X_{m}\times_{B^{m}}(Y\circ Z)^{*m}\\\
&\cong\int^{m}X_{m}\times_{B^{m}}(Y^{*m}\circ Z)\qquad\text{(by Lemma
\ref{lemmacompositionsymmon})}\\\
&\cong\int^{m,k}X_{m}\times_{B^{m}}(Y^{*m}_{k}\times_{B^{k}}Z^{*k})\qquad\text{(by
definition of composition)}\\\ &\cong\int^{k}(X\circ
Y)_{k}\times_{B^{k}}Z^{*k}\\\ &=(X\circ Y)\circ Z\,.\end{split}$

∎

######  Remark 2.9.

For any $Y$, $(-)\circ Y$ has a right adjoint [Kel05b, §3]. In general
$X\circ(-)$ does not have a right adjoint, but it does when $X$ is
concentrated in arity one [DH19, Lemma 2.9].

######  Remark 2.10.

If we replace $\mathsf{St}_{\Bbbk}$ by $\mathsf{AlgSt}_{\Bbbk}$ as the target
$2$-category, the composition product is no longer well defined since
$\mathsf{AlgSt}_{\Bbbk}$ is not bi-complete. However the composition is well
defined if $Y_{0}=\emptyset$. The reason is that the only colimits involved
turn out to be finite coproducts and quotients by finite group actions.
Proposition 2.7 and the full faithfulness of the embedding of
$\mathsf{AlgSt}_{\Bbbk}$ into $\mathsf{St}_{\Bbbk}$ then imply that the
composition product on the $2$-category of rooted ${\mathsf{S}}$-modules (in
$\mathsf{AlgSt}_{\Bbbk}$) over $B$ is $2$-naturally associative (whenever it
is defined) with unit $I_{1}$.

######  Example 2.11.

Let us finally spell out the example of $\mathsf{S}$-module over $B$ we are
interested in, which serves as a motivation for the theory we are presenting.
Here $B$ is the stack $\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})$ of
cyclotomic gerbes in ${\mathcal{B}G}$ defined in [AGV08, Definition 3.3.6].

As $n$ varies, the stacks $\mathcal{B}^{\mathrm{bal}}_{0,n}(G)$ together with
the evaluation map ${\rm ev}_{n}\colon\mathcal{B}^{\mathrm{bal}}_{0,n}(G)\to
B^{n}$ yield an $\mathsf{S}$-module over $B$ in $\mathsf{AlgSt}_{\Bbbk}$,
which we denote by $(\overline{\mathcal{B}},\overline{e})$ and we refer to it
as the $\mathsf{S}$-module of balanced twisted $G$-covers over genus $0$
curves.

##  3\. Grothendieck groups and their composition structure

Let us recall the definition of $K_{0}(\mathsf{AlgSt}_{\Bbbk})$ from [Eke09].
Note that there are different versions in the literature (see [BD07, Joy07,
Toë09]), all of which are mapped to by $K_{0}(\mathsf{AlgSt}_{\Bbbk})$.

######  Definition 3.1.

The Grothendieck group $K_{0}(\mathsf{AlgSt}_{\Bbbk})$ is the abelian group
generated by the isomorphism classes $\\{\mathcal{X}\\}$ of algebraic
$\Bbbk$-stacks subject to the relations

  1. (i)

$\\{\mathcal{X}\\}=\\{\mathcal{Y}\\}+\\{\mathcal{X}\smallsetminus\mathcal{Y}\\}$
if $\mathcal{Y}$ is a closed substack of $\mathcal{X}$, and

  2. (ii)

$\\{\mathcal{E}\\}=\\{\mathbb{A}^{r}\times_{\Bbbk}\mathcal{X}\\}$ if
$\mathcal{E}$ is a vector bundle of rank $r$ on $\mathcal{X}$.

The group $K_{0}(\mathsf{AlgSt}_{\Bbbk})$ has in fact a commutative ring
structure, given on generators by
$\\{\mathcal{X}\\}\\{\mathcal{Y}\\}=\\{\mathcal{X}\times_{\Bbbk}\mathcal{Y}\\}$.

######  Definition 3.2.

The Grothendieck group $K_{0}^{S_{n}}({\mathsf{AlgSt}_{\Bbbk}})$ is the
abelian group generated by the isomorphism classes $\\{\mathcal{X}\\}$ of
algebraic $\Bbbk$-stacks with an action of $S_{n}$ subject to the relations

  1. (i)

$\\{\mathcal{X}\\}=\\{\mathcal{Y}\\}+\\{\mathcal{X}\smallsetminus\mathcal{Y}\\}$
if $\mathcal{Y}$ is a closed substack of $\mathcal{X}$ invariant under the
action of $S_{n}$, and

  2. (ii)

$\\{\mathcal{E}\\}=\\{\mathbb{A}^{r}\times_{\Bbbk}\mathcal{X}\\}$ if
$\pi\colon\mathcal{E}\to\mathcal{X}$ is an $S_{n}$-equivariant vector bundle
of rank $r$ on $\mathcal{X}$, where the $S_{n}$-action on
$\mathbb{A}^{r}\times_{\Bbbk}\mathcal{X}$ is the extension of the given one on
$\mathcal{X}$ by the trivial action.

For our purposes we need to define the Grothendieck group associated to the
category of $\mathsf{S}$-modules over $B$, where
$(\mathsf{E},\diamond,I)=(\mathsf{AlgSt}_{\Bbbk},\times_{\Bbbk},\operatorname{Spec}\Bbbk)$
(we change the target category according to Remark 2.10). First of all, let us
define a morphism $F\colon(X,e)\to(X^{\prime},e^{\prime})$ (respectively an
isomorphism) of $\mathsf{S}$-modules over $B$ to be a natural transformation
$F\colon X\Rightarrow X^{\prime}$ (respectively a natural isomorphism) such
that the diagram

${X}$${X^{\prime}}$${{\mathsf{S}}_{B}}$$\scriptstyle{e}$$\scriptstyle{F}$$\scriptstyle{e^{\prime}}$

is commutative. As usual we say that two $\mathsf{S}$-modules over $B$,
$(X,e)$ and $(X^{\prime},e^{\prime})$, are isomorphic if there exists an
isomorphism $F\colon(X,e)\to(X^{\prime},e^{\prime})$ and we denote with
$\\{X,e\\}$ the isomorphism class of $(X,e)$. By a vector bundle in the
category of $\mathsf{S}$-modules over $B$ we mean a morphism
$\pi\colon(\mathcal{E},e^{\prime})\to(X,e)$ such that: for every $n$,
$\pi_{n}\colon\mathcal{E}_{n}\to X_{n}$ is a vector bundle (of some rank
$r(n)$) and it is $S_{n}$-equivariant.

######  Definition 3.3.

The Grothendieck group $K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}},B)$ is the
abelian group generated by the isomorphism classes $\\{X,e\\}$, of
$\mathsf{S}$-modules over $B$ in
$(\mathsf{AlgSt}_{\Bbbk},\times_{\Bbbk},\operatorname{Spec}\Bbbk)$, subject to
the relations

  1. (i)

$\\{X,e\\}=\\{Y,e_{|}\\}+\\{(X,e)\smallsetminus(Y,e_{|})\\}$, if $(Y,e_{|})$
is a closed sub-$\mathsf{S}$-module of $(X,e)$, and

  2. (ii)

$\\{\mathcal{E},e^{\prime}\\}=\\{(\mathbb{A}^{r(n)}\times_{\Bbbk}X_{n},e_{n})_{n}\\}$,
if $\pi\colon(\mathcal{E},e^{\prime})\to(X,e)$ is a vector bundle, where the
$S_{n}$-action on $\mathbb{A}^{r(n)}\times_{\Bbbk}X_{n}$ is the extension of
the given one on $X_{n}$ by the trivial action.

The Grothendieck group $K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}},B,r)$
associated to the category of rooted $\mathsf{S}$-modules over $B$ is defined
similarly and we omit the details.

Notice that $K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}},B)$ and
$K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}},B,r)$ have natural structures of
$K_{0}(\mathsf{AlgSt}_{\Bbbk})$-algebras induced by cartesian product.

