# On the tautological rings of stacks of twisted curves

Hsian-Hua Tseng  Department of Mathematics
Ohio State University
100 Math Tower, 231 West 18th Ave.
Columbus, OH 43210
USA<EMAIL_ADDRESS>
###### Abstract.

We introduce the tautological rings of moduli stacks of twisted curves and
establish some basic properties.

###### Key words and phrases:

Chow groups, Algebraic stacks, Tautological rings

###### 2020 Mathematics Subject Classification:

14H10, 14C15, 14C17

##  0\. Introduction

We work over an algebraically closed field of characteristic $0$.

###  0.1. Backgroud

Balanced twisted curves111Twisted curves are also called stacky curves or
orbifold curves. are nodal curves with specific types of stack structures
possibly at nodes and isolated smooth points. Étale locally at a smooth stacky
point, it looks like $[\mathbb{A}^{1}/\mu_{r}]$ for some $r$, where $\mu_{r}$
acts on $\mathbb{A}^{1}$ by multiplications. Étale locally at a stacky node,
it looks like $[(xy=0)/\mu_{s}]$ for some $s$, where $\mu_{s}$ acts222This
particular action is called balanced. via $\zeta\cdot(x,y)=(\zeta\cdot
x,\zeta^{-1}\cdot y)$.

Twisted curves arise naturally in [4], [1] as a solution to the problem of
compactifying spaces of maps from nodal curves to a proper Deligne-Mumford
stack. More precisely, to compactify a space of maps from smooth curves of a
proper Deligne-Mumford stack, one must add maps from nodal twisted curves, not
just maps from nodal curves.

Later on, compact moduli spaces of certain maps from balanced333The balanced
condition is used in certain axioms in Gromov-Witten theory. twisted curves to
a smooth proper Deligne-Mumford stack were used to construct Gromov-Witten
theory of Deligne-Mumford stack, see [3].

Let

$\mathfrak{M}_{g,n}^{tw}$

be the stack of prestable balanced twisted curves of genus $g$ with $n$ marked
gerbes. The existence of $\mathfrak{M}_{g,n}^{tw}$ and some basic properties
have been known to the experts. It is known that $\mathfrak{M}_{g,n}^{tw}$ is
a smooth Artin stack locally of finite type. A construction of
$\mathfrak{M}_{g,n}^{tw}$ using logarithmic geometry, and proofs of some of
its basic properties, can be found in [10].

A variant of $\mathfrak{M}_{g,n}^{tw}$ was introduced in [8, Section 2] for
the purpose of studying relations between higher genus Gromov-Witten theory
and genus $0$ Gromov-Witten theory of symmetric products. This variant was
also used in [5] to study genus $0$ Gromov-Witten theory of root gerbes. In
this note, this variant is denoted by $\mathfrak{M}_{g,I,m,a}^{tw,triv}$, see
(1.2) below.

###  0.2. This paper

In the study of Gromov-Witten theory of Deligne-Mumford stacks, the stack
$\mathfrak{M}_{g,n}^{tw}$ is mostly used as a placeholder. On the other hand,
$\mathfrak{M}_{g,n}^{tw}$ is a natural generalization of both the stack
$\overline{\mathcal{M}}_{g,n}$ of stable curves and the stack
$\mathfrak{M}_{g,n}$ of pretable curves. $\mathfrak{M}_{g,n}^{tw}$ has rich
geometry and deserves to be studied further. Motivated by this, this note
considers one geometric aspect of $\mathfrak{M}_{g,n}^{tw}$, namely the
tautological ring of $\mathfrak{M}_{g,n}^{tw}$. The approach taken here
closely follows the work [6].

Section 1 presents a definition of the tautological rings of
$\mathfrak{M}_{g,n}^{tw}$ as a subring of the Chow rings of
$\mathfrak{M}_{g,n}^{tw}$. In this construction, the aforementioned variant
$\mathfrak{M}_{g,I,m,a}^{tw,triv}$ introduced in [8, Section 2] plays an
important role. A key reason for working with
$\mathfrak{M}_{g,I,m,a}^{tw,triv}$ is that their universal curves have modular
descriptions, given by the morphisms of forgetting a non-stacky marked point.
This allows the tautological rings of $\mathfrak{M}_{g,I,m,a}^{tw,triv}$ be
defined using natural gluing and forgetful maps, see Definition 1.2 for more
details. The tautological rings of $\mathfrak{M}_{g,n}^{tw}$ are then defined
using tautological rings of $\mathfrak{M}_{g,I,m,a}^{tw,triv}$ via
restrictions, see Definition 1.5 for a precise description.

A natural class of elements in the tautological rings, namely decorated strata
classes, are also introduced, see Definition 1.7.

