# Tautological relations in moduli spaces of weighted pointed curves

Felix Janda

###### Abstract.

Pandharipande-Pixton have used the geometry of the moduli space of stable
quotients to produce relations between tautological Chow classes on the moduli
space $M_{g}$ of smooth genus $g$ curves. We study a natural extension of
their methods to the boundary and more generally to Hassett’s moduli spaces
$\overline{M}_{g,\mathbf{w}}$ of stable nodal curves with weighted marked
points.

Algebraic manipulation of these relations brings them into a Faber-Zagier type
form. We show that they give Pixton’s generalized FZ relations when all
weights are one. As a special case, we give a formulation of FZ relations for
the $n$-fold product of the universal curve over $M_{g}$.

##  1\. Introduction

###  1.1. Moduli spaces of curves with weighted markings

####  1.1.1. Definition

As a GIT variation of the Deligne-Mumford moduli space of stable marked
curves, for any $n$-tuple $\mathbf{w}=(w_{1},\cdots,w_{n})$ with
$w_{i}\in\mathbb{Q}\cap]0,1]$ Hassett [1] has defined a moduli space
$\overline{M}_{g,\mathbf{w}}$, parametrizing nodal semi-stable curves $C$ of
arithmetic genus $g$ with $n$ numbered marked points $(p_{1},\dotsc,p_{n})$ in
the smooth locus of $C$ satisfying two stability conditions:

  1. (1)

The points in a subset $S\subseteq\\{1,\dotsc,n\\}$ are allowed to come
together if and only if $\sum_{i\in S}w_{i}\leq 1$.

  2. (2)

$\omega_{C}\left(\sum_{i=1}^{n}w_{i}p_{i}\right)$ is ample.

The second condition implies that the total weight plus the number of nodes of
every genus $0$ component of $C$ must be strictly greater than 2.

The main cases we have in mind are the usual moduli space $\overline{M}_{g,n}$
of marked curves, which occurs when all the weights are equal to 1, the case
when $\sum_{i=1}^{n}w_{i}\leq 1$, which is a desingularization of the $n$-fold
product of the universal curve over $\overline{M}_{g}$, and the case when
$g=0$, $w_{1}=1$, $w_{2}=1$ and $\sum_{i=3}^{n}w_{i}\leq 1$, which gives the
Losev-Manin spaces [2]. Moduli spaces mixed pointed curves also naturally
appear when studying moduli spaces of stable quotients [3].

We will use various abbreviations for the weight data, like
$(\mathbf{w},1^{m})$ for the data with first entries given by $\mathbf{w}$ and
further $m$ entries of $1$.

####  1.1.2. Tautological classes

Using the universal curve
$\pi:\overline{C}_{g,\mathbf{w}}\to\overline{M}_{g,\mathbf{w}}$, the $n$
sections $s_{i}:\overline{M}_{g,\mathbf{w}}\to\overline{C}_{g,\mathbf{w}}$
corresponding to the markings and the relative dualizing sheaf $\omega_{\pi}$
we can define $\psi$ and $\kappa$ classes:

$\displaystyle\psi_{i}=c_{1}(s_{i}^{*}\omega_{\pi})$
$\displaystyle\kappa_{i}=\pi_{*}(c_{1}(\omega_{\pi})^{i+1})$

Notice that in the case of $\overline{M}_{g,n}$ the definition of $\kappa$
classes is different from the usual definition as in [4].111The $\kappa$
classes we use here appear naturally when pushing forward powers of $\psi$
classes along maps forgetting points of small weight whereas the usual
$\kappa$ classes are convenient when studying push-forwards of powers of
$\psi$ classes along maps forgetting points of weight equal to $1$. The fact
that we will mainly consider the first kind of push-forwards explains our
choice of $\kappa$ classes.

Each subset $S\subseteq\\{1,\dotsc,n\\}$ defines a diagonal class $D_{S}$ as
the class of the locus where all the points of $S$ coincide. By Condition (1)
the class $D_{S}$ is zero if and only if $\sum_{i\in S}w_{i}>1$.

As $\overline{M}_{g,n}$ the moduli space $\overline{M}_{g,\mathbf{w}}$ is
stratified according to the topological type of the curve and each stratum,
indexed by a dual graph $\Gamma$ (see [5, Appendix A] for a description of
dual graphs of strata of $\overline{M}_{g,n}$), is the image of a clutching
map

$\xi_{\Gamma}:\prod_{i}\overline{M}_{g_{i},\mathbf{w}_{i}}\to\overline{M}_{g,\mathbf{w}}.$

Here points which are glued together have weight 1. The map $\xi_{\Gamma}$ is
finite of degree $|\operatorname{Aut}(\Gamma)|$ 222To determine the number
$|\operatorname{Aut}(\Gamma)|$ of automorphisms of $\Gamma$ the graph $\Gamma$
should be regarded as a collection of distinct half-edges of which some are
glued together. For example when $n=0$ and $\Gamma$ consists of exactly one
vertex and one edge, there is exactly one non-trivial automorphism, which
interchanges the two half-edges; accordingly the map
$\xi_{\Gamma}:\overline{M}_{g-1,2}\to\overline{M}_{g}$ is a double cover..

###  1.2. Formulation of the relations

To state the Faber-Zagier-type relations on $\overline{M}_{g,\mathbf{w}}$ we
need to introduce several formal power series.

The hypergeometric series $A$ and $B$ already appeared in the original FZ
relations. They are defined by

$A(t)=\sum_{i=0}^{\infty}\frac{(6i)!}{(3i)!(2i)!}\left(\frac{t}{72}\right)^{i}=1+O(t^{1}),\quad
B(t)=\sum_{i=0}^{\infty}\frac{(6i)!}{(3i)!(2i)!}\frac{6i+1}{6i-1}\left(\frac{t}{72}\right)^{i}.$

We will actually not directly use $B$ in the definition of the relations but
$A$ and a family $C_{i}$ of series strongly related to $A$ and $B$ which were
already used in [6] for the proof of the equivalence between stable quotient
and FZ relations on $M_{g}$. They are defined recursively by

$C_{1}=C=\frac{B}{A},\quad
C_{i+1}=\left(12t^{2}\frac{\mathrm{d}}{\mathrm{d}t}-4it\right)C_{i}.$

Notice that $C_{i}$ is a multiple of $t^{i-1}$.

As in [6] these series appear in the study of the two variable functions

$\displaystyle\Phi(t,x)=$
$\displaystyle\sum_{d=0}^{\infty}\prod_{i=1}^{d}\frac{1}{1-it}\frac{(-1)^{d}}{d!}\frac{x^{d}}{t^{d}}$
$\displaystyle\frac{1}{2}+\delta=\gamma=$ $\displaystyle\sum_{i\geq
1}\frac{B_{2i}}{2i\cdot(2i-1)}t^{2i-1}+\log(\Phi),$

where the Bernoulli numbers $B_{k}$ are defined by

$\frac{t}{e^{t}-1}=\sum_{k=0}^{\infty}B_{k}\frac{t^{k}}{k!}.$

These two-variable functions appear both in the localization formula for
stable quotients (see Section 3 and [6]) and the $S$-matrix in the equivariant
genus $0$ Gromov-Witten theory of $\mathbb{P}^{1}$ (see Section 5).

Various linear bracket operators are used to insert Chow classes into the
power series in $t$. We define $\\{\\}_{\kappa}$, $\\{\\}_{\psi}$ and
$\\{\\}_{D_{S}}$ for $S\subseteq\\{1,\dotsc,n\\}$ by

$\displaystyle\\{t^{k}\\}_{\kappa}=$ $\displaystyle\kappa_{k}t^{k}$
$\displaystyle\\{t^{k}\\}_{\psi}=$ $\displaystyle\psi^{k}t^{k}$
$\displaystyle\\{t^{k}\\}_{D_{S}}=$
$\displaystyle\begin{cases}\psi_{i}^{k}t^{k}&\text{, if }S=\\{i\\}\\\
(-1)^{|S|-1}D_{S}\psi_{i}^{k-|S|+1}t^{k}&\text{, else, where }i\in
S\end{cases}$

and linear continuation. It will moreover be useful to define brackets
modified by a sign $\zeta\in\\{\pm 1\\}$, denoted by
$\\{\\}_{\kappa}^{\zeta}$, $\\{\\}_{\psi}^{\zeta}$ and
$\\{\\}_{D_{S}}^{\zeta}$ respectively, by composing the usual bracket operator
with the ring map induced by $t\mapsto\zeta t$. For a power series $F$ in $t$
we will use the notation $[F]_{t^{i}}$ for the $t^{i}$ coefficient of $F$.

######  Proposition 1.

For any codimension $r$ and the choice of a subset
$S\subseteq\\{1,\dotsc,n\\}$ such that $3r\geq g+1+|S|$ the class

$\displaystyle\Bigg{[}\sum_{\begin{subarray}{c}\Gamma\\\ \zeta:\Gamma\to\\{\pm
1\\}\end{subarray}}\frac{1}{|\operatorname{Aut}(\Gamma)|}\xi_{\Gamma*}\Big{(}\prod_{v\text{
vertex}}\exp(-\\{\log(A)\\}_{\kappa^{(v)}}^{\zeta(v)})\sum_{P\vdash
S_{v}}\prod_{i\in P}\\{C_{|i|}\\}_{D_{i}}^{\zeta(v)}\prod_{e\text{
edge}}\Delta_{e}\Big{)}\Bigg{]}_{t^{r-|E|}}$

in $A^{r}(\overline{M}_{g,\mathbf{w}})$ is zero, where the sum is taken over
all dual graphs $\Gamma$ of $\overline{M}_{g,\mathbf{w}}$ with vertices
colored by $\zeta$ with $+1$ or $-1$, the class $\kappa_{i}^{(v)}$ is the
$i$-th $\kappa$ class in the factor corresponding to $v$ and $S_{v}$ is $S$
restricted to the markings at $v$. The edge term $\Delta_{e}$ depends only on
the $\psi$ classes $\psi_{1}$, $\psi_{2}$ and colors
$\zeta_{i}=\zeta(v_{i})\in\\{\pm 1\\}$ at the vertices $v_{1}$, $v_{2}$ joined
by $e$ and is defined by

$2t(\psi_{1}+\psi_{2})\Delta_{e}=(\zeta_{1}+\zeta_{2})\\{A^{-1}\\}_{\psi_{1}}^{\zeta_{1}}\\{A^{-1}\\}_{\psi_{2}}^{\zeta_{2}}+\zeta_{1}\\{C\\}_{\psi_{1}}^{\zeta_{1}}+\zeta_{2}\\{C\\}_{\psi_{2}}^{\zeta_{2}}.$

To see that the series $\Delta_{e}$ is well-defined one can use the identity
$A(t)B(-t)+A(-t)B(t)+2=0$ [7]. The proof of Proposition 1 gives an alternative
more geometric proof.