######  Notation 1.

In the following we will consider also $\mathsf{S}$-modules over
$\operatorname{Spec}(\Bbbk)$, $X$, where the natural transformation
$X\Rightarrow\mathsf{S}_{\operatorname{Spec}(\Bbbk)}$ is the one induced by
the structure morphisms $X_{n}\to\operatorname{Spec}(\Bbbk)$, for all $n$, and
hence we omit it. The corresponding Grothendieck group
$K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}},\operatorname{Spec}(\Bbbk))$ will
be denoted $K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}})$.

For any $\Bbbk$-stack $B$ there is a morphism of
$K_{0}(\mathsf{AlgSt}_{\Bbbk})$-algebras

$K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}},B)\to
K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}})$

that is induced by associating any $\mathsf{S}$-modules over $B$, $(X,e)$,
with $X$. We will say that two elements $a,b\in
K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}},B)$ are congruent modulo $B$, in
symbol $a\equiv_{B}b$, if they have the same image under this morphism.

Let $(X,e)$ be an ${\mathsf{S}}$-module over $B$, and let
$(W,\varepsilon,\tilde{\varepsilon})$ be a rooted ${\mathsf{S}}$-module over
$B$ such that $W_{0}=\emptyset$. We observe that the class $\\{X\circ
W,\varepsilon^{\prime}\\}$ of the composition product (see Definition 2.5 and
Remark 2.10) depends only from the classes $\\{X,e\\}\in
K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}},B)$ and
$\\{W,\varepsilon,\tilde{\varepsilon}\\}\in
K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}},B,r)$. Furthermore we have the
following result.

######  Proposition 3.4.

Let $(K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}},B,r))_{1}$ be the sub-
algebra generated by the classes $\\{W,\varepsilon,\tilde{\varepsilon}\\}$ of
rooted $\mathsf{S}$-modules over $B$ with $W_{0}=\emptyset$. Then the
composition product descends to an operation

$\circ\colon
K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}},B)\times(K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}},B,r))_{1}\to
K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}},B)\,,$

which is $K_{0}(\mathsf{AlgSt}_{\Bbbk})$-linear in the first argument and such
that $\\{X,e\\}\circ\\{W,\varepsilon,\tilde{\varepsilon}\\}=\\{X\circ
W,\varepsilon^{\prime}\\}$, for any $(X,e)$,
$(W,\varepsilon,\tilde{\varepsilon})$ respectively $\mathsf{S}$-module and
rooted $\mathsf{S}$-module, over $B$.

In the case of varieties, this is part of a theorem announced in [GP06], but a
complete proof is given in [Bag19, Prop. 1.3.6]. Since the same arguments
apply in the present setting we refer to [Bag19] for the proof.

##  4\. Stratification

Let $G$ be a finite group and let $\mathcal{B}^{\rm bal}_{g,n}(G)$ be the
stack of balanced twisted $G$-covers of $n$-pointed curves of genus $g$
[ACV03]. Let us recall that $\mathcal{B}^{\rm bal}_{g,n}(G)$ is defined as the
stack of balanced twisted stable maps $\mathcal{K}^{\rm
bal}_{g,n}(\mathcal{B}G,0)$ (where $\mathcal{B}G$ is the classifying stack of
$G$), furthermore it is isomorphic to the stack of admissible $G$-covers
$\mathcal{A}dm_{g,n}(G)$ (see [ACV03], and Theorem 4.3.2 therein). Our aim is
to describe the class of $\mathcal{B}^{\rm bal}_{0,n}(G)$ in
$K_{0}(\mathsf{AlgSt}_{\Bbbk})$, in a suitable way. To this aim we first
introduce a stratification of $\mathcal{B}^{\rm bal}_{0,n}(G)$ according to
the combinatorial type of the covers. This will be achieved using the notion
of gerby tree associated to $\mathcal{B}G$.

Let us first recall the following definitions.

######  Definition 4.1 ([BM96]).

A graph $\tau$ is a triple $(F_{\tau},j_{\tau},R_{\tau})$, where

  * •

$F_{\tau}$ is a finite set, called the set of flags of $\tau$,

  * •

$j_{\tau}\colon F_{\tau}\to F_{\tau}$ is an involution, and

  * •

$R_{\tau}\subset F_{\tau}\times F_{\tau}$ is an equivalence relation.

Associated to every graph $\tau=(F_{\tau},j_{\tau},R_{\tau})$ there is a
triple of sets $(V_{\tau},E_{\tau},L_{\tau})$, which are defined as follows:

  * •

$V_{\tau}$ is the quotient of $F_{\tau}$ by $R_{\tau}$, it is called the set
of vertices of $\tau$,

  * •

$E_{\tau}$ is the set of orbits of $j_{\tau}$ with two elements, it is called
the set of edges of $\tau$,

  * •

$L_{\tau}$ is the set of fixed points of $j_{\tau}$, it is called the set of
leaves of $\tau$.

Finally, we associate to every vertex $v\in V_{\tau}$

  1. 1.

the subset $F_{\tau}(v)\subset F_{\tau}$ of flags whose equivalence class
under $R_{\tau}$ is $v$, and

  2. 2.

the valence $n(v):=\\#F_{\tau}(v)$ of $v$.

######  Definition 4.2.

The geometric realization $|\tau|$ of a graph $\tau$ is the topological space
constructed as follows. Let us consider the disjoint union

$T=\bigsqcup_{i\in L_{\tau}}[0_{i},{1/2}_{i})\sqcup\bigsqcup_{i\in
F_{\tau}\setminus L_{\tau}}[0_{i},{1/2}_{i}]\sqcup\bigsqcup_{v\in
V_{\tau}}\\{|v|\\}\,.$

For each $v\in V_{\tau}$, identify the elements of the set $\\{0_{i}\,|\,i\in
F_{\tau}(v)\\}$ with $|v|$. For each edge $\\{i_{1},i_{2}\\}\in E_{\tau}$
identify ${1/2}_{i_{1}}$ with ${1/2}_{i_{2}}$. In this way one obtains a set
$|\tau|$ together with a surjection $T\to|\tau|$. Then the topology on
$|\tau|$ is the quotient topology.

######  Definition 4.3.

A tree is a graph $\tau$ such that its geometric realization is simply
connected, i.e. $|\tau|$ is connected and $\\#V_{\tau}=\\#E_{\tau}+1$.

An $n$-tree is a tree $\tau$ together with a bijection $l_{\tau}\colon
L_{\tau}\to\\{1,\ldots,n\\}$.

A ($n$-)tree $\tau$ is stable if $n(v)\geq 3$, for any $v\in V_{\tau}$.

Let $\bar{\mathcal{I}}_{\mu}(\mathcal{B}G)$ be the stack of cyclotomic gerbes
in $\mathcal{B}G$ ([AGV08, Definition 3.3.6]) and let
$\bar{I}_{\mu}(\mathcal{B}G)$ be its coarse moduli space. Let us recall that
$\bar{\mathcal{I}}_{\mu}(\mathcal{B}G)=\sqcup_{r}\bar{\mathcal{I}}_{\mu_{r}}(\mathcal{B}G)$,
where $\bar{\mathcal{I}}_{\mu_{r}}(\mathcal{B}G)$ parametrizes pairs
$(\mathcal{G},\phi)$ of gerbes $\mathcal{G}$ banded by $\mu_{r}$ and
$\phi\colon\mathcal{G}\to\mathcal{B}G$ is a representable morphism.
Furthermore, by [AGV08, Prop. 3.4.1]
${\overline{\mathcal{I}}_{\bm{\mu}_{r}}({\mathcal{B}G})}$ is the
rigidification of the cyclotomic inertia stack
${{\mathcal{I}}_{\bm{\mu}_{r}}({\mathcal{B}G})}$ with respect to
$\bm{\mu}_{r}$, in particular $\bar{I}_{\mu}(\mathcal{B}G)$ can be canonically
identified with the set of conjugacy classes of $G$. Following [AGV08, §3.5]
we denote with
$\iota:\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})\to\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})$
the involution induced by the automorphisms $\mu_{r}\to\mu_{r}$,
$\zeta\mapsto\zeta^{-1}$, for $r\geq 1$.