Section 2 presents an additive spanning set of the tautological rings. Namely,
it is shown that the decorated strata classes span the tautological rings as
$\mathbb{Q}$-vector spaces. See Theorem 2.1 for the precise result.

###  0.3. Outlook

Naturally, the next goal in this direction is to understand the genus $0$
situation, thus extending the work [7] to twisted curves. More precisely, it
is expected that the Chow ring and tautological ring of
$\mathfrak{M}_{0,n}^{tw}$ coincide, and relations in the tautological ring of
$\mathfrak{M}_{0,n}^{tw}$ are all obtained from WDVV relations. This will be
discussed elsewhere.

A long-term, more ambitious goal, is to understand relations in the
tautological rings of $\mathfrak{M}_{g,n}^{tw}$.

###  0.4. Acknowledgment

The author thanks J. Schmitt for valuable discussions. The author also thanks
the referees for helpful suggestions. This work is supported in part by Simons
Foundation Collaboration Grant.

##  1\. Definitions of tautological rings

There are $n$ universal cyclic gerbes over $\mathfrak{M}_{g,n}^{tw}$
corresponding to marked points. Their fiber product over
$\mathfrak{M}_{g,n}^{tw}$, denoted by $\mathfrak{M}_{g,n}^{tw,triv}$,
parametrizes twisted curves with trivialized marked gerbes. The map

(1.1) $\mathfrak{M}_{g,n}^{tw,triv}\to\mathfrak{M}_{g,n}^{tw}$

is finite and étale.

###  1.1. Chow rings

By [10, Corollary 1.11], $\mathfrak{M}_{g,n}^{tw}$ and
$\mathfrak{M}_{g,n}^{tw,triv}$ are locally of finite type. The construction of
[6, Appendix A] applies to define the Chow groups444Chow groups are taken with
$\mathbb{Q}$-coefficients.

$\mathrm{CH}_{*}(\mathfrak{M}_{g,n}^{tw}),\quad\mathrm{CH}_{*}(\mathfrak{M}_{g,n}^{tw,triv}),$

which admit usual properties of Chow groups of finite type stacks. By the
analysis of stabilizers of twisted curves in [1], $\mathfrak{M}_{g,n}^{tw}$
and $\mathfrak{M}_{g,n}^{tw,triv}$ have affine stabilizer groups at geometric
points (except when $(g,n)=(1,0)$). Hence intersection products can be
defined, making these Chow groups into $\mathbb{Q}$-algebras.

The stack $\mathfrak{M}_{g,n}^{tw}$ is not connected. To specify component(s)
of $\mathfrak{M}_{g,n}^{tw}$ to work with, it is convenient to rephrase the
setup by introducing labels for marked points. Let $I$ be a set of $n$
elements. Let

$\mathfrak{M}_{g,I}^{tw}$

be the stack of prestable balanced twisted curves of genus $g$ and marked
gerbes labelled by $I$. Clearly there is an isomorphism
$\mathfrak{M}_{g,I}^{tw}\simeq\mathfrak{M}_{g,n}^{tw}$ given by choosing an
identification $I\to\\{1,2,...,n\\}$. For a function $m:I\to\mathbb{Z}_{>0}$,
let

$\mathfrak{M}_{g,I,m}^{tw}\subset\mathfrak{M}_{g,I}^{tw}$

be the locus parametrizing twisted curves whose stack structures at marked
points are given by the function $m$. The stacks

$\mathfrak{M}_{g,I}^{tw,triv},\quad\mathfrak{M}_{g,I,m}^{tw,triv}$

are defined by inverse images with respect to (1.1).

The stacks $\mathfrak{M}_{g,I,m}^{tw}$ and $\mathfrak{M}_{g,I,m}^{tw,triv}$
are smooth Artin stacks locally of finite type. It can be seen that
$\mathfrak{M}_{g,I,m}^{tw}$ and $\mathfrak{M}_{g,I,m}^{tw,triv}$ are
equidimensional. Furthermore, $\mathfrak{M}_{g,I,m}^{tw}$ and
$\mathfrak{M}_{g,I,m}^{tw,triv}$ have good filtrations by finite type
substacks, in the sense of [6, Definition A.2], given by loci of twisted
curves with bounds on the numbers of nodes.

We are interested in the Chow rings

$\mathrm{CH}^{*}(\mathfrak{M}_{g,I,m}^{tw}),\quad\mathrm{CH}^{*}(\mathfrak{M}_{g,I,m}^{tw,triv}).$

Since (1.1) is a fiber product of cyclic gerbes, with
$\mathbb{Q}$-coefficients there is an isomorphism

$\mathrm{CH}^{*}(\mathfrak{M}_{g,I,m}^{tw})\simeq\mathrm{CH}^{*}(\mathfrak{M}_{g,I,m}^{tw,triv})$

induced by pulling back via (1.1). In what follows, we work with
$\mathfrak{M}_{g,I,m}^{tw,triv}$.