In the case of $\overline{M}_{g,n}$ the relations of Proposition 1 are a
reformulation of the part of Pixton’s generalized FZ relations [7] with empty
partition $\sigma$ and coefficients $a_{i}$ only valued in $\\{0,1\\}$. The
set $S$ corresponds to the set of all $i$ such that $a_{i}=1$.

To obtain a set of relations analogous to Pixton’s relations we need to take
the closure of the relations of Proposition 1 under multiplication with $\psi$
and $\kappa$ classes and push forward under maps forgetting marked points of
weight 1. See for this also the discussion in Section 6.4 and [8, Section
3.5].

In total this gives a proof of [7, Conjecture 1] in Chow. We therefore have
verified Pixton’s remark [7] that it should be possible to adapt the stable
quotients method to prove that his generalized relations hold. In cohomology
[7, Conjecture 1] has already been established by a completely different
method in [8].

###  1.3. Plan of Paper

Section 2 introduces stable quotient moduli spaces of $\mathbb{P}^{1}$ with
weighted marked points, slightly generalizing the usual moduli spaces of
stable quotients. We define them, sketch their existence and review structures
on them. In Section 3 we review the virtual localization formulas both for
stable quotients and stable maps to $\mathbb{P}^{1}$. Section 4 contains a
proof of Proposition 1 restricted to powers of the universal curve over
$M_{g}$. The proof in this case is much simpler than the general case but many
parts of it can be referred to later on. Section 5 provides calculations of
localization sums using Givental’s method necessary in the proof of the
general relations. Finally Section 6 contains the proof of Proposition 1.

### Acknowledgments

The author would like to thank Rahul Pandharipande for the idea of considering
the universal curve on the moduli space of stable quotient and recognizing the
edge sums appearing in the localization calculation and Aaron Pixton for
explaining many aspects of the preprint [9] and FZ relations in general. The
author is grateful for discussions with Alina Marian, Dragos Oprea and Dimitri
Zvonkine on the moduli of stable quotients.

The author was supported by the Swiss National Science Foundation grant SNF
200021_143274.

##  2\. Moduli spaces of stable quotients

###  2.1. Introduction

The proof of Proposition 1 is based on the geometry of stable quotients. These
moduli spaces give an alternative compactification of the space of maps from
curves to Grassmannians and were first introduced in [3]. We will need a
combination of these spaces with Hassett’s spaces of weighted stable curves.
These spaces are different from the $\epsilon$-stable quotient spaces
introduced by Toda [10], where the stability conditions on quotient sheaf
instead of the points varies. Similar spaces
$\overline{M}_{g,\mathbf{w}}(\mathbb{P}^{m},d)$ in Gromov-Witten theory have
been studied in [11], [12] and [13].

The moduli space $\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$ parametrises
nodal curves $C$ of arithmetic genus $g$ with $n$ markings $p_{i}$ weighted by
$\mathbf{w}$ together with a quotient sequence

$0\to S\to\mathcal{O}_{C}\otimes\mathbb{C}^{2}\to Q\to 0$

satisfying several conditions:

  * •

The underlying curve with weighted marked points satisfies all the properties
of being stable but possibly the ampleness condition (2).

  * •

$S$ is locally free of rank 1.

  * •

$Q$ has degree $d$.

  * •

The torsion of $Q$ is outside the nodes and the markings of weight 1.

  * •

$\omega_{C}\left(\sum_{i=1}^{n}w_{i}p_{i}\right)\otimes
S^{\otimes(-\varepsilon)}$ is ample for any $0<\varepsilon\in\mathbb{Q}$.

Isomorphisms of stable quotients are defined by isomorphisms of weighted
stable curves such that the kernels of the quotient maps are related via pull-
back by the isomorphism. In Section 2.3 we give a precise definition for
families and sketch a proof that
$\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$ is a proper Deligne-Mumford
stack.

###  2.2. Structures

There is a universal curve $\overline{C}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$
over $\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$ with a universal quotient
sequence

$0\to\mathcal{S}\to\mathcal{O}_{\overline{C}_{g,\mathbf{w}}(\mathbb{P}^{1},d)}\otimes\mathbb{C}^{2}\to\mathcal{Q}\to
0.$

Moreover as in Gromov-Witten theory for each marking $i$ of weight $1$ there
is an evaluation map
$\operatorname{ev}_{i}:\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)\to\mathbb{P}^{1}$,
which is defined by noticing that

$0\to\mathcal{S}\to\mathcal{O}_{C}\otimes\mathbb{C}^{2}\to\mathcal{Q}\to 0$

restricted to a marking $p$ of weight $1$ gives a point in $\mathbb{P}^{1}$
since tensoring the above sequence with the residue field $k_{p}$ at $p$ gives
a $k_{p}$ valued point in the Grassmannian $\mathbb{G}(1,2)=\mathbb{P}^{1}$.

There is also a forgetful map
$\nu:\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)\to\overline{M}_{g,\mathbf{w}}$
forgetting the data of the quotient sequence and stabilizing unstable
components.

Finally, there is a comparison map
$c:\overline{M}_{g,\mathbf{w}}(\mathbb{P}^{1},d)\to\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$
from the moduli space of stable maps of degree $d$ to $\mathbb{P}^{1}$. It
contracts all components that would become unstable 333The only components
which are stable in the Gromov-Witten but not in the stable quotients theory
are non-contracted components of genus 0 with exactly one node and no marking.
and introduces torsion at the point the component is contracted to according
to the degree of the map restricted to that component.

The open substack
$Q_{g,\mathbf{w}}(\mathbb{P}^{1},d)\subset\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$
is defined to be the preimage of the moduli space of smooth curves
$M_{g,\mathbf{w}}$ under the forgetful map $\nu$.

###  2.3. Construction

We want to reduce the existence of moduli spaces of stable quotients with
weighted marked points to that of the usual moduli spaces of stable quotients.
For this we use ideas from [11] and [12]. Because it is not the main topic of
this article we will try to be as brief as possible.

We will allow in this section the weight data also to include $0$. So with
$\mathbf{w}$ we will denote an $n$-tuple $(w_{1},\dotsc,w_{n})$ with $0\leq
w_{i}\leq 1$ for all $i\in\\{1,\dotsc,n\\}$ here. We also consider more
generally stable quotients to $\mathbb{P}^{m}$.

######  Definition.

An object $(C,s_{1},\dotsc,s_{n},\mathcal{O}_{C}^{m+1}\to\mathcal{Q})$ in the
stack $\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{m},d)$ over a scheme $S$ is a
proper, flat morphism $\pi:C\to S$ together with $n$ sections $s_{i}$ and a
quotient sequence of quasi-coherent sheaves on $C$ flat over $S$

$0\to\mathcal{S}\to\mathcal{O}_{C}^{m+1}\to\mathcal{Q}\to 0$

such that

  1. (1)

The fibers of $\pi$ over geometric points are nodal connected curves of
arithmetic genus $g$.

  2. (2)

For any $S\subset\\{1,\dotsc,n\\}$ such that the intersection of $s_{i}$ for
all $i\in S$ is nonempty we must have $\sum_{i\in S}w_{i}\leq 1$ and if in
addition the intersection touches the singular locus of $\pi$ we must have
$\sum_{i\in S}w_{i}=0$.

  3. (3)

$\mathcal{S}$ is locally free of rank $1$.

  4. (4)

$\mathcal{Q}$ is locally free outside the singular locus of $\pi$ and of
degree $d$.

  5. (5)

$\omega_{\pi}(\sum_{i=1}^{n}w_{i}s_{i})\otimes\mathcal{S}^{\otimes(-\varepsilon)}$
is $\pi$ relatively ample for any $\varepsilon>0$.

Two families $(C,s_{1},\dotsc,s_{n},\mathcal{O}_{C}^{m+1}\to\mathcal{Q})$,
$(C^{\prime},s^{\prime}_{1},\dotsc,s^{\prime}_{n},\mathcal{O}_{C^{\prime}}^{m+1}\to\mathcal{Q}^{\prime})$
of stable quotients over $S$ are isomorphic if there exists an isomorphism
$\phi:C\to C^{\prime}$ over $S$ mapping $s_{i}$ to $s^{\prime}_{i}$ for all
$i\in\\{1,\dotsc,n\\}$ and such that $S$ and $\phi^{*}(S^{\prime})$ coincide
when viewed as subsheaves of $\mathcal{O}_{C^{\prime}}^{m+1}$.

Notice that there is no condition on the sections of weight $0$. Therefore the
space $\overline{Q}_{g,(\mathbf{w},0^{f})}(\mathbb{P}^{m},d)$ is isomorphic to
the $f$-fold power of the universal curve of
$\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{m},d)$ over
$\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{m},d)$.

In order to prove that $\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{m},d)$ is a
Deligne-Mumford stack it is enough to show that it is locally isomorphic to a
product of universal curves over $\overline{Q}_{g,1^{f}}(\mathbb{P}^{m},d)$
since this is a Deligne-Mumford stack as shown in [3] (by realizing it as a
stack quotient of a locally closed subscheme of a relative Quot scheme).

######  Lemma 1.

For each point $(C,s_{1},\dotsc,s_{n},\mathcal{O}_{C}^{m+1}\to\mathcal{Q})$ in
$\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{m},d)$ there exists an open
neighborhood which is isomorphic to an open substack of
$\overline{Q}_{g,\mathbf{w}^{\prime}}(\mathbb{P}^{m},d)$ where
$\mathbf{w}^{\prime}$ is $\\{0,1\\}$ valued.

###### Proof.

The argument is the same as in [11, Corollary 1.18]. ∎

Separatedness follows from the following lemma, which is analogous to [12,
Proposition 1.3.4].

######  Lemma 2.

The diagonal
$\Delta:\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{m},d)\to\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{m},d)\times\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{m},d)$
is representable, finite and separated.

###### Proof.

We proceed as in [12, Proposition 1.3.4].