The following definition is similar to [AJT16, Definition 2.14].

######  Definition 4.4.

A gerby tree $\tilde{\tau}$ associated to $\mathcal{B}G$ is given by a tree
$\tau=(F_{\tau},j_{\tau},R_{\tau})$ together with a map $\mathfrak{m}\colon
F_{\tau}\to\bar{I}_{\mu}(\mathcal{B}G)$ such that
$\mathfrak{m}(i)=\iota\circ\mathfrak{m}(i^{\prime})$, whenever the flags
$i,i^{\prime}\in F_{\tau}$ form an edge $\\{i,i^{\prime}\\}\in E_{\tau}$. We
will denote with $\mathfrak{o}\colon F_{\tau}\to\mathbb{Z}$ the function that
associates to any flag $i\in F_{\tau}$ the integer $r\in\mathbb{Z}$ such that
$\mathfrak{m}(i)\in\bar{I}_{\mu_{r}}(\mathcal{B}G)$.

A gerby $n$-tree is a gerby tree $\tilde{\tau}$ together with a bijection
$l_{\tau}\colon L_{\tau}\to\\{1,\ldots,n\\}$.

A gerby ($n$-)tree $\tilde{\tau}$ is stable, if the underlying ($n$-)tree is
stable.

######  Definition 4.5.

An isomorphism of graphs $\varphi\colon\tau\to\sigma$ is a bijective map
$\varphi_{F}\colon F_{\tau}\to F_{\sigma}$ such that $\varphi_{F}\circ
j_{\tau}=j_{\sigma}\circ\varphi_{F}$ and
$(\varphi_{F}\times\varphi_{F})(R_{\tau})=R_{\sigma}$. In particular
$\varphi_{F}$ induces a bijection $\varphi_{V}\colon V_{\tau}\to V_{\sigma}$
commuting with the projections and a bijection $\varphi_{L}\colon L_{\tau}\to
L_{\sigma}$.
An isomorphism of $n$-trees
$\varphi\colon(\tau,l_{\tau})\to(\sigma,l_{\sigma})$ is an isomorphism of the
underlying trees $\varphi\colon\tau\to\sigma$ such that
$l_{\tau}=l_{\sigma}\circ\varphi_{F}$. The automorphism group of a tree $\tau$
(resp. an $n$-tree $(\tau,l_{\tau})$) will be denoted ${\rm Aut}(\tau)$ (resp.
${\rm Aut}(\tau,l_{\tau})$).

An isomorphism of gerby trees (resp. $n$-trees)
$\tilde{\varphi}\colon\tilde{\tau}\to\tilde{\sigma}$ is an isomorphism of the
underlying trees (resp. $n$-trees) $\varphi\colon\tau\to\sigma$ such that
$\mathfrak{m}_{\sigma}\circ\varphi_{F}=\mathfrak{m}_{\tau}$. The automorphism
group of a gerby ($n$-)tree $\tilde{\tau}$ will be denoted ${\rm
Aut}(\tilde{\tau})$.

Given a gerby tree $\tilde{\tau}$, there are notions of $\tilde{\tau}$-marked
pre-stable curves and of $\tilde{\tau}$-marked twisted stable maps to
$\mathcal{B}G$ ([AJT16], Definition 2.18 and 2.20). Let us recall their
definitions.

######  Definition 4.6.

A $\tilde{\tau}$-marked pre-stable curve over a scheme $T$ is a triple
$(\mathcal{C},\Sigma,\xi)$, where

  * •

$\mathcal{C}=(\mathcal{C}_{v})_{v\in V_{\tilde{\tau}}}$ is a collection of
twisted curves of genus $0$ over $T$,

  * •

$\Sigma=(\Sigma_{i})_{i\in F_{\tilde{\tau}}}$ is a collection of disjoint
closed sub-stacks $\Sigma_{i}\subset\mathcal{C}_{v}$, where $i\in
F_{\tilde{\tau}}(v)$,

  * •

$\xi=(\xi_{\\{i,i^{\prime}\\}})_{\\{i,i^{\prime}\\}\in E_{\tilde{\tau}}}$ is a
collection of isomorphisms
$\xi_{\\{i,i^{\prime}\\}}\colon\Sigma_{i}\to\Sigma_{i^{\prime}}$.

Furthermore the previous data is required to satisfy the following conditions:

  1. (1)

for any $i\in F_{\tilde{\tau}}$, $\Sigma_{i}$ is contained in the smooth locus
of $\mathcal{C}_{v}\to T$;

  2. (2)

for any $i\in F_{\tilde{\tau}}$, the composition
$\Sigma_{i}\subset\mathcal{C}_{v}\to T$ is an étale gerbe $\Sigma_{i}\to T$
banded by $\mu_{r}$, where $r=\mathfrak{o}(i)$;

  3. (3)

for any $\\{i,i^{\prime}\\}\in E_{\tilde{\tau}}$, $\xi_{\\{i,i^{\prime}\\}}$
is an isomorphism over $T$ inverting the band.

######  Definition 4.7.

Let $\tilde{\tau}$ be a stable gerby tree. A $\tilde{\tau}$-marked twisted
stable map to $\mathcal{B}G$ over a scheme $T$ is given by:

  * •

a $\tilde{\tau}$-marked pre-stable curve $(\mathcal{C},\Sigma,\xi)$;

  * •

a collection $(f_{v})_{v\in V_{\tilde{\tau}}}$ of morphisms
$f_{v}\colon\mathcal{C}_{v}\to\mathcal{B}G$;

  * •

for any $\\{i,i^{\prime}\\}\in E_{\tilde{\tau}}$, $2$-isomorphisms
$f_{\\{i,i^{\prime}\\}}\colon{f_{v}}_{|\Sigma_{i}}\Rightarrow{f_{v^{\prime}}}_{|\Sigma_{i^{\prime}}}\circ\xi_{\\{i,i^{\prime}\\}}$,
where $i$ is in the class $v$ and $i^{\prime}$ is in $v^{\prime}$.

Moreover the following condition is required:

  1. for any $i\in F_{\tilde{\tau}}$, ${f_{v}}_{|\Sigma_{i}}$ is an object of ${\bar{\mathcal{I}}}_{\mu}(\mathcal{B}G)(T)$ over the component indexed by $\mathfrak{m}(i)\in{\bar{I}}_{\mu}(\mathcal{B}G)$.

We denote with $\mathcal{B}_{\tilde{\tau}}(G)$ the stack of
$\tilde{\tau}$-marked twisted stable maps to $\mathcal{B}G$. Furthermore, let
$\mathcal{B}_{\tilde{\tau}}^{\rm sm}(G)$ be the sub-stack of
$\mathcal{B}_{\tilde{\tau}}(G)$ consisting of twisted stable maps with smooth
domain, that is the curves $\mathcal{C}_{v}$, for $v\in V_{\tilde{\tau}}$, in
Definition 4.6 are smooth.

For any gerby $n$-tree ${\tilde{\tau}}$, let
$\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})^{(\tilde{\tau})}$ be
defined by the equalizer diagram

${\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})^{(\tilde{\tau})}}$${\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})^{F_{\tilde{\tau}}}}$${\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})^{F_{\tilde{\tau}}}}$${\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})^{F_{\tilde{\tau}}\setminus
L_{\tilde{\tau}}}\times\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})^{L_{\tilde{\tau}}}}$$\scriptstyle{(\iota\circ-,\operatorname{id}_{L_{\tilde{\tau}}})}$$\scriptstyle{-\circ
j_{\tilde{\tau}}}$$\scriptstyle{(\operatorname{id}_{F_{\tilde{\tau}}\setminus
L_{\tilde{\tau}}},\operatorname{id}_{L_{\tilde{\tau}}})}$

where
$\iota:\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})\to\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})$
is the involution defined in [AGV08, §3.5]. By definition of
$\mathcal{B}_{\tilde{\tau}}(G)$, there is a cartesian diagram

(3) ${\mathcal{B}_{\tilde{\tau}}(G)}$${\prod_{v\in
V_{\tilde{\tau}}}\mathcal{B}^{\mathrm{bal}}_{0,F_{\tilde{\tau}}(v)}(G)}$${\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})^{(\tilde{\tau})}}$${\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})^{F_{\tilde{\tau}}}}$

where the right vertical arrow is determined by the natural evaluation
morphisms
$e_{i}:\mathcal{B}^{\mathrm{bal}}_{0,F_{\tilde{\tau}}(v)}(G)\to\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})$,
for $i\in F_{\tilde{\tau}}(v)$.