###  1.2. Tautological rings

In the definition of tautological rings for moduli spaces of stable curves,
the modular interpretation of their universal curves plays an important role.
For moduli stacks of prestable curves, such a moduli interpretation is not
valid. This issue is overcome in [6]. We follow their recipe to treat the case
of twisted curves, as follows.

Let $\mathcal{A}$ be a commutative semigroup with unit
$\mathbb{0}\in\mathcal{A}$ such that

  1. (1)

$\mathcal{A}$ has indecomposable zero: for $x,y\in\mathcal{A}$, if
$x+y=\mathbb{0}$, then we must have $x=y=\mathbb{0}$;

  2. (2)

$\mathcal{A}$ has finite decomposition: if $a\in\mathcal{A}$, then the set
$\\{(a_{1},a_{2})|a_{1}+a_{2}=a\\}\subset\mathcal{A}\times\mathcal{A}$ is
finite.

######  Example 1.1.

The example of $\mathcal{A}$ that matters to us the most is the following:
$\mathcal{A}_{\mathbb{0}}=\\{\mathbb{0},\mathbb{1}\\}$ with
$\mathbb{0}+\mathbb{0}=\mathbb{0},\mathbb{0}+\mathbb{1}=\mathbb{1}+\mathbb{0}=\mathbb{1}+\mathbb{1}=\mathbb{1}$.

Fix an $\mathcal{A}$. Let $a\in\mathcal{A}$. We consider the stack

(1.2) $\mathfrak{M}_{g,I,m,a}^{tw,triv}$

of twisted curves with labellings of irreducible components by elements of
$\mathcal{A}$. This stack is defined in [8, Section 2], where it is denoted by
$\mathfrak{M}_{g,I,m,a}$. Roughly speaking, an object of
$\mathfrak{M}_{g,I,m,a}^{tw,triv}$ is a twisted curve
$(\mathcal{C},\\{p_{i}\\}_{i\in I})\in\mathfrak{M}_{g,I,m}^{tw,triv}$ together
with a map $\mathcal{C}_{v}\mapsto a_{\mathcal{C}_{v}}$ from the set of
irreducible components $\\{\mathcal{C}_{v}\\}$ of the normalization of
$\mathcal{C}$ to $\mathcal{A}$ such that $a_{\mathcal{C}_{v}}$ sum up to $a$.
The following stability condition is required: for each irreducible component
$\mathcal{C}_{v}$, either $a_{\mathcal{C}_{v}}\neq\mathbb{0}$, or
$\mathcal{C}_{v}$ together with its special points is a stable twisted curve.

There is a natural morphism

$\mathfrak{M}_{g,I,m,a}^{tw,triv}\to\mathfrak{M}_{g,I,m}^{tw,triv}$

that forgets the $\mathcal{A}$-valuation.

It follows from [8, Proposition 2.0.2] and properties of
$\mathfrak{M}_{g,I,m}^{tw,triv}$ that $\mathfrak{M}_{g,I,m,a}^{tw,triv}$ is a
smooth Artin stack locally of finite type, with good filtrations by finite
type substacks.

Consider the stack
$\mathfrak{M}_{g,I\cup\\{\bullet\\},m^{\prime},a}^{tw,triv}$ where
$m^{\prime}:I\cup\\{\bullet\\}\to\mathbb{Z}_{>0}$ is given by
$m^{\prime}|_{I}=m$ and $m^{\prime}(\bullet)=1$ (i.e. the marked point
$\bullet$ has trivial stack structure). Then there is a morphism

(1.3)
$\pi_{\bullet}:\mathfrak{M}_{g,I\cup\\{\bullet\\},m^{\prime},a}^{tw,triv}\to\mathfrak{M}_{g,I,m,a}^{tw,triv},$

given by forgetting the marked point $\bullet$ and contracting resulting
unstable components with $\mathcal{A}$-valuations $\mathbb{0}$. It follows
from [8, Proposition 2.1.1] that (1.3) is the universal curve.

We now recall the natural gluing maps with target
$\mathfrak{M}_{g,I,m,a}^{tw,triv}$. Let

(1.4) $\Gamma=(V,H,E,L,g:V\to\mathbb{Z}_{\geq 0},v:H\to V,\iota:H\to H)$

be a prestable graph of genus $g$ with $n$ markings. The required properties
of $\Gamma$ are

  1. (1)

$V$ is the set of vertices, $g:V\to\mathbb{Z}_{\geq 0}$ is the genus
assignment;

  2. (2)

$H$ is the set of half-edges, $v:H\to V$ is the vertex assignment, and
$\iota:H\to H$ is an involution;

  3. (3)

the set of $2$-cycles of $\iota$ in $H$ is by definition the set of edges $E$
(self-edges at vertices are permitted);