Let $(C,s_{1},\dotsc,s_{n},\mathcal{O}_{C}^{m+1}\to\mathcal{Q})$ and
$(C^{\prime},s^{\prime}_{1},\dotsc,s^{\prime}_{n},\mathcal{O}_{C^{\prime}}^{m+1}\to\mathcal{Q}^{\prime})$
be two stable quotients over a base scheme $S$. We need to show that the
category

$\mathrm{Isom}((C,s_{1},\dotsc,s_{n},\mathcal{O}_{C}^{m+1}\to\mathcal{Q}),(C^{\prime},s^{\prime}_{1},\dotsc,s^{\prime}_{n},\mathcal{O}_{C^{\prime}}^{m+1}\to\mathcal{Q}^{\prime}))$

is represented by scheme finite and separated over $S$.

The images $T$ and $T^{\prime}$ of the maps $(\mathcal{O}_{C}^{m+1})^{*}\to
S^{*}$ and $(\mathcal{O}_{C^{\prime}}^{m+1})^{*}\to(S^{\prime})^{*}$ are given
by $T=S^{*}(-D)$, $T^{\prime}=(S^{\prime})^{*}(-D^{\prime})$ for effective
divisors $D$, $D^{\prime}$ of some degree $d^{\prime}\leq d$ on $C$
respectively $C^{\prime}$. From this we can construct $d^{\prime}$ additional
sections $s_{n+1},\dotsc,s_{n+d^{\prime}}$ for $C$ and
$s^{\prime}_{n+1},\dotsc,s^{\prime}_{n+d^{\prime}}$ for $C^{\prime}$. As in
[14, Proposition 1.3.4] at least étale locally $d-d^{\prime}$ 444Actually
$(m+1)(d-d^{\prime})$ further sections $s_{n+d^{\prime}},\dotsc,s_{n+d}$ for
$C$ and $s^{\prime}_{n+d^{\prime}},\dotsc,s^{\prime}_{n+d}$ for $C^{\prime}$
can be constructed by choosing suitable hyperplanes $H_{i}$ in
$\mathbb{C}^{m+1}$ and marking sections at which the quotient is locally free
and the quotient sequence coincides with the sequence corresponding to the
inclusion of $H_{i}$ in $\mathbb{C}^{m+1}$.

By the definition of stable quotients the resulting marked curves
$(C,s_{1},\dotsc,s_{n+d})$ and
$(C^{\prime},s^{\prime}_{1},\dotsc,s^{\prime}_{n+d})$ are
$(\mathbf{w},\varepsilon^{d})$-stable. This gives a closed embedding

$\displaystyle\mathrm{Isom}((C,s_{1},\dotsc,s_{n},\mathcal{O}_{C}^{m+1}\to\mathcal{Q}),(C^{\prime},s^{\prime}_{1},\dotsc,s^{\prime}_{n},\mathcal{O}_{C^{\prime}}^{m+1}\to\mathcal{Q}^{\prime}))\hookrightarrow$
$\displaystyle\bigsqcup_{\sigma\in
S_{d^{\prime}}}\mathrm{Isom}((C,s_{1},\dotsc,s_{n+d}),(C^{\prime},s^{\prime}_{1},\dotsc,s^{\prime}_{n},s^{\prime}_{n+\sigma(1)},\dotsc,s^{\prime}_{n+\sigma(d^{\prime})},s^{\prime}_{n+d^{\prime}+1},\dotsc,s^{\prime}_{n+d})).$

Since by [1] the right hand side is a scheme finite and separated over $S$ so
is the left hand side. ∎

######  Lemma 3.

There is a surjective comparison map
$c:\overline{M}_{g,\mathbf{w}}(\mathbb{P}^{m},d)\to\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{m},d)$.

###### Proof.

This is similar to [10, Lemma 2.23]. ∎

######  Lemma 4.

For $\mathbf{w}^{\prime}\leq\mathbf{w}$ there is a surjective reduction
morphism

$\rho_{\mathbf{w}^{\prime}\mathbf{w}}:\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{m},d)\to\overline{Q}_{g,\mathbf{w}^{\prime}}(\mathbb{P}^{m},d).$

###### Proof.

This follows from Lemma 3 and the corresponding result for stable maps (see
[12, Proposition 1.2.1]). ∎

Since $\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{m},d)$ is proper for
$\mathbf{w}=1^{n}$ the preceding lemma implies that
$\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{m},d)$ is proper in general.

##  3\. Localization

The virtual localization formulas ([15]) for
$\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{m},d)$ and
$\overline{M}_{g,\mathbf{w}}(\mathbb{P}^{m},d)$ are the main tool we use to
derive stable quotient relations. The existence of the necessary virtual
fundamental class $[\overline{M}_{g,\mathbf{w}}(\mathbb{P}^{m},d)]^{vir}$ for
$\overline{M}_{g,\mathbf{w}}(\mathbb{P}^{m},d)$ has been shown in [11] and
[12]. For the existence of a $2$-term obstruction theory of
$\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{m},d)$ the same arguments as in [3,
Section 3.2] can be used. They depend on existence of a $\nu$-relative
$2$-term obstruction theory of the $\mathrm{Quot}$ scheme and the non-
singularity of the Hassett moduli spaces of weighted curves. We will now
follow Section 7 of [3].

In this paper we will only look at torus actions on moduli spaces of stable
quotients or stable maps of $\mathbb{P}^{1}$ which are induced from the
diagonal action of $\mathbb{C}^{*}$ on $\mathbb{C}^{2}$ given by
$([x_{0}:x_{1}],\lambda)\mapsto[x_{0}:\lambda x_{1}]$. By $s$ we will denote
the pull-back of the equivariant class $s\in
A_{\mathbb{C}^{*}}^{1}(\mathrm{pt})$ defined as the first equivariant Chern
class of the trivial rank 1 bundle on a point space with weight one action of
$\mathbb{C}^{*}$ on it.

The equivariant cohomology of $\mathbb{P}^{1}$ is generated as an algebra over
$\mathbb{Q}[s]$ by the equivariant classes $[0]$, $[\infty]$ of the two fixed
points $0$ and $\infty$. These classes satisfy (among others) the relation
$[0]-[\infty]=s$. Localizing by $s$, the classes $[0]$ and $[\infty]$ give a
basis of $\mathbb{Q}[s,s^{-1}]$ as a free $\mathbb{Q}[s,s^{-1}]$-module.

###  3.1. Fixed loci

The fixed loci for the action of $\mathbb{C}^{*}$ on
$\overline{M}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$ and
$\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$ are very similar. They are
both given by the data of

  1. (1)

a graph $\Gamma=(V,E)$,

  2. (2)

a coloring $\zeta:V\to\\{\pm 1\\}$,

  3. (3)

a genus assignment $g:V\to\mathbb{Z}_{\geq 0}$,

  4. (4)

a map $d:V\sqcup E\to\mathbb{Z}_{\geq 0}$,

  5. (5)

a point assignment $p:\\{1,\dots,n\\}\to V$,

such that $\Gamma$ is connected and contains no self-edges, two vertices
directly connected by an edge do not have the same color $\zeta$,

$\displaystyle g=$ $\displaystyle h^{1}(\Gamma)+\sum_{v\in V}g(v),$
$\displaystyle d=$ $\displaystyle\sum_{i\in V\sqcup E}d(i),\quad d|_{E}\geq
1,$

and one further condition which depends on whether we look at
$\overline{M}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$ or
$\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$.

To state the stable quotient condition we need the following definition: A
vertex $v\in V$ is called non-degenerate if the inequality

$2g(v)-2+n(v)+\varepsilon d(v)>0$

holds for any $\varepsilon>0$. Here $n(v)$ is the number of edges at $v$ plus
the number of preimages under $p$ weighted by the corresponding weight.

Then for the combinatorial data on the stable quotients side we demand each
vertex to be non-degenerate, whereas for the stable maps data the additional
condition is $d|_{V}=0$.

In the combinatorial data for $\overline{M}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$
the vertices $v$ of $\Gamma$ correspond to curve components contracted to the
fixed point of $\mathbb{P}^{1}$ specified by $\zeta$, $0$ for $\zeta(v)=1$ and
$\infty$ for $\zeta(v)=-1$. The edges correspond to multiple covers of
$\mathbb{P}^{1}$ ramified at $0$ and $\infty$ with degree given by the map
$d$. For $\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$ the vertices of
$\Gamma$ correspond to components $C$ of the curve over which the subsheaf $S$
is an ideal sheaf of the trivial subsheaf given by one of the two 1
dimensional fixed subspaces of $\mathbb{C}^{2}$555These correspond to the two
fixed points in $\mathbb{P}^{1}$, $\zeta$ specifies which subspace of
$\mathbb{C}^{2}$ was chosen and $-d$ the degree of $S$. Edges correspond to
multiple covers of $\mathbb{P}^{1}$ ramified in the two torus-fixed points.

The fixed locus corresponding to the combinatorial data is, up to a finite
map, isomorphic to the product

$\prod_{v\in
V}\overline{M}_{g(v),(\mathbf{w}(v),\varepsilon^{d(v)})}/S_{d(v)},$

where $\mathbf{w}(v)$ is a $(|p^{-1}(v)|+|\\{e\text{ adjacent to
}v\\}|)$-tuple, which we will index by $p^{-1}(v)\sqcup\\{e\text{ adjacent to
}v\\}$, such that $w(v)_{i}=w_{i}$ if $i\in p^{-1}(v)$ and $w(v)_{e}=1$ for
adjacent edges $e$. The symmetric group $S_{d(v)}$ permutes the
$\varepsilon$-stable points. The product should be taken only over all non-
degenerate vertices.

###  3.2. The formula

The virtual localization formula expresses the virtual fundamental class as a
sum of the (virtual) fundamental classes of the fixed loci $X$ weighted by the
inverse of the equivariant Euler class of the corresponding virtual normal
bundle $N_{X}$. In order to make sense of this inverse it is necessary to
localize the the equivariant cohomology ring by $s$.

The inverse of the Euler class of $N_{X}$ is in both cases a product

$\operatorname{Cont}_{\Gamma}:=\prod_{v}\operatorname{Cont}(v)\prod_{e\text{
edge}}\operatorname{Cont}(e),$

for certain vertex and edge contributions depending only on the data of the
graph corresponding to the vertex or edge. The contributions
$\operatorname{Cont}(e)$ and $\operatorname{Cont}(v)$ for $v$ degenerate are
pulled back from the equivariant cohomology of a point. We will not need to
know the exact form of the edge and unstable vertex contributions here apart
from the fact that the contribution of a vertex $v$ with $d(v)=g(v)=0$ and
$|\mathbf{w}(v)|=1$ is equal to $1$.