For any stable gerby $n$-tree, the gluing construction of [AGV08] (see Section
5 and Appendix A) yields a morphism

(4) $\xi_{\tilde{\tau}}\colon\mathcal{B}_{\tilde{\tau}}(G)\to\mathcal{B}^{\rm
bal}_{0,n}(G)\,.$

######  Proposition 4.8.

The morphism $\xi_{\tilde{\tau}}$ in (4) is representable and proper.

###### Proof.

Let us choose an ordering of the edges of $\tilde{\tau}$,
$E_{\tilde{\tau}}=\\{\\{i_{1},i_{1}^{\prime}\\},\ldots,\\{i_{k},i_{k}^{\prime}\\}\\}$.
Hence $F_{\tilde{\tau}}\setminus
L_{\tilde{\tau}}=\\{i_{1},i_{1}^{\prime},\ldots,i_{k},i_{k}^{\prime}\\}$ and
$j_{\tilde{\tau}}(i_{\ell})=i_{\ell}^{\prime}$, for $\ell=1,\ldots,k$. Notice
that, since $\tau$ is a tree, $\\#V_{\tilde{\tau}}=k+1$ and, for any $\ell$,
$[i_{\ell}]\not=[i_{\ell}^{\prime}]$. In particular there is an $\ell$ such
that either the vertex $[i_{\ell}]$ or $[i_{\ell}^{\prime}]$ belongs only to
the edge $\\{i_{\ell},i_{\ell}^{\prime}\\}$ and we suppose, without loss of
generality, that $[i_{k}^{\prime}]$ satisfies this condition.

Let us proceed by induction on $k\geq 1$ (if $k=0$, $\xi_{\tilde{\tau}}$ is
the identity). For $k=1$ we have that

${\mathcal{B}}_{\tilde{\tau}}(G)\cong{\mathcal{B}}^{\rm
bal}_{0,F_{\tilde{\tau}}([i_{1}])}(G)\tensor[_{\check{e}_{i_{1}}}]{\times}{{}_{e_{i_{1}}^{\prime}}}{\mathcal{B}}^{\rm
bal}_{0,F_{\tilde{\tau}}([i_{1}^{\prime}])}(G)\,,$

where $\check{e}_{i_{1}}:=\iota\circ e_{i_{1}}$, and $\xi_{\tilde{\tau}}$ is
the morphism in [AGV08], Proposition 5.2.1.(1). Therefore $\xi_{\tilde{\tau}}$
is representable by the aforementioned proposition. Finally,
$\xi_{\tilde{\tau}}$ is proper since the stacks $\mathcal{B}^{\rm
bal}_{0,n}(G)$ are proper [AV02], Theorem 1.4.1, in particular they are
separated and also $\mathcal{B}_{\tilde{\tau}}(G)$ is proper.

Let us suppose that the result is true for gerby trees with $k-1$ edges and
let $\tilde{\tau}$ be a gerby $n$-tree with $\\#E_{\tilde{\tau}}=k$. Let
${\tilde{\tau}}_{1}$ be the gerby tree with:

  * •

$F_{{\tilde{\tau}}_{1}}:=[i_{1}]\cup\ldots\cup[i_{k}]\subset
F_{\tilde{\tau}}$;

  * •

$j_{{\tilde{\tau}}_{1}}$ equal to $j_{\tilde{\tau}}$ on
$F_{{\tilde{\tau}}_{1}}\setminus\\{i_{k}\\}$ and
$j_{{\tilde{\tau}}_{1}}(i_{k})=i_{k}$;

  * •

$R_{{\tilde{\tau}}_{1}}=R_{\tilde{\tau}}\cap(F_{{\tilde{\tau}}_{1}}\times
F_{{\tilde{\tau}}_{1}})$;

  * •

$\mathfrak{m}_{{\tilde{\tau}}_{1}}={\mathfrak{m}_{\tilde{\tau}}}_{|F_{{\tilde{\tau}}_{1}}}$.

Furthermore, let $\tilde{\tau}_{2}$ be the gerby tree with:

  * •

$F_{{\tilde{\tau}}_{2}}:=[i_{k}^{\prime}]\subset F_{\tilde{\tau}}$;

  * •

$j_{{\tilde{\tau}}_{2}}$ equal to the identity;

  * •

$R_{{\tilde{\tau}}_{2}}=F_{{\tilde{\tau}}_{2}}\times F_{{\tilde{\tau}}_{2}}$;

  * •

$\mathfrak{m}_{{\tilde{\tau}}_{2}}={\mathfrak{m}_{\tilde{\tau}}}_{|F_{{\tilde{\tau}}_{2}}}$.

Since $\tilde{\tau}_{1}$ has $k-1$ edges,
$\xi_{\tilde{\tau}_{1}}\colon\mathcal{B}_{\tilde{\tau}_{1}}(G)\to\mathcal{B}^{\rm
bal}_{0,m}(G)$ is representable and proper, $m:=\\#L_{\tilde{\tau}_{1}}$.
Moreover,

$\mathcal{B}_{\tilde{\tau}}(G)\cong\mathcal{B}_{\tilde{\tau}_{1}}(G)\tensor[_{\check{e}_{i_{k}}}]{\times}{{}_{e_{i_{k}^{\prime}}}}\mathcal{B}_{\tilde{\tau}_{2}}(G)\,,$

where $\check{e}_{i_{k}}:=\iota\circ e_{i_{k}}$, and under this identification
$\xi_{\tilde{\tau}}$ is the composition of the following morphisms:

$\mathcal{B}_{\tilde{\tau}_{1}}(G)\tensor[_{\check{e}_{i_{k}}}]{\times}{{}_{e_{i_{k}^{\prime}}}}\mathcal{B}_{\tilde{\tau}_{2}}(G)\stackrel{{\scriptstyle(\xi_{\tilde{\tau}_{1}},{\rm
Id})}}{{\longrightarrow}}\mathcal{B}_{0,m}^{\rm
bal}(G)\tensor[_{\check{e}_{i_{k}}}]{\times}{{}_{e_{i_{k}^{\prime}}}}\mathcal{B}_{\tilde{\tau}_{2}}(G)\to\mathcal{B}_{0,n}^{\rm
bal}(G)\,,$

where the morphism to the right is the gluing one of [AGV08], Proposition
5.2.1 (1). ∎

Furthermore we have the following result.

######  Proposition 4.9.

The morphism $\xi_{\tilde{\tau}}$ in (4) is a closed immersion.

To prove the previous proposition we need a preliminary result.

######  Lemma 4.10.

The automorphism group of a stable $n$-tree $(\tau,l_{\tau})$ is trivial.

###### Proof.

We proceed by induction on the number $\\#E_{\tau}$ of edges. If
$\\#E_{\tau}=0$ the claim is obvious. So, let us assume that $\\#E_{\tau}=k+1$
and that the result holds true for trees with $k$ edges. Let us fix a
numbering of the edges,
$E_{\tau}=\\{\\{i_{1},i_{1}^{\prime}\\},\ldots,\\{i_{k+1},i_{k+1}^{\prime}\\}\\}$,
as in the proof of Proposition 4.8, that is such that
$j_{\tau}(i_{\ell})=i_{\ell}^{\prime}$, for $\ell=1,\ldots,k+1$, and
$[i_{k+1}^{\prime}]$ belongs only to the edge
$\\{i_{k+1},i_{k+1}^{\prime}\\}$. Notice that, since $\tau$ is stable,
$F_{\tau}([i_{k+1}^{\prime}])\cap L_{\tau}\not=\emptyset$.