  4. (4)

the set of fixed points of $\iota$ is by definition the set of legs $L$ (which
correspond to the $n$ markings);

  5. (5)

the pair $(V,E)$ defines a connected graph $\Gamma$ satisfying the genus
condition

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

For a prestable graph $\Gamma$, we fix an identification $L\simeq I$ between
the set of legs and marking labels. In addition, we need to choose a map

(1.5) $m_{\Gamma}:H\to\mathbb{Z}_{\geq 0}$

so that $m_{\Gamma}|_{L}=m$ and

$m_{\Gamma}(\iota(h))=m_{\Gamma}(h)$

for $h\in H$, which ensures that stack structures at nodes are balanced. We
also need to choose an $\mathcal{A}$-valuation $a:V(\Gamma)\to\mathcal{A}$
satisfying $\sum_{v\in V(\Gamma)}a(v)=a$.

For a vertex $v\in V(\Gamma)$, let $H(v)\subset H$ be the set of half-edges at
$v$ and $L(v)\subset H(v)$ the set of legs at $v$. $I(v)\subset I$ is defined
by the identification $L\simeq I$.

Associated to $(\Gamma,m_{\Gamma},a)$ is a gluing morphism

(1.6)
$\xi_{(\Gamma,m_{\Gamma},a)}:\mathfrak{M}_{(\Gamma,m_{\Gamma},a)}^{tw,triv}=\prod_{v\in
V(\Gamma)}\mathfrak{M}_{g(v),H(v),m_{\Gamma}|_{H(v)},a(v)}^{tw,triv}\to\mathfrak{M}_{g,I,m,a}^{tw,triv},$

which is defined as follows. $\xi_{(\Gamma,m_{\Gamma},a)}$ sends a collection
$(\mathcal{C}_{v},\\{p_{h}\\}_{h\in H(v)})_{v\in V}$ to the curve
$(\mathcal{C},\\{p_{i}\\}_{i\in I})$ obtained by identifying the markings
$p_{h},p_{\iota(h)}$ for each pair $(h,\iota(h))$ of half-edges forming an
edge of $\Gamma$. Here the gerbe structures at $p_{h}$ and $p_{\iota(h)}$ are
identified via the homomorphism
$\mu_{m_{\Gamma}(h)}\to\mu_{m_{\Gamma}(\iota(h))}$, $\zeta\mapsto\zeta^{-1}$.
Since gerbes at nodes are not trivialized, over its image,
$\xi_{(\Gamma,m_{\Gamma},a)}$ is the product of universal gerbes at nodes
(which correspond to edges of $\Gamma$) and has degree $\prod_{(h,\iota(h))\in
E}\frac{1}{m_{\Gamma}(h)}$, see [2, Section 5.2].

The following definition is a direct extension of [6, Definition 1.3], which
is modelled on the treatment of the tautological rings of moduli stacks of
stable curves in [9].

######  Definition 1.2.

The tautological rings

$\\{\mathrm{R}^{*}(\mathfrak{M}_{g,I,m,a}^{tw,triv})\\}_{g,I,m,a}$

is the smallest system of unital $\mathbb{Q}$-subalgebras of the Chow rings
$\\{\mathrm{CH}^{*}(\mathfrak{M}_{g,I,m,a}^{tw,triv})\\}_{g,I,m,a}$ which is
closed under taking pushforwards by the natural forgetful maps (1.3) and
gluing maps (1.6).

Consider the semigroup $\mathcal{A}_{\mathbb{0}}$ in Example 1.1. Let
$\mathbb{1}\in\mathcal{A}_{\mathbb{0}}$. Consider the subset

$\mathfrak{Z}_{g,I,m}^{tw,triv}\subset\mathfrak{M}_{g,I,m,\mathbb{1}}^{tw,triv}$

consisting of $\mathcal{A}$-valued curves $(\mathcal{C},\\{p_{i}\\}_{i\in
I},(a_{\mathcal{C}_{v}})_{v})$ such that one of the values
$a_{\mathcal{C}_{v}}$ is equal to $\mathbb{0}$. The following Lemma is a
direct adaptation of [6, Proposition 2.6], whose proof can be easily adapted
here as well.

######  Lemma 1.3.

$\mathfrak{Z}_{g,I,m}^{tw,triv}$ is closed. The composition

$\mathfrak{U}_{g,I,m}^{tw,triv}=\mathfrak{M}_{g,I,m,\mathbb{1}}^{tw,triv}\setminus\mathfrak{Z}_{g,I,m}^{tw,triv}\hookrightarrow\mathfrak{M}_{g,I,m,\mathbb{1}}^{tw,triv}\to\mathfrak{M}_{g,I,m}^{tw,triv}$

of the inclusion and the natural map defines an isomorphism

$\mathfrak{U}_{g,I,m}^{tw,triv}\simeq\mathfrak{M}_{g,I,m}^{tw,triv}.$

The following extends [6, Corollary 2.7].