The non-degenerate vertex contribution is pulled back from

$A_{\mathbb{C}^{*}}^{*}(\overline{M}_{g(v),(\mathbf{w}(v),\varepsilon^{d(v)})}/S_{d(v)})\otimes\mathbb{Q}[s,s^{-1}].$

It is given by

(1)
$\operatorname{Cont}(v)=(\zeta(v)s)^{g(v)-d(v)-1}\sum_{j=0}^{\infty}\frac{c_{i}(\mathbb{F}_{d(v)})}{(\zeta(v)s)^{i}}\prod_{e}\frac{\zeta(v)s}{\omega_{e}^{v}-\psi_{e}},$

where $\mathbb{F}_{d}$ is the K-theory class
$\mathbb{F}_{d}=\mathbb{E}^{*}-\mathbb{B}_{d}-\mathbb{C}$, $\prod_{e}$ denotes
a product over edges adjacent to $v$ and

$\omega_{e}^{v}=\frac{\zeta(v)s}{d(e)}.$

The dual $\mathbb{E}^{*}$ of the Hodge bundle $\mathbb{E}$ has fiber

$(H^{0}(C,\omega_{C}))^{*}$

over a marked curve $(C,p_{i})$, the rank $d$ bundle $\mathbb{B}_{d}$ on
$\overline{M}_{g,(\mathbf{w},\varepsilon^{d})}/S_{d}$ has the fiber

$H^{0}(C,\mathcal{O}_{C}(p_{n+1}+\dotsb+p_{n+d})|_{p_{n+1}+\dotsb+p_{n+d}})$

over a marked curve $(C,p_{i})$. The bundle $\mathbb{B}_{d}$ can also be
thought as an $S_{d}$ invariant bundle living on
$\overline{M}_{g,(\mathbf{w},\varepsilon^{d})}$. In [3] the Chern classes of
this lifted bundle have been computed:

$c(-\mathbb{B}_{d})=\prod_{i=n+1}^{n+d}\frac{1}{1-\psi_{n+i}-\sum_{j=n+1}^{n+i}D_{ij}}.$

Notice the similarity to the $\Phi$ function. The Chern classes for
$\mathbb{B}_{d}$ on $\overline{M}_{g,(\mathbf{w},\varepsilon^{d})}/S_{d}$ can
be calculated by push-forward along the finite projection map and dividing by
the degree $d!$.

###  3.3. Comparison

The $\mathbb{C}^{*}$ actions on
$\overline{M}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$ and
$\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$ are compatible with the
comparison map
$c:\overline{M}_{g,\mathbf{w}}(\mathbb{P}^{1},d)\to\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$
and $c$ hence maps fixed loci to fixed loci. The image of a locus
corresponding to some combinatorial data is given by contracting all
degenerate vertices, adding the values of $d$ of the contracted vertices and
edges to the degree of the image vertex of the contraction, and replacing the
point assignment $p$ by its composition with the contraction.

The Gromov-Witten stable quotient comparison says that the contribution of a
stable quotient fixed locus to the virtual fundamental class of
$\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$ is the sum of push-forwards of
the contributions of all stable maps loci in the preimage of $c$ to the
virtual fundamental class of $\overline{M}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$.
This in particular implies that

$c_{*}([\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)]^{vir})=[\overline{M}_{g,\mathbf{w}}(\mathbb{P}^{1},d)]^{vir}.$

##  4\. The open locus

We will first restrict ourselves to the special case of $n$-fold tensor powers
$M_{g|n}$ of the universal curve $C_{g}$ over $M_{g}$. This case occurs when
the weights are sufficiently small (i.e. $\sum_{i=1}^{n}w_{i}<1$) and we
restrict ourselves to the locus corresponding to smooth curves.

###  4.1. Statement of the stable quotient relations

Let us define the bracket operator $\\{\\}_{\Delta}$ on
$\mathbb{Q}[[t,p_{1},\dotsc,p_{n}]]$ in terms of the operators from the
introduction by

$\displaystyle\\{f(t)\\}_{\Delta}=$ $\displaystyle-\\{f(t)\\}_{\kappa}$
$\displaystyle\left\\{f(t)\prod_{i\in S}p_{i}\right\\}=$
$\displaystyle\\{f(t)\\}_{D_{S}}$
$\displaystyle\\{p_{i}^{e}f(t,p_{1},\dotsc,p_{n})\\}_{\Delta}=$
$\displaystyle\\{p_{i}f(t,p_{1},\dotsc,p_{n})\\}_{\Delta},$

if $e>0$. For example $\\{t^{2}p_{1}^{3}\\}_{\Delta}=t^{2}\psi_{1}^{2}$ and
$\\{t^{2}p_{1}p_{2}\\}_{\Delta}=-D_{12}\psi_{1}=-D_{12}\psi_{2}$.

######  Proposition 2.

The relations given by

$\sum_{\zeta\in\\{\pm 1\\}}\zeta^{g-1}\left[\exp\left(-\frac{1}{2}\zeta
p+\\{\exp(-pD)\gamma(\zeta
t,x)\\}_{\Delta}\right)\right]_{t^{r}x^{d}\mathbf{p}^{\mathbf{a}}}=0,$

with the differential operator $D=tx\frac{\mathrm{d}}{\mathrm{d}x}$ and
$p=p_{1}+\dots+p_{n}$ hold in $M_{g|n}$ under the condition
$g-2d-1+|\mathbf{a}|<r$.

###  4.2. Proof of the stable quotient relations

We generalize here the localization method of [6].

For each $i\in\\{1,\dots,n\\}$ we can define a class $s_{i}\in
A^{1}(M_{g|n}(\mathbb{P}^{1},d))$ as the pull-back of $c_{1}(\mathcal{S})$
from the universal curve $C_{g|n}(\mathbb{P}^{1},d)$ via the $i$-th section.

For given nonnegative integers $a_{i}$ let us look at the class

$\mathbf{s}^{\mathbf{a}}=\prod_{i=1}^{n}(-s_{i})^{a_{i}}\in
A^{|\mathbf{a}|}(M_{g|n}(\mathbb{P}^{1},d)).$

The strategy is to study the $\mathbb{C}^{*}$ action from Section 3, to lift
$\mathbf{s}^{\mathbf{a}}$ to equivariant cohomology and to write down the
localization formula calculating the push-forward of this class to the
equivariant cohomology of $M_{g|n}$. As we have seen the general form of the
localization formula is a sum of contributions from the fixed loci. For each
fixed locus a class from its equivariant Chow ring _localized by the
localization variable_ $s$ pushed forward via the inclusion of the fixed
locus. We get the relations from the fact that the rational functions in the
localization variable we obtain must actually be polynomials in $s$ after
summing over all fixed loci.

######  Remark 1.

In [3] and [6] the same strategy was pursued but other related classes were
chosen to be pushed forward. In similar spirit we could add factors of the
form $\pi_{*}(s_{n+1}c_{1}(\omega_{\pi})^{b})$, where
$\pi:Q_{g|n+1}(\mathbb{P}^{1},d)\to Q_{g|n}(\mathbb{P}^{1},d)$ is the
forgetful map and $\omega_{\pi}$ is the relative dualising sheaf, to the class
we are pushing forward. However because of the commutative diagram

$Q_{g|n+1}(\mathbb{P}^{1},d)$$M_{g|n+1}$$Q_{g|n}(\mathbb{P}^{1},d)$$M_{g|n}$$\nu$$\pi$$\nu$$\pi$

and the fact that $c_{1}(\omega_{\pi})=\nu^{*}(\psi_{n+1})$ these will be
contained in the completed set of stable quotient relations.666To see this for
more than one factor (say $m$ factors) one needs to first interpret the
product as a class on the $m$-fold tensor power $X$ of
$Q_{g|n+1}(\mathbb{P}^{1},d)$ over $Q_{g|n}(\mathbb{P}^{1},d)$ and use the
birational map from $Q_{g|n+m}(\mathbb{P}^{1},d)$ to $X$.

In [6] only factors with $a=1$ were used to derive the FZ relations on
$M_{g}$. The results from Section 4.3.2 imply that allowing higher values for
$a$ would not have led to more stable quotient relations.

The action of $\mathbb{C}^{*}$ on $\mathbb{P}^{1}$ is induced by the action of
$\mathbb{C}^{*}$ on $\mathbb{C}^{2}$ given by
$([z_{0},z_{1}],\lambda)\mapsto[z_{0},\lambda z_{1}]$. This naturally induces
$\mathbb{C}^{*}$ actions not only on $Q_{g|n}(\mathbb{P}^{1},d)$ but also on
the universal curve $C_{g|n}(\mathbb{P}^{1},d)$ and the universal sheaf
$\mathcal{S}$. This gives a natural lift of the $s_{i}$ to equivariant
cohomology and therefore also a natural lift of $\mathbf{s}^{\mathbf{a}}$. We
will not choose this lift but instead

$\tilde{\mathbf{s}}^{\mathbf{a}}=\prod_{i=1}^{n}\left(-s_{i}-\frac{1}{2}s\right)^{a_{i}}\in
A_{\mathbb{C}^{*}}^{|\mathbf{a}|}(M_{g|n}(\mathbb{P}^{1},d)),$

where the $s_{i}$ are the natural lifts.

Let us consider the localization formula for this equivariant lift applied to
the push-forward

$\nu_{*}(\mathbf{s}^{\mathbf{a}}\cap[M_{g|n}(\mathbb{P}^{1},d)]^{vir})\in
A_{2g+2d-2+n-|\mathbf{a}|}(M_{g|n}).$

Let us shorten this by writing $\nu_{*}^{vir}(\mathbf{s}^{\mathbf{a}})$ for
this push-forward after capping with the virtual fundamental class.