Let now $\varphi\in{\rm Aut}(\tau,l_{\tau})$. By definition, $\varphi(f)=f$
for any $f\in F_{\tau}([i_{k+1}^{\prime}])\cap L_{\tau}$, hence
$\varphi([i_{k+1}^{\prime}])=[i_{k+1}^{\prime}]$ and
$\varphi(i_{k+1}^{\prime})=i_{k+1}^{\prime}$ (in other words $\varphi$ is the
identity on $[i_{k+1}^{\prime}]$). Let now $\tau_{1}$ be the tree defined as
in the proof of Proposition 4.8, with $F_{\tau_{1}}=F_{\tau}\setminus
F_{\tau}([i_{k+1}^{\prime}])$, and let $l_{\tau_{1}}$ be a $n_{1}$-marking,
where $n_{1}:=\\#L_{\tau_{1}}$. Notice that $(\tau_{1},l_{\tau_{1}})$ is a
stable $n_{1}$-tree and that the restriction of $\varphi$ to $F_{\tau_{1}}$ is
an automorphism of $(\tau_{1},l_{\tau_{1}})$. By induction we conclude that
$\varphi={\rm Id}$. ∎

###### Proof.

(of Proposition 4.9) The claim follows essentially by the same proofs of
Proposition (10.11) and Corollary (10.22) in Chapter XII of [ACG11]. The main
ingredients in these proofs are: the existence of Kuranishi families (see
[ACG11], Chapter XVI, Section 9), and the fact that the automorphism group of
any stable gerby $n$-tree $(\tilde{\tau},l_{\tau})$ is trivial (Lemma 4.10). ∎

The image of $\mathcal{B}_{\tilde{\tau}}^{\rm sm}(G)$ under
$\xi_{\tilde{\tau}}$ is the locus of (balanced) twisted $G$-covers of
combinatorial type $\tilde{\tau}$, which we denote $\mathcal{B}^{\rm
bal}(G,\tilde{\tau})$. Then we have the following corollary of Proposition
4.9.

######  Corollary 4.11.

The image $\xi_{\tilde{\tau}}(\mathcal{B}_{\tilde{\tau}}(G))$ is closed in
$\mathcal{B}^{\rm bal}_{0,n}(G)$ and $\mathcal{B}^{\rm bal}(G,\tilde{\tau})$
is open in $\xi_{\tilde{\tau}}(\mathcal{B}_{\tilde{\tau}}(G))$.

###### Proof.

Notice that $\mathcal{B}^{\rm sm}_{\tilde{\tau}}(G)$ is the pre-image, under
the top arrow in (3), of the open and dense locus of $\prod_{v\in
V_{\tilde{\tau}}}\mathcal{B}^{\mathrm{bal}}_{0,F_{\tilde{\tau}}(v)}(G)$
corresponding to $G$-covers of smooth curves. Hence $\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)$ is open in $\mathcal{B}_{\tilde{\tau}}(G)$ and
$\mathcal{B}^{\rm bal}(G,\tilde{\tau})$ is open in the image of
$\xi_{\tilde{\tau}}$ because it is the complement of a closed locus, by
Proposition 4.8. ∎

##  5\. Recursive relations

In this section we prove our main result, Theorem 5.4, which yields recursive
formulae that express each class $\\{\mathcal{B}^{\mathrm{bal}}_{0,n}(G)\\}\in
K_{0}(\mathsf{AlgSt}_{\Bbbk})$ in terms of the classes of the open loci in
$\mathcal{B}^{\mathrm{bal}}_{0,m}(G)$, for $m\leq n$, corresponding to stable
maps with smooth domain. To this aim we need some preliminary definitions and
results. First of all, following [Bag22, Definition 4.2], let us give the
following definition.

######  Definition 5.1.

Let $\mathcal{X}$ be an algebraic $\Bbbk$-stack. A locally closed
decomposition of $\mathcal{X}$ is a morphism
$f\colon\mathcal{Y}\to\mathcal{X}$ of algebraic $\Bbbk$-stacks that satisfies
the following conditions:

  * •

the restriction of $f$ to each connected component of $\mathcal{Y}$ is a
locally closed immersion;

  * •

$f$ is bijective on closed points.

Let us recall that a locally closed immersion is a morphism that can be
factored as $j\circ i$, where $i$ is a closed immersion and $j$ is an open
immersion.

In the following we will use the next result, which is well known, but we
provide an elementary proof for completeness.

######  Lemma 5.2.

Let $f\colon\mathcal{Y}\to\mathcal{X}$ be a locally closed decomposition. Let
$\mathcal{Y}_{1},\ldots,\mathcal{Y}_{m}$ be the connected components of
$\mathcal{Y}$. Then the following equality holds true in
$K_{0}(\mathsf{AlgSt}_{\Bbbk})$:

$\\{\mathcal{X}\\}=\sum_{k=1}^{m}\\{f(\mathcal{Y}_{k})\\}\,.$

###### Proof.

By definition, $f(\mathcal{Y}_{1}),\ldots,f(\mathcal{Y}_{m})$ are locally
closed substacks of $\mathcal{X}$ such that

$\mathcal{X}=\bigsqcup_{k=1}^{m}f(\mathcal{Y}_{k})\,.$

We proceed by induction on $m$. Let us assume $m>1$, since the case $m=1$ is
trivial. Now, for $\overline{f(\mathcal{Y}_{m})}$ being the closure of
$f(\mathcal{Y}_{m})$, using the properties of $K_{0}(\mathsf{AlgSt}_{\Bbbk})$
and the fact that $f(\mathcal{Y}_{m})$ is open in
$\overline{f(\mathcal{Y}_{m})}$, we have the following equalities:

$\begin{split}\\{\mathcal{X}\\}&=\\{\overline{f(\mathcal{Y}_{m})}\\}+\\{\mathcal{X}\setminus\overline{f(\mathcal{Y}_{m})}\\}\\\
&=\\{{f(\mathcal{Y}_{m})}\\}+\\{\overline{f(\mathcal{Y}_{m})}\setminus
f(\mathcal{Y}_{m})\\}+\\{\mathcal{X}\setminus\overline{f(\mathcal{Y}_{m})}\\}\,.\end{split}$

Now we observe that

$\overline{f(\mathcal{Y}_{m})}\setminus
f(\mathcal{Y}_{m})=\bigsqcup_{k=1}^{m-1}\left(\overline{f(\mathcal{Y}_{m})}\setminus
f(\mathcal{Y}_{m})\right)\cap
f(\mathcal{Y}_{k})=\bigsqcup_{k=1}^{m-1}\overline{f(\mathcal{Y}_{m})}\cap
f(\mathcal{Y}_{k})$

and

$\mathcal{X}\setminus\overline{f(\mathcal{Y}_{m})}=\bigsqcup_{k=1}^{m-1}\left(\mathcal{X}\setminus\overline{f(\mathcal{Y}_{m})}\right)\cap
f(\mathcal{Y}_{k})\,,$

which are locally closed decompositions of the corresponding right hand sides.
By induction we deduce that

$\begin{split}\\{\mathcal{X}\\}&=\\{{f(\mathcal{Y}_{m})}\\}+\sum_{k=1}^{m-1}\\{\overline{f(\mathcal{Y}_{m})}\cap
f(\mathcal{Y}_{k})\\}+\sum_{k=1}^{m-1}\\{\left(\mathcal{X}\setminus\overline{f(\mathcal{Y}_{m})}\right)\cap
f(\mathcal{Y}_{k})\\}\\\
&=\\{{f(\mathcal{Y}_{m})}\\}+\sum_{k=1}^{m-1}\\{f(\mathcal{Y}_{k})\\}\,.\end{split}$

∎

######  Proposition 5.3.

For any integer $n\geq 3$, let $\tilde{\Gamma}_{0,n}$ be a set of
representatives of the isomorphism classes of stable gerby $n$-trees
associated to $\mathcal{B}G$. Then the following equalities hold true in
$K_{0}^{S_{n}}({\mathsf{AlgSt}_{\Bbbk}})$:

(5)
$\displaystyle\sum_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}\\{V_{\tilde{\tau}}\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\\}$ $\displaystyle=$
$\displaystyle\\{\mathcal{B}\\}\circ\left(\\{I_{1}\\}+D\\{\overline{\mathcal{B}}\\}\right)_{n}\,;$
(6)
$\displaystyle\sum_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}\\{E_{\tilde{\tau}}\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\\}$ $\displaystyle=$ $\displaystyle\\{I_{2}\\}\circ
D\\{\overline{\mathcal{B}}\\}_{n}\,;$ (7)
$\displaystyle\sum_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}\\{(F_{\tilde{\tau}}\setminus
L_{\tilde{\tau}})\times\mathcal{B}^{\rm sm}_{\tilde{\tau}}(G)\\}$
$\displaystyle=$
$\displaystyle\\{I_{2}\times_{B^{2}}(D\overline{\mathcal{B}}\ast
D\overline{\mathcal{B}})\\}_{n}\,.$

###### Proof.

To prove (5) we show that there is a locally closed decomposition

(8)
$\coprod_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}V_{\tilde{\tau}}\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\to\coprod_{m}\left(\mathcal{B}_{m}\times_{B^{m}}\coprod_{\begin{subarray}{c}\underline{k}\in\mathbb{N}^{m}\\\
|\underline{k}|=n\end{subarray}}{\rm Sh}(\underline{k})\times(I_{1}\coprod
D\overline{\mathcal{B}})_{\underline{k}}\right)/S_{m}\,,$

where

$(I_{1}\coprod D\overline{\mathcal{B}})_{\underline{k}}=(I_{1}\coprod
D\overline{\mathcal{B}})_{k_{1}}\times\ldots\times(I_{1}\coprod
D\overline{\mathcal{B}})_{k_{m}}\,,$

the claim then follows since (by (2) in Definition 2.5) the class in
$K_{0}(\mathsf{AlgSt}_{\Bbbk})$ of the target of (8) is the right hand side of
(5). Notice that for any $n$ the set $\tilde{\Gamma}_{0,n}$ is finite and
$3\leq m\leq n$ in (8).

Let $w\in V_{\tilde{\tau}}$ and let $m=|F_{\tilde{\tau}}(w)|$. We first
construct a morphism

(9) $\Phi_{w}\colon S_{m}\times\\{w\\}\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\to\mathcal{B}_{m}\times_{B^{m}}\coprod_{\begin{subarray}{c}\underline{k}\in\mathbb{N}^{m}\\\
|\underline{k}|=n\end{subarray}}{\rm Sh}(\underline{k})\times(I_{1}\coprod
D\overline{\mathcal{B}})_{\underline{k}}\,,$

where, with a slight abuse of notation, the $S_{m}$ in the left hand side
denotes the set of bijections $\\{1,\ldots,m\\}\to F_{\tilde{\tau}}(w)$. To
this aim, let $\theta\in S_{m}$ be such a bijection and let
$f_{i}:=\theta(i)$, for $i=1,\ldots,m$. Composing the restriction to
$\mathcal{B}_{\tilde{\tau}}(G)^{\rm sm}$ of the morphism

in (3) with the projection to the factor indexed by $w$ we obtain a morphism
$\mathcal{B}_{\tilde{\tau}}(G)^{\rm
sm}\to\mathcal{B}_{0,F_{\tilde{\tau}}(w)}(G)$, which gives, via $\theta$, the
first component $\mathcal{B}_{\tilde{\tau}}(G)^{\rm sm}\to\mathcal{B}_{m}$ of
(9).

The second component of (9) is defined as follows. Let $\tilde{\tau}_{w}$ be
the graph $(F_{\tilde{\tau}}\setminus
F_{\tilde{\tau}}(w),j_{\tilde{\tau}_{w}},R_{\tilde{\tau}_{w}})$, where

$j_{\tilde{\tau}_{w}}(f):=\begin{cases}j_{\tilde{\tau}}(f)&{\rm
if}\,\,j_{\tilde{\tau}}(f)\not\in F_{\tilde{\tau}}(w)\,,\\\ f&\,{\rm
otherwise}\,,\end{cases}$

and $R_{\tilde{\tau}_{w}}$ is the restriction of $R_{\tilde{\tau}}$ to
$F_{\tilde{\tau}}\setminus F_{\tilde{\tau}}(w)$. Notice that the geometric
realization $|\tilde{\tau}_{w}|$ of $\tilde{\tau}_{w}$ is the complement in
$|\tilde{\tau}|$ of

$\bigcup_{f\in L_{\tilde{\tau}}\cap
F_{\tilde{\tau}}(w)}[0_{f},{1/2}_{f})\bigcup\bigcup_{f\in
F_{\tilde{\tau}}(w)\setminus L_{\tilde{\tau}}}[0_{f},{1/2}_{f}]\,,$

in particular the connected components of $|\tilde{\tau}_{w}|$ are in
bijection with the flags in $F_{\tilde{\tau}(w)}\setminus L_{\tilde{\tau}}$.
For any $i=1,\ldots,m$ such that $f_{i}\in F_{\tilde{\tau}(w)}\setminus
L_{\tilde{\tau}}$, let $\tilde{\tau}_{w}^{i}$ the sub-graph of
$\tilde{\tau}_{w}$ that contains $j_{\tilde{\tau}}(f_{i})$ and whose geometric
realization is a connected component of $|\tilde{\tau}_{w}|$. For any
$i=1,\ldots,m$, let

$k_{i}:=\begin{cases}1&{\rm if}\,\,f_{i}\in L_{\tilde{\tau}}\,,\\\
|L_{\tilde{\tau}^{i}_{w}}|-1&\,\,{\rm otherwise}\,,\end{cases}$

so that $k_{1}+\ldots+k_{m}=n$. The marking $l_{\tilde{\tau}}\colon
L_{\tilde{\tau}}\leftrightarrow\\{1,\ldots,n\\}$ (recall that $\tilde{\tau}$
in an $n$-tree) induces a unique marking of the leaves of
$\tilde{\tau}_{w}^{i}$ such that $j_{\tilde{\tau}}(f_{i})$ is marked with
$k_{i}+1$ and the other leaves are marked from $1$ to $k_{i}$ preserving the
order determined by $l_{\tilde{\tau}}$. This data determines a
$(k_{1},\ldots,k_{m})$-shuffle $\sigma$ in the following way: setting
$k_{0}=0$, for any $i=1,\ldots,m$, if $k_{i}>1$,
$\sigma(k_{i-1}+1)<\sigma(k_{i-1}+2)<\ldots<\sigma(k_{i-1}+k_{i})$ are the
indices of the first $k_{i}$ leaves of $\tilde{\tau}^{i}_{w}$ with respect to
the marking $l_{\tilde{\tau}}$; if $k_{i}=1$,
$\sigma(k_{i-1}+1)=l_{\tilde{\tau}}(f_{i})$.

Finally, for any $i=1,\ldots,m$ such that $k_{i}>1$, the subgraph
$\tilde{\tau}_{w}^{i}$ is stable, hence we have a morphism
$\mathcal{B}_{\tilde{\tau}}(G)^{\rm sm}\to\prod_{v\in
V_{\tilde{\tau}_{w}^{i}}}\mathcal{B}_{0,F_{\tilde{\tau}}(v)}(G)$, that is the
composition of the restriction to $\mathcal{B}_{\tilde{\tau}}(G)^{\rm sm}$ of
the morphism $\mathcal{B}_{\tilde{\tau}}(G)\to\prod_{v\in
V_{\tilde{\tau}}}\mathcal{B}_{0,F_{\tilde{\tau}}(v)}(G)$ in (3) with the
projection on the product of factors corresponding to the vertices $v\in
V_{\tilde{\tau}_{w}^{i}}$. Then the gluing construction of [AGV08] (Section 5
and Appendix A), together with the enumeration of the leaves of
$\tilde{\tau}_{w}^{i}$ as described before, yields a morphisms
$\mathcal{B}_{\tilde{\tau}}(G)^{\rm sm}\to(D\overline{\mathcal{B}})_{k_{i}}$,
where the indices $i$ are such that $k_{i}>1$. In the remaining cases where
$k_{i}=1$ the composition of the first component
$\mathcal{B}_{\tilde{\tau}}(G)^{\rm sm}\to\mathcal{B}_{m}$ in (9) with the
evaluation ${\rm ev}_{i}\colon\mathcal{B}_{m}\to B$ yields a map
$\mathcal{B}_{\tilde{\tau}}(G)^{\rm sm}\to(I_{1})_{1}$. In this way we define
the third component of $\Phi_{w}$ in (9).