######  Corollary 1.4.

The universal curve over $\mathfrak{M}_{g,I,m}^{tw,triv}$ is given by

$\pi_{\bullet}:\mathfrak{M}_{g,I\cup\\{\bullet\\},m,\mathbb{1}}^{tw,triv}\setminus\pi_{\bullet}^{-1}(\mathfrak{Z}_{g,I,m}^{tw,triv})\to\mathfrak{M}_{g,I,m,\mathbb{1}}^{tw,triv}\setminus\mathfrak{Z}_{g,I,m}^{tw,triv}$

######  Definition 1.5.

The tautological ring

$\mathrm{R}^{*}(\mathfrak{M}_{g,I,m}^{tw,triv})\subset\mathrm{CH}^{*}(\mathfrak{M}_{g,I,m}^{tw,triv})$

is defined to be the image of the restriction of
$\mathrm{R}^{*}(\mathfrak{M}_{g,I,m,\mathbb{1}}^{tw,triv})$ to the open
substack from Lemma 1.3.

###  1.3. Stratum classes

We define some basic classes in the tautological ring
$\mathrm{R}^{*}(\mathfrak{M}_{g,I,m,a}^{tw,triv})$. Recall the universal curve
$\pi_{\bullet}$ in (1.3). Let

(1.7)
$\sigma_{i}:\mathfrak{M}_{g,I,m,a}^{tw,triv}\to\mathfrak{M}_{g,I\cup\\{\bullet\\},m^{\prime},a}^{tw,triv},\quad
i\in I$

be the section corresponding to the marking labelled by $i\in I$. Note that
$\sigma_{i}$ exists because marked gerbes are trivialized. Let
$\omega_{\pi_{\bullet}}$ be the relative dualizing sheaf of $\pi_{\bullet}$.

######  Definition 1.6.

$\begin{split}&\psi_{i}:=\sigma_{i}^{*}c_{1}(\omega_{\pi_{\bullet}})\in\mathrm{CH}^{1}(\mathfrak{M}_{g,I,m,a}^{tw,triv}),\quad
i\in I\\\
&\kappa_{m}:=(\pi_{\bullet})_{*}(\psi_{\bullet}^{m+1})\in\mathrm{CH}^{m}(\mathfrak{M}_{g,I,m,a}^{tw,triv}).\end{split}$

######  Definition 1.7 (Decorated stratum classes).

Let $\Gamma$ be a prestable graph of genus $g$ with $n$ markings. Let
$m_{\Gamma}$ be as in (1.5) and $a:V(\Gamma)\to\mathcal{A}$ an
$\mathcal{A}$-valuation with total value $a\in\mathcal{A}$. A decoration on
$(\Gamma,m_{\Gamma},a)$ is a class on
$\mathfrak{M}_{(\Gamma,m_{\Gamma},a)}^{tw,triv}=\prod_{v\in
V(\Gamma)}\mathfrak{M}_{g(v),H(v),m_{\Gamma}|_{H(v)},a(v)}^{tw,triv}$ of the
form

(1.8) $\alpha=\prod_{v\in V}\left(\prod_{i\in
H(v)}\psi_{v,i}^{a_{i}}\prod_{j\geq
1}\kappa_{v,j}^{b_{v,j}}\right)\in\mathrm{CH}^{*}(\mathfrak{M}_{(\Gamma,m_{\Gamma},a)}^{tw,triv}).$

Here
$\psi_{v,i},\kappa_{v,j}\in\mathrm{CH}^{*}(\mathfrak{M}_{g(v),H(v),m_{\Gamma}|_{H(v)},a(v)}^{tw,triv})$
are $\psi$ and $\kappa$ classes of the moduli space corresponding to the
vertex $v\in V(\Gamma)$. The decorated stratum class
$[(\Gamma,m_{\Gamma},a),\alpha]$ is defined to be the pushforward

(1.9)
$[(\Gamma,m_{\Gamma},a),\alpha]:=(\xi_{(\Gamma,m_{\Gamma},a)})_{*}\alpha\in\mathrm{CH}^{*}(\mathfrak{M}_{g,I,m,a}^{tw,triv}).$

##  2\. An additive spanning set of tautological rings

The main goal of this section is to prove the following extension of [6,
Theorem 1.4] and [9, Proposition 11]:

######  Theorem 2.1.

The tautological ring $\mathrm{R}^{*}(\mathfrak{M}_{g,I,m,a}^{tw,triv})$ is
spanned by decorated stratum classes (1.9) as a $\mathbb{Q}$-vector space.

The treatment here follows those of [6] and [9].