Because we assume the strict inequality $\sum_{i=1}^{n}w_{i}<1$ there are only
two fixed loci with respect to the torus actions in the description of Section
3.1. Concretely in this special case they correspond to elements
$(C,p_{1},\dotsc,p_{n})$ of $M_{g|n}$ with quotient sequence

$0\to\mathcal{O}_{C}(-p_{n+1}-\dotsb-
p_{n+d})\to\mathcal{O}_{C}\to\mathcal{O}_{C}^{2}\to Q\to 0$

with $\mathcal{O}_{C}\to\mathcal{O}_{C}^{2}$ induced from one of the two torus
invariant inclusions of $\mathbb{C}$ as a coordinate axis in $\mathbb{C}^{2}$.
Here $\mathcal{O}_{C}(-p_{n+1}-\dotsb-p_{n+d})$ is an ideal sheaf of
$\mathcal{O}_{C}$ of degree $d$. Both fixed point loci can be identified with
$M_{g|n+d}/S_{d}$ where the symmetric group $S_{d}$ permutes the last $d$
markings.

Since the graphs corresponding to the fixed point loci have each only one
vertex and no edge we can calculate the inverse of the equivariant Euler class
of the normal bundle to each fixed locus using (1). It is given by

$(\zeta s)^{g-d-1}\sum_{j=0}^{\infty}\frac{c_{i}(\mathbb{F}_{d})}{(\zeta
s)^{i}},$

where $\zeta$ is $+1$ and $-1$ for $0$ and $\infty$ respectively.

Applying the fixed point formula we obtain for the $s^{c}$ part of the push-
forward

(2)
$[\nu_{*}^{vir}(\tilde{\mathbf{s}}^{\mathbf{a}})]_{s^{c}}=\frac{1}{d!}\sum_{\zeta}\varepsilon_{*}\left[\prod_{i=1}^{n}\left(-s_{i}t-\frac{1}{2}\zeta\right)^{a_{i}}\sum_{j=0}^{\infty}t^{j}\zeta^{g-d-1-j}c_{j}(\mathbb{F}_{d})\right]_{t^{g-1-d-c+|\mathbf{a}|}},$

where $\varepsilon:M_{g|n+d}\to M_{g|n}$ forgets the last $d$ markings. We
have here by abuse of notation denoted similarly defined classes on
$M_{g|n+d}$ with the same name as on $M_{g|n}(\mathbb{P}^{1},d)$. The
expression $(-s_{i}t-\frac{1}{2}\zeta)$ comes from the fact that the torus
acts trivially on the subspace of $\mathbb{C}^{2}$ given by $0$ and with
weight $1$ on the subspace given by $\infty$. The equivariant lift of the
$s_{i}$ was chosen in order to have this symmetric expression.

Since the push-forward must be an honest equivariant class the classes (2)
must be zero if $c<0$.

Let us package these relations into a power series. We have

(3)
$\prod_{i=1}^{n}a_{i}!\cdot[\nu_{*}^{vir}(\tilde{\mathbf{s}}^{\mathbf{a}})]_{s^{c}}=\sum_{\zeta}\zeta^{g-d-1}\varepsilon_{*}[\exp(-T_{1})T_{2}]_{t^{r}x^{d}p^{\mathbf{a}}},$

with

$\displaystyle T_{1}=$
$\displaystyle\sum_{i=1}^{n}\left(s_{i}t+\frac{1}{2}\zeta\right)p_{i},$
$\displaystyle T_{2}=$
$\displaystyle\sum_{j=0}^{\infty}\sum_{d=0}^{\infty}(\zeta
t)^{j}c_{j}(\mathbb{F}_{d})\frac{x^{d}}{d!}.$

Since

$s_{i}=\sum_{j=n+1}^{n+d}D_{ij}$

we can rewrite $T_{1}$ as

$T_{1}=\sum_{i=1}^{n}\sum_{j=n+1}^{n+d}D_{ij}p_{i}+\frac{1}{2}\zeta\sum_{i=1}^{n}p_{i}.$

####  4.2.1. Relations between diagonal classes

In order to better understand $c(\mathbb{F}_{d})$ let us collect the universal
relations between classes in $A^{*}(M_{g|n})$ here.

The basic relations are

$\displaystyle D_{ij}\psi_{i}=D_{ij}\psi_{j}=-D_{ij}^{2}$ $\displaystyle
D_{ij}D_{ik}=D_{ijk}$

for pairwise different $i$, $j$, $k$ (compare to [16]). Let $D_{S,a}\in
A^{a}(M_{g|n})$ by defined for any $S\subset\\{1,\dotsc,n\\}$ and $a\geq|S|-1$
by

$D_{S,a}=\begin{cases}\psi_{i}^{a}&\text{, if }S=\\{i\\}\\\
(-1)^{|S|-1}D_{S}\psi_{i}^{a-|S|+1}&\text{, else, where
}i\in\\{1,\dotsc,d\\}.\end{cases}$

Then $D_{S,a}D_{T,b}=D_{S\cup T,a+b}$ if $S\cap T\neq\emptyset$.

######  Lemma 5.

Each monomial $M$ in diagonal and cotangent line classes in $A^{*}(M_{g|n})$
can be written as a product

$M=\prod_{S\in P}D_{S,a(S)+|S|-1}$

for some partition $P$ of $\\{1,\dotsc,d\\}$ and a function
$a:P\to\mathbb{Z}_{\geq 0}$. This product decomposition is unique if we only
use the above relations between diagonal and cotangent line classes.

######  Remark 2.

If the partition $P$ is the one element set partition, we say that $M$ is
connected. We call the factors of the decomposition (or just the elements of
$P$) the connected components of $M$.

The push-forward under the forgetful map $\pi:M_{g|n}\to M_{g|n-1}$ is given
by

$\pi_{*}D_{S,a}=\begin{cases}0&\text{, if }n\notin S\\\ \kappa_{a-1}&\text{,
if }|S\cap\\{n\\}|=1\\\ -D_{S\setminus\\{n\\},a-1}&\text{, else.}\end{cases}$

Here and in the rest of this article $\kappa_{-1}$ is defined to be zero.

####  4.2.2. Simplest relations

Let us first consider the stable quotient relations in the case of
$\mathbf{a}=0$. Then they are simply

$0=\sum_{\zeta}\zeta^{g-d-1}\varepsilon_{*}\left[\sum_{d=0}^{\infty}\sum_{j=0}^{\infty}(\zeta
t)^{j}c_{j}(\mathbb{F}_{d})\frac{x^{d}}{d!}\right]_{t^{r}x^{d}}$

for

(4) $r>g-1-2d.$

By the definition of $\mathbb{F}_{d}$ as a K-theoretic difference of
$\mathbb{E}^{*}$, a trivial rank $1$ bundle and $\mathbb{B}_{d}$ the inner sum
breaks into two factors. The part corresponding to $\mathbb{E}^{*}$ is pulled
back from $M_{g|n}$ and does not depend on $d$. Using Mumford’s formula [17]
we can therefore rewrite the relations as:

$\sum_{\zeta}\zeta^{g-d-1}\left[\exp\Big{(}-\sum_{i\geq
1}\frac{B_{2i}}{2i(2i-1)}\kappa_{2i-1}(t\zeta)^{2i-1}\Big{)}\varepsilon_{*}\sum_{d=0}^{\infty}\sum_{j=0}^{\infty}(\zeta
t)^{j}c_{j}(-\mathbb{B}_{d})\frac{x^{d}}{d!}\right]_{t^{r}x^{d}}$

vanishes if (4) holds.

To deal with the second factor we formally expand it as a power series in $x$,
$t$ and the classes $D_{S,a}$ for various $S\subset\\{n+1,\dotsc,n+d\\}$ and
$a\in\mathbb{Z}_{\geq 0}$. By the exponential formula and the facts from
Section 4.2.1 we have that

$\varepsilon_{*}\sum_{d=0}^{\infty}\sum_{j=0}^{\infty}(\zeta
t)^{j}c_{j}(-\mathbb{B}_{d})\frac{x^{d}}{d!}=\exp\left(\varepsilon_{*}\sum_{d=1}^{\infty}\sum_{r=0}^{\infty}S_{d}^{r}D_{\\{n+1,\dotsc,n+d\\},r}(\zeta
t)^{r}\frac{x^{d}}{d!}\right)$

where

$\log\left(\sum_{d=0}^{\infty}\prod_{i=1}^{d}\frac{1}{1-it}\frac{x^{d}}{d!}\right)=\sum_{d=1}^{\infty}\sum_{r=0}^{\infty}S_{d}^{r}t^{r}\frac{x^{d}}{d!}.$

With $\varepsilon_{*}D_{\\{n+1,\dotsc,n+d\\},r}=(-1)^{d-1}\kappa_{r-d}$ and
noticing the similarities between the series defining $S_{d}^{r}$ and
$\log(\Phi)$ we obtain

$\varepsilon_{*}\sum_{d=0}^{\infty}\zeta^{d}\sum_{j=0}^{\infty}\frac{(\zeta
t)^{j}}{t^{d}}c_{j}(-\mathbb{B}_{d})\frac{x^{d}}{d!}=\exp\left(-\\{\log(\Phi(\zeta
t))\\}_{\kappa}\right)$

and so the stable quotient relations in the case $\mathbf{a}=0$ are

$\left[\sum_{\zeta}\zeta^{g-1}\exp\left(-\\{\gamma(\zeta
t)\\}_{\kappa}\right)\right]_{t^{r}x^{d}}=0$

under Condition (4).

####  4.2.3. General relations

We will investigate how monomials in the $s_{i}$ affect the push-forward under
$\varepsilon$ of monomials supported only on the last $d$ points.

Notice that for each partition $P$ of $\\{n+1,\dotsc,n+d\\}$ we have

(5) $\exp\left(-\sum_{i=1}^{n}s_{i}\right)=\prod_{S\in
P}\exp\left(-\sum_{i=1}^{n}\sum_{j\in S}D_{ij}\right),$

and each factor is pulled back via the map forgetting all points in
$\\{n+1,\dotsc,n+d\\}$ not in $S$.