By construction $\Phi_{w}$ is $S_{m}$-equivariant, hence it induces a morphism

$\\{w\\}\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\to\left(\mathcal{B}_{m}\times_{B^{m}}\coprod_{\begin{subarray}{c}\underline{k}\in\mathbb{N}^{m}\\\
|\underline{k}|=n\end{subarray}}{\rm Sh}(\underline{k})\times(I_{1}\coprod
D\overline{\mathcal{B}})_{\underline{k}}\right)/S_{m}\,,$

which canonically induces the morphism (8).

In order to prove that (8) is a locally closed decomposition it suffices to
show that it has the following two properties: it is bijective on closed
points; (9) is a locally closed immersion (because the quotient morphism by a
finite group is open and closed). The first property can be verified, for
example, as in [Bag19, Proposition3.2.2]. For the second one, let $\theta\in
S_{m}$ and let

$\Phi_{w,\theta}\colon\\{\theta\\}\times\\{w\\}\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\to\mathcal{B}_{m}\times_{B^{m}}(I_{1}\coprod
D\overline{\mathcal{B}})_{\underline{k}}$

be the restriction of $\Phi_{w}$, where we have adopted the same notation as
before, so that $\underline{k}$ refers to the shuffle determined by $\theta$.
In the next discussion $w$ and $\theta$ are fixed, so we omit them. Let us
rewrite diagram (3) as follows:

(10)
${\mathcal{B}_{\tilde{\tau}}(G)}$${\mathcal{B}_{m}\times\prod_{v\not=w}\mathcal{B}^{\mathrm{bal}}_{0,F_{\tilde{\tau}}(v)}(G)}$${\mathcal{B}_{m}\times\prod_{i:k_{i}>1}\prod_{v\in
V_{\tilde{\tau}^{i}_{w}}}\mathcal{B}^{\mathrm{bal}}_{0,F_{\tilde{\tau}^{i}_{w}}(v)}(G)}$${\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})^{(\tilde{\tau})}}$${B^{m}\times
B^{F_{\tilde{\tau}_{w}}}}$${B^{m}\times\prod_{i:k_{i}>1}B^{F_{\tilde{\tau}^{i}_{w}}}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

This yields an isomorphism

$\mathcal{B}_{\tilde{\tau}}(G)\cong\mathcal{B}_{m}\times_{B^{m}}\prod_{i:k_{i}>1}\mathcal{B}_{\tilde{\tau}^{i}_{w}}(G)\,.$

On the other hand we have the natural identification

$\mathcal{B}_{m}\times_{B^{m}}(I_{1}\coprod
D\overline{\mathcal{B}})_{\underline{k}}\cong\mathcal{B}_{m}\times_{B^{m}}\prod_{i:k_{i}=1}B\times\prod_{i:k_{i}>1}D\overline{\mathcal{B}}_{k_{i}}\,.$

The morphism $\Phi_{w,\theta}$ is the composition of the open immersion

$\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\subset\mathcal{B}_{\tilde{\tau}}(G)\cong\mathcal{B}_{m}\times_{B^{m}}\prod_{i:k_{i}>1}\mathcal{B}_{\tilde{\tau}^{i}_{w}}(G)$

followed by

$\mathrm{pr}_{\mathcal{B}_{m}}\times\prod_{i:k_{i}=1}\mathrm{ev}_{i}\circ\mathrm{pr}_{\mathcal{B}_{m}}\times\prod_{i:k_{i}>1}\xi_{\tilde{\tau}^{i}_{w}}\circ\mathrm{pr}_{\mathcal{B}_{\tilde{\tau}^{i}_{w}}(G)}\colon\mathcal{B}_{m}\times_{B^{m}}\prod_{i:k_{i}>1}\mathcal{B}_{\tilde{\tau}^{i}_{w}}(G)\to\mathcal{B}_{m}\times_{B^{m}}\prod_{i:k_{i}=1}B\times\prod_{i:k_{i}>1}D\overline{\mathcal{B}}_{k_{i}}\,,$

where $\mathrm{pr}$ denotes the projection to the corresponding factor. The
claim now follows from Proposition 4.9.

The proof of (6) and (7) are similar to that of (5), so we sketch them. The
right hand side of (6) is the class of

$\left((I_{2})_{2}\times_{B^{2}}\coprod_{\begin{subarray}{c}\underline{k}\in\mathbb{N}^{2}\\\
|\underline{k}|=n\end{subarray}}{\rm Sh}(\underline{k})\times
D\overline{\mathcal{B}}_{\underline{k}}\right)/S_{2}\,,$

while the right hand side of (7) is the class of

$(I_{2})_{2}\times_{B^{2}}\coprod_{\begin{subarray}{c}\underline{k}\in\mathbb{N}^{2}\\\
|\underline{k}|=n\end{subarray}}{\rm Sh}(\underline{k})\times
D\overline{\mathcal{B}}_{\underline{k}}\,,$

where we do not pass to the quotient by $S_{2}$.

The choice of a flag $f\in F_{\tilde{\tau}}\setminus L_{\tilde{\tau}}$
corresponds to the oriented edge
$\overrightarrow{\mathfrak{e}}=(f,j_{\tilde{\tau}}(f))$ of $\tilde{\tau}$. For
any such oriented edge $\overrightarrow{\mathfrak{e}}$, if we cut
$|\tilde{\tau}|$ at the point corresponding to ${1/2}_{f}$, we obtain two
subtrees $\tilde{\tau}^{1}$ and $\tilde{\tau}^{2}$ of $\tilde{\tau}$, where
$f\in F_{\tilde{\tau}^{1}}$ and $j_{\tilde{\tau}}(f)\in F_{\tilde{\tau}^{2}}$.
Let $k_{1}+1$ (resp. $k_{2}+1$) be the number of leaves of $\tilde{\tau}^{1}$
(resp. $\tilde{\tau}^{2}$), then $k_{1}+k_{2}=n$. As before this decomposition
of $\tilde{\tau}$ determines a shuffle in ${\rm Sh}(\underline{k})$ and gluing
the various irreducible components of the objects in $\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)$ according to $\tilde{\tau}^{1}$ and $\tilde{\tau}^{2}$
yields a morphism

(11) $\\{\overrightarrow{\mathfrak{e}}\\}\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\to(I_{2})_{2}\times_{B^{2}}\coprod_{\begin{subarray}{c}\underline{k}\in\mathbb{N}^{2}\\\
|\underline{k}|=n\end{subarray}}{\rm Sh}(\underline{k})\times
D\overline{\mathcal{B}}_{\underline{k}}\,.$

In this way we obtain a locally closed decomposition

$\coprod_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}(F_{\tilde{\tau}}\setminus
L_{\tilde{\tau}})\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\to(I_{2})_{2}\times_{B^{2}}\coprod_{\begin{subarray}{c}\underline{k}\in\mathbb{N}^{2}\\\
|\underline{k}|=n\end{subarray}}{\rm Sh}(\underline{k})\times
D\overline{\mathcal{B}}_{\underline{k}}$

that proves (7). At the same time, exchanging the orientation of
$\overrightarrow{\mathfrak{e}}$ corresponds to the natural $S_{2}$-action on
the target of (11), so we also have a locally closed decomposition

$\coprod_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}E_{\tilde{\tau}}\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\to\left((I_{2})_{2}\times_{B^{2}}\coprod_{\begin{subarray}{c}\underline{k}\in\mathbb{N}^{2}\\\
|\underline{k}|=n\end{subarray}}{\rm Sh}(\underline{k})\times
D\overline{\mathcal{B}}_{\underline{k}}\right)/S_{2}$

that proves (6).

∎

######  Theorem 5.4.

Let $(\overline{\mathcal{B}},\overline{e})$ be the ${\mathsf{S}}$-module
defined in Example 2.11 and let $({\mathcal{B}},e)$ be the ${\mathsf{S}}$-sub-
module given by the open locus corresponding to the stable maps with smooth
domain. Then the following relation holds true:

(12)
$\\{\overline{\mathcal{B}}\\}\,\equiv_{B}\,\\{\mathcal{B}\\}\circ\left(\\{I_{1}\\}+D\\{\overline{\mathcal{B}}\\}\right)+\\{I_{2}\\}\circ
D\\{\overline{\mathcal{B}}\\}-\\{I_{2}\times_{B^{2}}(D\overline{\mathcal{B}}\ast
D\overline{\mathcal{B}})\\}\,,$

where $B$ is $\overline{\mathcal{I}}_{\bm{\mu}}({\mathcal{B}G})$ and both
sides of the relation belong to
$K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}},B)$ (see Notation 1).