Because decorated stratum classes are tautological, their span is a vector
subspace of the tautological ring. Since the tautological rings are defined to
be the smallest system of unital $\mathbb{Q}$-subalgebras which is closed
under pushforwards by (1.3) and (1.6), we must prove two things:

  1. (1)

The span of decorated stratum classes is a $\mathbb{Q}$-subalgebra, i.e. it is
closed under intersection product.

  2. (2)

The span of decorated stratum classes is closed under pushforwards by (1.3)
and (1.6), hence the span must be the smallest such system.

The proof involves descriptions of pullbacks and pushforwards of decorated
stratum classes under natural maps. We spell out the details in our case,
following the treatment in [6, Section 3.2].

###  2.1. Closure under intersection product

We recall the combinatorial set-up. Let $A$ be an $\mathcal{A}$-valued
prestable graph.

######  Definition 2.2.

([6, Definition 3.5]) An $A$-structure on an $\mathcal{A}$-valued prestable
graph $\Gamma$, denoted by

$\Gamma\to A,$

consists of choices of subgraphs $\\{\Gamma_{v}\\}_{v\in V(A)}$ and maps
$V(\Gamma)\to V(A)$ and $H(A)\to H(\Gamma)$ satisfying

  1. (1)

The total $\mathcal{A}$-value of $\Gamma_{v}$ is $a(v)$, where
$a:V(A)\to\mathcal{A}$ is the $\mathcal{A}$-valuation of $A$.

  2. (2)

The map $V(\Gamma)\to V(A)$ is surjective, and the inverse image of $v\in
V(A)$ are precisely the vertices of $\Gamma_{v}$.

  3. (3)

The $\mathcal{A}$-valuation of $\Gamma$ is given by the composition of
$V(\Gamma)\to V(A)$ and $a:V(A)\to\mathcal{A}$.

  4. (4)

The map $H(A)\to H(\Gamma)$ is injective, and identifies half-edges $H(v)$ of
$A$ with legs of $\Gamma_{v}$.

  5. (5)

The maps $V(\Gamma)\to V(A)$ and $H(A)\to H(\Gamma)$ respect the incidence
relation of half-edges and vertices, and the pairs of half-edges forming
edges. (In particular, $E(A)\subset E(\Gamma)$.)

Given an $A$-structure $\Gamma\to A$ on $\Gamma$, the map
$m_{\Gamma}:H(\Gamma)\to\mathbb{Z}_{\geq 0}$ and the injection $H(A)\to
H(\Gamma)$ defines the map

$m_{A}:H(A)\to\mathbb{Z}_{\geq 0}.$

Then there is a natural gluing map

$\xi_{(\Gamma\to
A,m_{\Gamma},a)}:\mathfrak{M}_{(\Gamma,m_{\Gamma},a)}^{tw,triv}\to\mathfrak{M}_{(A,m_{A},a)}^{tw,triv}.$

We have

  1. (1)

If $H(A)\to H(\Gamma)$ maps $i\in H(v)\subset H(A)$ to $j\in H(w)\subset
H(\Gamma)$, then $\xi_{(\Gamma\to A,m_{\Gamma},a)}^{*}\psi_{v,i}=\psi_{w,j}$.

  2. (2)

$\xi_{(\Gamma\to A,m_{\Gamma},a)}^{*}\kappa_{v,l}=\sum_{w\in
V(\Gamma_{v})}\kappa_{w,l}$.

These can be proven by ways analogous to their counterparts for moduli stacks
of prestable curves, see [6, Section 3.2]. These properties allow, in a
straightforward manner, a description of the pull-back $\xi_{(\Gamma\to
A,m_{\Gamma},a)}^{*}\alpha$ of a decoration $\alpha$ on
$\mathfrak{M}_{(A,m_{A},a)}^{tw,triv}$ as in (1.8).

Suppose a prestable graph $\Gamma$ admits $A$ and $B$-structures
$f_{A}:\Gamma\to A$ and $f_{B}:\Gamma\to B$. As in [6, Section 3.2], the pair
$(f_{A},f_{B})$ is called a generic $(A,B)$-structure on $\Gamma$ if each
half-edge of $\Gamma$ corresponds to either a half-edge of $A$ or a half-edge
of $B$. In other words, the injections $H(A)\to H(\Gamma)$ and $H(B)\to
H(\Gamma)$ have disjoint images and their union is the whole $H(\Gamma)$.

With a generic $(A,B)$ structure on $\Gamma$, the maps
$m_{A}:H(A)\to\mathbb{Z}_{\geq 0}$ and $m_{B}:H(B)\to\mathbb{Z}_{\geq 0}$
uniquely determine a map $m_{\Gamma}:H(\Gamma)\to\mathbb{Z}_{\geq 0}$.