Moreover notice that if $M$ is a connected monomial supported in the last $d$
marked points with
$\varepsilon_{*}M=[-\\{f(t)\\}_{\kappa}]_{t^{r}}=[\\{f(t)\\}_{\Delta}]_{t^{r}}$,
we have

(6)
$[\varepsilon_{*}((-s_{i}p_{i})^{e}x^{d}M)]_{x^{d}}=[\varepsilon_{*}((-dD_{i,n+1}p_{i})^{e}x^{d}M)]_{x^{d}}=[\\{(p_{i}D)^{e}(x^{d}f(t))\\}_{\Delta}]_{t^{r}x^{d}}.$

From (5), (6) and the identity

$\exp(pD)\exp(f(t,x))=\exp(\exp(pD)f(t,x))$

we obtain the general stable quotient relations

$\sum_{\zeta}\zeta^{g-1}\left[\exp\left(-\frac{1}{2}\zeta
p+\\{\exp(pD)\gamma(\zeta
t,x)\\}_{\Delta}\right)\right]_{t^{r}x^{d}\mathbf{p}^{\mathbf{a}}}=0,$

if

(7) $r>g-1-2d+|\mathbf{a}|.$

###  4.3. Evaluation of the relations

####  4.3.1. Minor simplification

Notice that in the relations of Proposition 2 the summands in the $\zeta$ sum
are equal up to a sign

$(-1)^{g-1+r+|\mathbf{a}|}.$

Therefore the relations are in fact zero if $g+r+|\mathbf{a}|\equiv
0\pmod{2}$, and in the case

(8) $g-1+r+|\mathbf{a}|\equiv 0\pmod{2}$

we can reformulate them to

$\left[\exp\left(-\frac{1}{2}p+\\{\exp(pD)\gamma(t,x)\\}_{\Delta}\right)\right]_{t^{r}x^{d}\mathbf{p}^{\mathbf{a}}}=0.$

####  4.3.2. Differential algebra

In this section we will establish that it is enough to consider the stable
quotient relations in the case that $a_{i}<2$ for all $i\in\\{1,\dotsc,m\\}$.

The series $\delta=D\gamma-\frac{1}{2}$ satisfies the differential equation

$D\delta=-\delta^{2}+x+\frac{1}{4},$

as can be derived from the differential equation

$D(\Phi-D\Phi)=-x\Phi,$

which is satisfied by $\Phi$ as seen by looking at its series definition.

Reformulating the relations with $\delta$ gives

$\left[\exp\left(-\\{\gamma\\}_{\kappa}+\sum_{i=1}^{\infty}\frac{1}{i!}\\{p^{i}D^{i-1}\delta\\}_{\Delta}\right)\right]_{t^{r}x^{d}\mathbf{p}^{\mathbf{a}}}=0$

if (7) and (8) hold.

Let us consider

$G=\frac{\frac{\partial^{2}}{\partial p_{j}^{2}}F}{F},$

with

$F=\exp\left(-\\{\gamma\\}_{\kappa}+\sum_{i=1}^{\infty}\frac{1}{i!}\\{p^{i}D^{i-1}\delta\\}_{\Delta}\right).$

We have that

$G=\frac{\partial^{2}}{\partial
p_{j}^{2}}\sum_{i=1}^{\infty}\frac{1}{i!}\\{p^{i}D^{i-1}\delta\\}_{\Delta}+\left(\frac{\partial}{\partial
p_{j}}\sum_{i=1}^{\infty}\frac{1}{i!}\\{p^{i}D^{i-1}\delta\\}_{\Delta}\right)^{2}.$

We can move the bracket out of the square in the second summand since the part
of the brackets in both factors which is not annihilated by
$\frac{\partial}{\partial p_{j}}$ contains at least a factor $p_{j}$. So we
obtain

$\displaystyle G=$
$\displaystyle\\{\exp(pD)D\delta+\left(\exp(pD)\delta\right)^{2}\\}_{\Delta_{j}}=\\{\exp(pD)(D\delta+\delta^{2})\\}_{\Delta_{j}}$
$\displaystyle=$
$\displaystyle\left\\{\exp(pD)\left(x+\frac{1}{4}\right)\right\\}_{\Delta_{j}}=\frac{1}{4}+x\\{\exp(pt)\\}_{\Delta_{j}},$

where the operator $\\{\\}_{\Delta_{j}}$ is defined by
$\\{f\\}_{\Delta_{j}}=p_{j}^{-1}\\{p_{j}f\\}_{\Delta}$. Therefore we have that

$\frac{\partial^{2}}{\partial
p_{j}^{2}}F=\left(\frac{1}{4}+x\\{\exp(pt)\\}_{\Delta_{j}}\right)F$

and can express the relations for $a_{j}\geq 2$ in terms of lower relations
multiplied by monomials in cotangent line and diagonal classes.

####  4.3.3. Substitution

The differential equation satisfied by $-t\gamma$ has been studied by Ionel in
[18]. In [6] her results were extended to give formulas for $D^{i}\gamma$. We
will collect some of their results on $\gamma$ and its derivatives here.

With the new variables

$u=\frac{t}{\sqrt{1+4x}},\qquad y=\frac{-x}{1+4x}$

one can write

$\gamma=\frac{1}{t}(t\gamma)(0,x)+\frac{1}{4}\log(1+4y)+\sum_{k=1}^{\infty}\sum_{j=0}^{k}c_{k,j}u^{k}y^{j},$

for some coefficients $c_{k,j}$, which are defined to vanish outside the
summation region above. Furthermore, we have for the derivatives of $\delta$

$D^{i-1}\delta=(1+4y)^{-\frac{i}{2}}\left(\sum_{j=0}^{i-1}b_{j}^{i}u^{i-1}y^{j}-\sum_{k=0}^{\infty}\sum_{j=0}^{k+i}c_{k,j}^{i}u^{k+i}y^{j}\right)=:(1+4y)^{-\frac{i}{2}}\delta_{i},$

for some coefficients $b_{j}^{i}$, $c_{k,j}^{i}$.

We will also need a result by Ionel relating coefficients of a power series
$F$ in $x$ and $t$ before and after the variable substitution:

$[F]_{t^{r}x^{d}}=(-1)^{d}[(1+4y)^{\frac{r+2d-2}{2}}F]_{u^{r}y^{d}}$

Let us now apply these formulas to the relations. Using the fact that
$\kappa_{-1}=0$ we get

$\left[(1+4y)^{e}\exp\left(-\left\\{\sum_{k=1}^{\infty}\sum_{j=0}^{k}c_{k,j}u^{k}y^{j}\right\\}_{\kappa}+\sum_{i=1}^{\infty}\frac{1}{i!}\\{p^{i}\delta_{i}\\}_{\Delta}\right)\right]_{u^{r}y^{d}\mathbf{p}^{\mathbf{a}}}=0,$

under conditions (7) and (8) with the exponent

$e=\frac{r+2d-2}{2}-\frac{\kappa_{0}}{4}-\frac{|\mathbf{a}|}{2}=\frac{r+2d-1-g-|\mathbf{a}|}{2}.$

####  4.3.4. Extremal coefficients

It is noticeable that in the series

$\sum_{k=1}^{\infty}\sum_{j=0}^{k}c_{k,j}u^{k}y^{j},\quad\sum_{j=0}^{i-1}b_{j}^{i}u^{i-1}y^{j}-\sum_{k=0}^{\infty}\sum_{j=0}^{k+i}c_{k,j}^{i}u^{k+i}y^{j}$

appearing in the formulas for $\gamma$ and $D^{i-1}\delta$ the $y$ degree is
bounded from above by the $u$ degree. We will only be interested in the
extremal coefficients. Here the $A$ and $C_{i}$ come into the picture.

In [18] it has been proven that

$\log(A(t))=\sum_{k=1}^{\infty}c_{k,k}t^{k}$

and in [6] it is shown that

$2^{i}C_{i}(t)=b_{i-1}^{i}t^{i-1}-\sum_{k=0}^{\infty}c_{k,k+i}^{i}t^{k+i}.$

We see that in the exponential factor of the relations for each summand the
$y$ degree is bounded from above by the $u$ degree. Furthermore the exponent
$e$ of the $(1+4y)$ factor is integral and positive by (8) and (7). This
implies that the relation is zero unless

(9) $3r\geq g+1+|\mathbf{a}|$

holds. With the following lemma we can extract the extremal part of the
relations.

######  Lemma 6.

Fix any $\mathbb{Q}$ algebra $A$, any $F\in A[y]$ and any
$c\in\mathbb{Z}_{\geq 0}$. The relations

$[(1+4y)^{d}F]_{y^{d}}=0$

for all $d>c$ imply $F=0$.

###### Proof.

The relations can be rewritten to

$\left[\left(\frac{1}{y}+4\right)^{d}F\right]_{y^{0}}=0.$

The lemma follows by the fact that $F$ is a polynomial in $y$ and linear
algebra. ∎

Using the lemma and the formulas for the extremal coefficients we obtain that
the relations

$\left[\exp(-\\{\log(A)\\}_{\kappa})\exp\left(\sum_{i=1}^{\infty}\frac{1}{i!}\\{p^{i}C_{i}\\}_{\Delta}\right)\right]_{z^{r}\mathbf{p}^{\mathbf{a}}}=0$

hold under the conditions (8) and (9).

Ignoring terms of higher order in the $p_{i}$ the second factor can be
rewritten

$\displaystyle\exp\left(\sum_{i=1}^{\infty}\frac{1}{i!}\\{p^{i}C_{i}\\}_{\Delta}\right)\equiv\exp\left(\sum_{\emptyset\neq
S\subseteq\\{1,\dotsc,n\\}}\\{p^{S}C_{|S|}\\}_{\Delta}\right)$
$\displaystyle\equiv\sum_{S\subseteq\\{1,\dotsc,n\\}}p^{S}\sum_{P\vdash
S}\prod_{i\in P}\\{C_{|i|}\\}_{D_{i}}.$

Thus we finally obtain that the FZ type relations

$\left[\exp(-\\{\log(A)\\}_{\kappa})\sum_{P\vdash S}\prod_{i\in
P}\\{C_{|i|}\\}_{D_{i}}\right]_{z^{r}}=0,$

hold for any $S\subseteq\\{1,\dotsc,n\\}$ if (8) and (9).

##  5\. Localization sums

In the derivation of the more general stable quotient relations we will need
to deal with two types of localization sums related to the nodes and marked
points respectively. To keep the length of the proof of the stable quotient
relations more reasonable we will deal with them here. The sums are genus
independent and have been studied more generally by Givental [19]. We apply
here his method in a special case. See also [20] and [21].