###### Proof.

Since the relation (12) is an equality in
$K_{0}^{\mathsf{S}}({\mathsf{AlgSt}_{\Bbbk}})$, it suffices to prove that
$\\{\overline{\mathcal{B}}_{n}\\}\in K_{0}(\mathsf{AlgSt}_{\Bbbk})$ coincides
with the part in degree $n$ of the right hand side, for any $n\in\mathbb{N}$.
To this aim, let $\tilde{\Gamma}_{0,n}$ be a set of representatives of the
isomorphism classes of stable gerby $n$-trees associated to $\mathcal{B}G$.
Since $\overline{\mathcal{B}}_{n}$ is the disjoint union of the locally closed
substacks $\mathcal{B}^{\rm bal}(G,\tilde{\tau})$ (Corollary 4.11), for
$\tilde{\tau}\in\tilde{\Gamma}_{0,n}$, we obtain the following equalities in
$K_{0}(\mathsf{AlgSt}_{\Bbbk})$:

(13)
$\\{\overline{\mathcal{B}}_{n}\\}=\sum_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}\\{\mathcal{B}^{\rm
bal}(G,\tilde{\tau})\\}=\sum_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}\\{\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\\}\,,$

where, in the second equality, we have used the fact that $\mathcal{B}^{\rm
bal}(G,\tilde{\tau})$ and $\mathcal{B}^{\rm sm}_{\tilde{\tau}}(G)$ are
isomorphic via the morphism $\xi_{\tilde{\tau}}$ defined in the previous
section.

For any $\tilde{\tau}\in\tilde{\Gamma}_{0,n}$ there is a canonical injection

(14) $F_{\tilde{\tau}}\hookrightarrow
V_{\tilde{\tau}}\,\sqcup\,E_{\tilde{\tau}}\,\sqcup\,L_{\tilde{\tau}}$

that maps $F_{\tilde{\tau}}$ into the complement of a (canonical) point
$\ast\in V_{\tilde{\tau}}\,\sqcup\,E_{\tilde{\tau}}\,\sqcup\,L_{\tilde{\tau}}$
([GP06, (4.2)]). On the one hand (13) yields

$\sum_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}\\{(\ast\,\sqcup\,F_{\tilde{\tau}})\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\\}=\\{\overline{\mathcal{B}}_{n}\\}+\sum_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}\\{F_{\tilde{\tau}}\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\\}\,,$

on the other hand, identifying $\ast\,\sqcup\,F_{\tilde{\tau}}$ with
$V_{\tilde{\tau}}\,\sqcup\,E_{\tilde{\tau}}\,\sqcup\,L_{\tilde{\tau}}$ via
(14) we obtain the next equality:

$\begin{split}\\{\overline{\mathcal{B}}_{n}\\}+\sum_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}\\{F_{\tilde{\tau}}\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\\}=&\sum_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}\\{V_{\tilde{\tau}}\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\\}+\\\
&\sum_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}\\{E_{\tilde{\tau}}\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\\}+\\\
&\sum_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}\\{L_{\tilde{\tau}}\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\\}\,.\end{split}$

As $F_{\tilde{\tau}}=(F_{\tilde{\tau}}\setminus L_{\tilde{\tau}})\sqcup
L_{\tilde{\tau}}$, the previous equality reduces to

$\begin{split}\\{\overline{\mathcal{B}}_{n}\\}+\sum_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}\\{(F_{\tilde{\tau}}\setminus
L_{\tilde{\tau}})\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\\}=&\sum_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}\\{V_{\tilde{\tau}}\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\\}+\\\
&\sum_{\tilde{\tau}\in\tilde{\Gamma}_{0,n}}\\{E_{\tilde{\tau}}\times\mathcal{B}^{\rm
sm}_{\tilde{\tau}}(G)\\}\,.\end{split}$

The theorem now follows directly from Proposition 5.3. ∎

######  Corollary 5.5.

For any $n\geq 3$, the class $\\{\overline{\mathcal{B}}_{n}\\}\in
K_{0}^{S_{n}}({\mathsf{AlgSt}_{\Bbbk}})$ is determined by the classes
$\\{I_{1}\\},\\{I_{2}\\},\\{{\mathcal{B}}_{3}\\},\ldots,\\{{\mathcal{B}}_{n}\\}$.

###### Proof.

We show that $\\{\overline{\mathcal{B}}_{n}\\}$ is determined by
$\\{I_{1}\\},\\{I_{2}\\},\\{\overline{\mathcal{B}}_{3}\\},\ldots,\\{\overline{\mathcal{B}}_{n-1}\\}$,
hence the claim follows by induction and the equality
$\\{\overline{\mathcal{B}}_{3}\\}=\\{{\mathcal{B}}_{3}\\}$. From (12) we have
that

$\\{\overline{\mathcal{B}}_{n}\\}=\\{\mathcal{B}\\}\circ\left(\\{I_{1}\\}+D\\{\overline{\mathcal{B}}\\}\right)_{n}+\\{I_{2}\\}\circ
D\\{\overline{\mathcal{B}}\\}_{n}-\\{I_{2}\times_{B^{2}}(D\overline{\mathcal{B}}\ast
D\overline{\mathcal{B}})\\}_{n}\,.$

By definition of the composition,

$\\{\mathcal{B}\\}\circ\left(\\{I_{1}\\}+D\\{\overline{\mathcal{B}}\\}\right)_{n}=\left\\{\int^{m}\mathcal{B}_{m}\times_{B^{m}}(I_{1}\coprod
D\overline{\mathcal{B}})^{\ast m}(n)\right\\}\,,$

where

$(I_{1}\coprod D\overline{\mathcal{B}})^{\ast
m}(n)=\coprod_{\begin{subarray}{c}(k_{1},\ldots,k_{m})\in\mathbb{N}^{m}\\\
k_{1}+\ldots+k_{m}=n\end{subarray}}{\rm
Sh}(k_{1},\ldots,k_{m})\times(I_{1}\coprod
D\overline{\mathcal{B}})_{k_{1}}\times\ldots\times(I_{1}\coprod
D\overline{\mathcal{B}})_{k_{m}}\,.$

Since $\mathcal{B}_{m}=\emptyset$, for $m<3$, and $(I_{1}\coprod
D\overline{\mathcal{B}})_{k}=\emptyset$, if $k=0$, in the previous equality we
have non-trivial contributions only for $k_{i}\leq n-2$, $\forall
i=1,\ldots,m$. From this it follows that
$\\{\mathcal{B}\\}\circ\left(\\{I_{1}\\}+D\\{\overline{\mathcal{B}}\\}\right)_{n}$
is determined by $\\{I_{1}\\}$, $\\{\mathcal{B}_{n}\\}$ (corresponding to the
case $m=n$, $k_{1}=\ldots=k_{n}=1$) and by $\\{\overline{\mathcal{B}}_{k}\\}$,
for $k=3,\ldots,n-1$.

For the second summand we have the following equality:

$\\{I_{2}\\}\circ
D\\{\overline{\mathcal{B}}\\}_{n}=\left\\{\int^{m}(I_{2})_{m}\times_{B^{m}}(D\overline{\mathcal{B}})^{\ast
m}(n)\right\\}\,.$

Since $(I_{2})_{m}=\emptyset$, if $m\not=2$, and
$(D\overline{\mathcal{B}})_{k}=\emptyset$, for $k<2$, we deduce, as before,
that $\\{I_{2}\\}\circ D\\{\overline{\mathcal{B}}\\}_{n}$ is determined by
$\\{I_{2}\\}$ and by $\\{\overline{\mathcal{B}}_{k}\\}$, for $k=3,\ldots,n-1$.
The third summand is similar to the second one, hence we omit the details. ∎

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