######  Proposition 2.3.

Let $A$ and $B$ be $\mathcal{A}$-valued prestable graphs for
$\mathfrak{M}_{g,I,m,a}$. Let $m_{A}$ and $m_{B}$ be as in (1.5). Then the
fiber product of the gluing maps
$\xi_{(A,m_{A},a)}:\mathfrak{M}_{(A,m_{A},a)}^{tw,triv}\to\mathfrak{M}_{g,I,m,a}^{tw,triv}$
and
$\xi_{(B,m_{B},a)}:\mathfrak{M}_{(B,m_{B},a)}^{tw,triv}\to\mathfrak{M}_{g,I,m,a}^{tw,triv}$
is given by the disjoint union

$\coprod_{\Gamma\in\mathcal{G}_{A,B}}\mathfrak{M}_{(\Gamma,m_{\Gamma},a)}^{tw,triv}$

taken over the set $\mathcal{G}_{A,B}$ of isomorphism classes of generic
$(A,B)$ structures on prestable graphs $\Gamma$, with two projection maps
given by $\xi_{(\Gamma\to A,m_{\Gamma},a)}$ and $\xi_{(\Gamma\to
B,m_{\Gamma},a)}$. Moreover, the top Chern class of the excess bundle

$E_{(\Gamma,m_{\Gamma},a)}=\xi_{(\Gamma\to
A,m_{\Gamma},a)}^{*}\mathcal{N}_{\xi_{(A,m_{A},a)}}/\mathcal{N}_{\xi_{(\Gamma\to
B,m_{\Gamma},a)}}$

is given by

(2.1) $c_{top}(E_{(\Gamma,m_{\Gamma},a)})=\prod_{e=(h,h^{\prime})\in E(A)\cap
E(B)\subset E(\Gamma)}(-\psi_{h}-\psi_{h^{\prime}})/m_{\Gamma}(h).$

The proof of this Proposition is similar to that of [6, Proposition 3.6]. The
only thing to notice is the factor $1/m_{\Gamma}(h)$ in (2.1), which accounts
for the fact that the gerbe at nodes are not trivialized.

The projection formula yields the following description of products of
decorated stratum classes:

(2.2)
$[(A,m_{A},a),\alpha]\cdot[(B,m_{B},a),\beta]=\sum_{\Gamma\in\mathcal{G}_{A,B}}(\xi_{(\Gamma,m_{\Gamma},a)})_{*}(\xi_{(\Gamma\to
A,m_{\Gamma},a)}^{*}\alpha\cdot\xi_{(\Gamma\to B,m_{\Gamma},a)}^{*}\beta\cdot
c_{top}(E_{(\Gamma,m_{\Gamma},a)}).$

The above discussion shows that the right-hand side of (2.2) is contained in
the span of decorated stratum classes. This shows that the span is closed
under intersection product.

###  2.2. Closure under pushforwards

####  2.2.1. Gluing maps

Pushing forward via (1.6) is straightforward to understand. Let $\Gamma_{0}$
be a graph, $m_{\Gamma_{0}}:H(\Gamma_{0})\to\mathbb{Z}_{\geq 0}$, and
$a_{0}:V(\Gamma_{0})\to\mathcal{A}$ an $\mathcal{A}$-valuation. Suppose for
each $v\in V(\Gamma_{0})$, we have the data of a genus $g(v)$ graph
$\Gamma_{v}$, a map $m_{\Gamma_{v}}:H(\Gamma_{v})\to\mathbb{Z}_{\geq 0}$, and
an $\mathcal{A}$-valuation $a_{v}:V(\Gamma_{v})\to\mathcal{A}$, satisfying the
following conditions:

  1. (1)

For each $v\in V(\Gamma_{0})$, there is an embedding of the set of half-edges
$H(\Gamma_{0})|_{v}\subset H(\Gamma_{v})$ so that $m_{\Gamma_{v}}$ restricts
to $m_{\Gamma_{0}}|_{H(\Gamma_{0})|_{v}}$.

  2. (2)

For each $v\in V(\Gamma_{0})$, $\sum_{v^{\prime}\in
V(\Gamma_{v})}a_{v}(v^{\prime})=a_{0}(v)$.

For each $v\in V(\Gamma_{0})$, we can replace $v$ by the graph $\Gamma_{v}$
and replace half-edges in $H(\Gamma_{0})|_{v}$ by their images in
$H(\Gamma_{v})$. This yields a new graph $\Gamma$ with $V(\Gamma)=\cup_{v\in
V(\Gamma_{0})}V(\Gamma_{v})$ and $H(\Gamma)=\cup_{v\in
V(\Gamma_{0})}H(\Gamma_{v})$. Define $m_{\Gamma}$ by
$m_{\Gamma}|_{H(\Gamma_{v})}=m_{\Gamma_{v}}$ and define
$a:V(\Gamma)\to\mathcal{A}$ by $a|_{V(\Gamma_{v})}=a_{v}$.