Let $N_{\mathbb{C}^{*}}^{*}(\mathbb{P}^{1})$ be the Novikov ring of
$\mathbb{P}^{1}$ with values in $\mathbb{C}[s,s^{-1}]$. We define a formal
Frobenius manifold structure on
$X=A_{\mathbb{C}^{*}}^{*}(\mathbb{P}^{1})\otimes
N_{\mathbb{C}^{*}}^{*}(\mathbb{P}^{1})$ over
$N_{\mathbb{C}^{*}}^{*}(\mathbb{P}^{1})$ using equivariant Gromov-Witten
theory. The $N_{\mathbb{C}^{*}}^{*}(\mathbb{P}^{1})$-module $X$ is free of
dimension two with idempotent basis $\\{\phi_{0},\phi_{\infty}\\}$ for
$\phi_{i}=[i]/e_{i}$, where $e_{i}$ is the equivariant Euler class of the
tangent space of $\mathbb{P}^{1}$ at $i$. We denote the corresponding
coordinate functions by $y_{0}$, $y_{\infty}$. This gives a basis
$\\{\frac{\partial}{\partial y_{0}},\frac{\partial}{\partial y_{\infty}}\\}$
of the space of vector fields on $X$. The metric is given in this basis by

$g=\begin{pmatrix}s^{-1}&0\\\
0&-s^{-1}\end{pmatrix}=\begin{pmatrix}e_{0}^{-1}&0\\\
0&e_{\infty}^{-1}\end{pmatrix}.$

The primary equivariant Gromov-Witten potential is

$\sum_{n=0}^{\infty}\frac{1}{n!}\sum_{d=0}^{\infty}x^{d}\int_{[\overline{M}_{0,n}(\mathbb{P}^{1},d)]^{vir}}\prod_{i=1}^{n}\operatorname{ev}_{i}^{*}(y_{0}\phi_{0}+y_{\infty}\phi_{\infty})=\frac{y_{0}^{3}}{6s}-\frac{y_{\infty}^{3}}{6s}+xe^{w},$

where we have set $w=(y_{0}-y_{\infty})/s$ (compare to [22]). Thus we have

$\alpha=\begin{pmatrix}\mathrm{d}y_{0}+\frac{x}{e_{0}}e^{w}\mathrm{d}w&\frac{x}{e_{\infty}}e^{w}\mathrm{d}w\\\
\frac{x}{e_{0}}e^{w}\mathrm{d}w&\mathrm{d}y_{\infty}+\frac{x}{e_{\infty}}e^{w}\mathrm{d}w\end{pmatrix},$

for the matrix of one forms $\alpha$ defined by

$\alpha=\sum_{i}A_{i}\mathrm{d}y_{i}$

from the matrices $A_{i}$ defined by the quantum product

$\frac{\partial}{\partial y_{i}}\star\frac{\partial}{\partial
y_{a}}=\sum_{b}[A_{i}]_{a}^{b}\frac{\partial}{\partial y_{b}}.$

Using $\alpha$ we can compactly write down the differential equation for the
$S$-matrix

$(t\mathrm{d}-\alpha)S=0,$

with initial condition $S(0,0)=\mathrm{Id}$.

If we set

$S=\begin{pmatrix}S_{0}^{0}&S_{\infty}^{0}\\\
S_{0}^{\infty}&S_{\infty}^{\infty}\end{pmatrix}$

this gives the system

$t\mathrm{d}S_{i}^{j}=S_{i}^{j}\mathrm{d}y_{j}+\sum_{k}S_{i}^{k}\frac{x}{e_{k}}e^{w}\mathrm{d}w,$

with solution

$S_{i}^{j}=e^{\frac{y_{i}}{t}}\left(\left(\frac{1+\zeta_{i}\zeta_{j}}{2}-tx\frac{\mathrm{d}}{\mathrm{d}x}\right)\Phi\right)\left(-\frac{t}{e_{i}},\frac{xe^{w}}{e_{i}^{2}}\right),$

with the signs $\zeta_{i}=\frac{e_{i}}{s}\in\\{\pm 1\\}$.

There is a set of canonical coordinates $u^{i}$ on $X$ defined by the
localization sums

$u^{i}=\sum_{n=0}^{\infty}\sum_{d=0}^{\infty}x^{d}\sum_{\Gamma\in
G_{0,n+2}^{u^{i}}(\mathbb{P}^{1},d)}\operatorname{Cont}_{\Gamma}\left(e_{i}\frac{\partial^{2}F_{0}}{(\partial
y_{i})^{2}}\right),$

where $G_{0,n+2}^{u^{i}}(\mathbb{P}^{1},d)$ is the set of $n+2$-point (each of
weight 1) degree $d$ localization graphs with the first two points on a single
component contracted to $i$. By the Gromov-Witten stable quotient comparison
and since there is only one fixed locus on the stable quotient side we have

$u^{i}(0,0)=e_{i}e_{i}^{-d-1}\sum_{d=1}^{\infty}\frac{x^{d}}{d!}\int_{\overline{M}_{0,2|d}}\sum_{j=0}^{\infty}\frac{c_{j}(\mathbb{F}_{d})}{e_{i}^{j}},$

where $\overline{M}_{0,2|d}=\overline{M}_{0,(1^{2},\varepsilon^{d})}$ is a
Losev-Manin space. Since the Hodge bundle is trivial on $\overline{M}_{0,2|d}$
this simplifies to

$u^{i}(0,0)=e_{i}e_{i}^{-d-1}\sum_{d=1}^{\infty}\frac{x^{d}}{d!}\int_{\overline{M}_{0,2|d}}\sum_{j=0}^{\infty}\frac{c_{j}(-\mathbb{B}_{d})}{e_{i}^{j}}.$

Recalling the definition of $\mathbb{B}_{d}$ we can write the integrand on the
right hand side as a sum of monomials in $\psi$ and diagonal classes. The
integral of such a monomial vanishes unless it is the diagonal where all $d$
points come together. The constants $-C_{d}^{-1}$ which are defined from
$\log(\Phi)$ by

$\log(\Phi(t,x))=\sum_{d=1}^{\infty}\sum_{r=-1}^{\infty}C_{d}^{r}t^{r}\frac{x^{d}}{d!}$

exactly count these contributions. So we get

(10) $u^{i}(0,0)=-\sum_{d=1}^{\infty}C_{d}^{-1}\frac{x^{d}}{d!}e_{i}^{-2d+1}.$

Using the $S$-matrix and the $u^{i}$ we can now calculate the series we are
interested in. We will not regard $S$ and the $u^{i}$ outside $(0,0)$, so let
us write from now on $S_{i}^{j}$ for $S_{i}^{j}(0,0)$ and $u^{i}$ for
$u^{i}(0,0)$.

The first series $P^{ij}$ will be needed to deal with stable quotient
localization chains containing one of the weight $1$ marked points. We have

$P^{ij}(t,x):=\frac{1+\zeta_{i}\zeta_{j}}{2}+\sum_{d=1}^{\infty}x^{d}\sum_{\Gamma\in
G_{0,2}^{P^{ij}}(\mathbb{P}^{1},d)}\frac{e_{i}e^{u^{i}/\omega_{\Gamma}}}{\omega_{\Gamma}-t}\operatorname{Cont}_{\Gamma}\left(\frac{\partial
F_{0}}{\partial y_{i}\partial
y_{j}}\right)=e^{u^{i}/t}\zeta_{i}\zeta_{j}S_{i}^{j}(-t),$

where $G_{0,2}^{P^{ij}}(\mathbb{P}^{1},d)$ is the set of all localization
graphs with the first marking on a valence 2 vertex at $i$ and the second
marking at $j$ and $\omega_{\Gamma}$ is the $\omega$ as in (1) corresponding
to the flag at the first marking.

The second series is needed to deal with chains at the nodes of the curve. We
have

$\displaystyle E^{ij}(t_{1},t_{2},x):=\sum_{d=1}^{\infty}x^{d}\sum_{\Gamma\in
G_{0,2}^{E^{ij}}(\mathbb{P}^{1},d)}\frac{e_{i}e^{u^{i}/\omega_{\Gamma,1}}}{\omega_{\Gamma,1}-t_{1}}\frac{e_{j}e^{u^{j}/\omega_{\Gamma,2}}}{\omega_{\Gamma,2}-t_{2}}\operatorname{Cont}_{\Gamma}\left(\frac{\partial
F_{0}}{\partial y_{i}\partial y_{j}}\right)$
$\displaystyle=\frac{s}{t_{1}+t_{2}}\left(\frac{\zeta_{i}+\zeta_{j}}{2}-e^{u^{i}/t_{1}+u^{j}/t_{2}}\zeta_{i}\zeta_{j}\sum_{\mu}S_{i}^{\mu}(-t_{1})\zeta_{\mu}S_{j}^{\mu}(-t_{2})\right),$

where $G_{0,2}^{E^{ij}}(\mathbb{P}^{1},d)$ is the set of all localization
graphs with both markings on valence 2 vertices, the first marking mapped to
$i$, the second marking to $j$, and $\omega_{\Gamma,1}$, $\omega_{\Gamma,2}$
are $\omega$ as in Section 3.2 corresponding to the flag at the first and
second marking respectively.

Let us now simplify the expressions for $P^{ij}(t,x)$ and
$E^{ij}(t_{1},t_{2},x)$ using the explicit $S$-matrix and canonical
coordinates at 0. For $P^{ij}(t,x)$ we obtain

$\displaystyle P^{ij}(t,x)=$
$\displaystyle\exp\left(-\sum_{d=1}^{\infty}C_{d}^{-1}t^{-1}\frac{x^{d}}{d!}e_{i}^{-2d}\right)\left(\left(\frac{1+\zeta_{i}\zeta_{j}}{2}-t\zeta_{i}\zeta_{j}x\frac{\mathrm{d}}{\mathrm{d}x}\right)\Phi\right)\left(\frac{t}{e_{i}},\frac{x}{e_{i}^{2}}\right)$
$\displaystyle=$
$\displaystyle\left(\left(\frac{1}{2}-\zeta_{i}\zeta_{j}\delta\right)\Phi^{\prime}\right)\left(\frac{t}{e_{i}},\frac{x}{e_{i}^{2}}\right),$

where $\Phi^{\prime}$ is defined by

$\Phi^{\prime}(t,x)=\exp\left(-\sum_{d=1}^{\infty}C_{d}^{-1}t^{-1}\frac{x^{d}}{d!}\right)\Phi(t,x)=\exp\left(\sum_{d=1}^{\infty}\sum_{r=0}^{\infty}C_{d}^{r}t^{r}\frac{x^{d}}{d!}\right).$

Similarly we obtain

$\displaystyle\frac{t_{1}+t_{2}}{s}E^{ij}(t_{1},t_{2},x)=$
$\displaystyle\frac{\zeta_{i}+\zeta_{j}}{2}+$
$\displaystyle\Phi^{\prime}\left(\frac{t_{1}}{e_{i}},\frac{x}{e_{i}^{2}}\right)\Phi^{\prime}\left(\frac{t_{2}}{e_{j}},\frac{x}{e_{j}^{2}}\right)\left(\zeta_{i}\delta\left(\frac{t_{1}}{e_{i}},\frac{x}{e_{i}^{2}}\right)+\zeta_{j}\delta\left(\frac{t_{2}}{e_{j}},\frac{x}{e_{j}^{2}}\right)\right).$

##  6\. The general case

We now extend the relations of Section 4 to the boundary and allow nearly
arbitrary weights. More precisely we will assume that:

(11) If for some $S\subset\\{1,\dotsc,n\\}$ we have $\sum_{i\in S}w_{i}=1$, we
must have $|S|=1$.