This construction describes a natural gluing map

$\prod_{v\in
V(\Gamma_{0})}\mathfrak{M}_{(\Gamma_{v},m_{\Gamma_{v}},a_{v})}^{tw,triv}\to\mathfrak{M}_{(\Gamma,m_{\Gamma},a)}^{tw,triv}.$

The pushforward of a class $\prod_{v\in
V(\Gamma_{0})}[(\Gamma_{v},m_{\Gamma_{v}},a_{v}),\alpha_{v}]$ under this
gluing map is given by

$\frac{1}{\prod_{h\in
E(\Gamma_{0})}m_{\Gamma_{0}}(h)}[(\Gamma,m_{\Gamma},a),\alpha]$

where $\alpha$ is formed by combining $\alpha_{v}$ using $V(\Gamma)=\cup_{v\in
V(\Gamma_{0})}V(\Gamma_{v})$.

####  2.2.2. Forgetful maps

We begin with an analogue of [6, Proposition 3.11].

Let $[(\Gamma,m_{\Gamma},a),\alpha]\in
R^{*}(\mathfrak{M}_{g,I\cup\\{\bullet\\},m^{\prime},a}^{tw,triv})$ with
$\alpha=\prod_{v\in V(\Gamma)}\alpha_{v}$. Let $v\in V(\Gamma)$ be the vertex
incident to $\bullet$ and let $\Gamma^{\prime}$ be the graph obtained from
$\Gamma$ by forgetting $\bullet$ and stabilizing if the vertex $v$ becomes
unstable. $v$ become unstable exactly when $g(v)=0$, $a(v)=\mathbb{0}$, and
$v$ has two other half-edges. By stability, the curve component corresponding
to $v$ is a genus $0$ stable twisted curve with three marked points. By
assumption the marked point corresponding to $\bullet$ has trivial stack
structure: $m_{\Gamma}(\bullet)=1$. Stabilization contracts this curve
component, hence the other marked points have opposite stack structure. Hence
$m_{\Gamma^{\prime}}:H(\Gamma^{\prime})\to\mathbb{Z}_{\geq 0}$ is well-defined
from $m_{\Gamma}$. $a^{\prime}:V(\Gamma^{\prime})\to\mathcal{A}$ is naturally
defined from $a$.

######  Proposition 2.4.

  1. (1)

If $v$ remains stable after forgetting $\bullet$, then

$(\pi_{\bullet})_{*}[(\Gamma,m_{\Gamma},a),\alpha]=(\xi_{(\Gamma^{\prime},,m_{\Gamma^{\prime}},a^{\prime})})_{*}\left((\pi_{v})_{*}\alpha_{v}\cdot\prod_{w\neq
v}\alpha_{w}\right),$

where $\pi_{v}$ is the forgetful map of $\bullet$ of the vertex $v$.

  2. (2)

Suppose $v$ becomes unstable after forgetting $\bullet$. If $\alpha_{v}\neq
1$, then $[(\Gamma,m_{\Gamma},a),\alpha]=0$. If $\alpha_{v}=1$, then

$(\pi_{\bullet})_{*}[(\Gamma,m_{\Gamma},a),\alpha]=[(\Gamma^{\prime},m_{\Gamma^{\prime}},a^{\prime}),\prod_{w\neq
v}\alpha_{w}].$

The proof of Proposition 2.4 is similar to that of [6, Proposition 3.11] and
is omitted.

We can study pushforwards of products of $\kappa$ and $\psi$ classes under
$\pi_{\bullet}$, as follows. A product

$\prod_{b}\kappa_{b}^{e_{b}}\cdot\prod_{i\in
I}\psi_{i}^{l_{i}}\cdot\psi_{\bullet}^{l_{\bullet}}\in
R^{*}(\mathfrak{M}_{g,I\cup\\{\bullet\\},m^{\prime},a}^{tw,triv})$

can be written as

$(\pi_{\bullet})^{*}(\prod_{b}\kappa_{b}^{e_{b}}\cdot\prod_{i\in
I}\psi_{i}^{l_{i}})\cdot\psi_{\bullet}^{l_{\bullet}}+\text{boundary terms}.$

This uses a straightforward analogue of [6, Proposition 3.10] and known
intersection formulas on
$\mathfrak{M}_{g,I\cup\\{\bullet\\},m^{\prime},a}^{tw,triv}$. Applying
$(\pi_{\bullet})_{*}$ yields

$\left(\prod_{b}\kappa_{b}^{e_{b}}\cdot\prod_{i\in
I}\psi_{i}^{l_{i}}\right)\cdot\kappa_{l_{\bullet}-1}+(\pi_{\bullet})_{*}(\text{boundary
terms}).$

The boundary terms can be handled by induction together with Proposition 2.4.

We conclude that $(\pi_{\bullet})_{*}$ preserves the span of decorated stratum
classes.

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