We still obtain relations for any weight data since one can always modify
$\mathbf{w}$ a bit such that the moduli space $\overline{M}_{g,\mathbf{w}}$
does not change (whereas $\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)$ will
change). It is even not necessary to allow that there exist points of weight 1
at all but it is interesting to see different ways of obtaining the same
relations.

###  6.1. Statement of the stable quotient relations

Let $\mathcal{G}$ be the set of stable graphs describing the strata on
$\overline{M}_{g,\mathbf{w}}$. The data of $\mathcal{G}$ in particular
includes a map $p$ from $\\{1,\dotsc,n\\}$ to the set of vertices as in
Section 3.1. Let us assume that the first $n^{\prime}$ points are of weight
different from $1$ and the remaining $n^{\prime\prime}$ points are of weight
$1$.

######  Proposition 3.

For a codimension $r$, a degree $d$, an $n$-tuple $\mathbf{a}$ such that
$g-2d-1+|\mathbf{a}|<r-|E|$, in $A^{r}(\overline{M}_{g,\mathbf{w}})$ the
relation

$\displaystyle 0=\Bigg{[}\sum_{\begin{subarray}{c}\Gamma\in\mathcal{G}\\\
\zeta:\Gamma\to\\{\pm
1\\}\end{subarray}}\frac{1}{\operatorname{Aut}(\Gamma)}\xi_{\Gamma*}\Big{(}\prod_{v}\mathrm{Vertex^{3}}_{v}^{\zeta(v)}\prod_{e}$
$\displaystyle\mathrm{Edge^{3}}_{e}^{\zeta(v_{1}),\zeta(v_{2})}$
$\displaystyle\prod_{i=n^{\prime}+1}^{n}$
$\displaystyle\mathrm{Point^{3}}(\psi_{i}\zeta(p(i))t,p_{i})\Big{)}\Bigg{]}_{t^{r-|E|}x^{d}\mathbf{p}^{\mathbf{a}}}$

holds, where $\mathrm{Vertex^{3}}_{v}^{\zeta}$ is a product

$\mathrm{Vertex^{3}}_{v}^{\zeta}=\zeta^{g(v)-1}\exp\left(-\frac{1}{2}\zeta
p_{(v)}+\\{\exp(p_{(v)}D)\gamma(t\zeta,x)\\}_{\Delta^{(v)}}+V^{\zeta}\right)$

in $A^{*}(\overline{M}_{g(v),\mathbf{w}_{v}})[x,t,\mathbf{p}]$, where
$p_{(v)}=\sum_{n^{\prime}\geq i\in p^{-1}(v)}p_{i}$, $\\{\\}_{\Delta^{(v)}}$
is defined identically to $\\{\\}_{\Delta}$ but $\kappa_{j}$ is replaced by
$\kappa_{j}^{(v)}$ — the $\kappa_{j}$ class at $v$ — and

$V^{\zeta}=-\sum_{i\geq
1}\frac{B_{2i}}{2i\cdot(2i-1)}\frac{1}{t^{2}}\sum_{\Delta\in\mathcal{DG}}\frac{1}{\operatorname{Aut}(\Delta)}\xi_{\Delta,*}\left(\frac{\psi_{a}^{2i-1}+\psi_{b}^{2i-1}}{\psi_{a}+\psi_{b}}\right)(\zeta
t)^{2i-1}.$

Here $\mathcal{DG}\subseteq\mathcal{G}$ is the set of graphs corresponding to
divisor classes, i.e. graphs with only one edge, and $\psi_{a}$, $\psi_{b}$
are the two cotangent line classes corresponding to the unique node
corresponding to a divisor. The edge and point series are given by

$\displaystyle
t(\psi_{1}+\psi_{2})\mathrm{Edge^{3}}_{e}^{\zeta_{1},\zeta_{2}}=(\Phi^{\prime}(t\zeta_{1}\psi_{1})\Phi^{\prime}(t\zeta_{2}\psi_{2}))^{-1}\frac{\zeta_{1}+\zeta_{2}}{2}+$
$\displaystyle\zeta_{1}\delta(t\zeta_{1}\psi_{1},x)+\zeta_{2}\delta(t\zeta_{2}\psi_{2},x),$

where $\Phi^{\prime}$ is from Section 5, and

$\mathrm{Point^{3}}(t,p)=1+p\delta(t,x).$

###  6.2. Proof of the stable quotient relations

In Section 4.2 we looked at the integrand $\mathbf{s}^{\mathbf{b}}$ coming
from powers of the pulled back first Chern class of the universal sheaf
$\mathcal{S}$ over $C_{g|n}(\mathbb{P}^{1},d)$. This works also well for
$\overline{Q}_{g|\mathbf{w}}(\mathbb{P}^{1},d)$ if no point is of weight $1$.
For the points of weight $1$ we however need to choose different classes since
by the stability conditions the torsion of $\mathcal{S}$ must be away from the
sections of points of weight $1$. Instead we will pull back classes from
$\mathbb{P}^{1}$ via evaluation maps.

For an $n$-tuple $\mathbf{a}$, which can be split into an $n^{\prime}$-tuple
$\mathbf{a}^{\prime}$ and an $n^{\prime\prime}$-tuple
$\mathbf{a}^{\prime\prime}$, we will thus study the equivariant class

$\varphi(\mathbf{a}):=\tilde{\mathbf{s}}^{\mathbf{a}^{\prime}}\prod_{i=n^{\prime}+1}^{n}\mathrm{ev}_{i}^{*}\left(\left(\frac{[0]+[\infty]}{2}\right)^{a_{i}}\right)\in
A_{\mathbb{C}^{*}}^{|\mathbf{a}|}(\overline{Q}_{g,\mathbf{w}}(\mathbb{P}^{1},d)),$

where $[0]$ and $[\infty]$ are the equivariant classes of $0$ and $\infty$
respectively, and $\tilde{\mathbf{s}}^{\mathbf{a}^{\prime}}$ is the
equivariant class from Section 4.2. Since
$A_{\mathbb{C}^{*}}^{*}(\mathbb{P}^{1})\otimes\mathbb{Q}[s,s^{-1}]$ is a two
dimensional $\mathbb{Q}[s,s^{-1}]$ module we do not lose any relations by
considering only the case when $\mathbf{a}^{\prime\prime}$ is $\\{0,1\\}$
valued.

As before we consider the $s^{c}$ part of the push-forward of
$\varphi(\mathbf{a})$ for $c<0$ when using the virtual localization formula.

In order to gain overview over the in $d$ monotonously growing number of fixed
loci we sort them according to the stratum of $\overline{M}_{g,\mathbf{w}}$
they push forward to. Let us consider stable graphs $\Gamma=(V,E)$ of
$\overline{M}_{g,\mathbf{w}}$ together with a coloring $\zeta:V\to\\{\pm 1\\}$
and a degree assignment $d:V\sqcup
E\sqcup\\{n^{\prime}+1,\dots,n\\}\to\mathbb{Z}_{\geq 0}$. This data records
coloring and the $d|_{V}$ from a stable quotient graph, the total degrees of
the chains which destabilize to a node or a weight $1$ marked point when
pushing forward to $\overline{M}_{g,\mathbf{w}}$. To get back to a stable
quotient graph one needs to choose for each edge and each weight $1$ marked
point of degree $d$ a splitting
$d=d_{e_{1}}+d_{v_{1}}+\dotsb+d_{v_{\ell-1}}+d_{e_{\ell}}$ of $d$
corresponding to the degrees on the chain (see Figure 1). Because of the
conditions on the coloring of a stable graph there is a mod 2 condition on the
length $\ell$ of the chains depending on the color at the connection vertices
or $\mathbf{a}$. If we had not imposed (11) we would also have chains for each
set of marked points of total weight exactly $1$.

Figure 1. An edge and a point
chain$0$$+1$$\infty$$-1$$\mathbb{P}^{1}$$\zeta$$\overline{M}_{g(v),\mathbf{w}(v)}$$d_{e_{1}}$$d_{v_{1}}$$d_{e_{2}}$$d_{v_{2}}$$d_{e_{3}}$$d_{v_{3}}$$d_{e_{4}}$$d_{v_{4}}$$d_{e_{5}}$$d_{v_{5}}$$d_{e_{6}}$$\overline{M}_{g(w),\mathbf{w}(w)}$$\overline{M}_{g(v),\mathbf{w}(v)}$$d_{e_{1}}$$d_{v_{1}}$$d_{e_{2}}$$d_{v_{2}}$$d_{e_{3}}$$d_{v_{3}}$$d_{e_{4}}$$d_{v_{4}}$$d_{e_{5}}$point

The fixed loci corresponding to such a decorated stable graph are, up to a
finite map, isomorphic to products

$\prod_{v}\overline{M}_{g(v),(\mathbf{w}(v),\varepsilon^{d(v)})}/S_{d(v)}$

times a number of Losev-Manin factors $\overline{M}_{0,2|d}/S_{d}$
corresponding to the vertices inside the chains.

For the localization calculation we will also have to consider the pull-back
of the integrand $\varphi(\mathbf{a})$ to each fixed locus. The factors
$\operatorname{ev}_{i}^{*}(([0]+[\infty])^{a_{i}})$ of $\varphi(\mathbf{a})$
merely restrict the coloring of the stable quotient graphs and their
contribution is pulled back from the equivariant cohomology of a point. The
other factors $s_{i}^{a_{i}}$ need to be partitioned along the factors of the
stable quotient fixed locus. However it is easy to see that in order for the
contribution to be nonzero $s_{i}^{a_{i}}$ has to land at the factor
corresponding to the vertex $p(i)$.

After all these preconsiderations let us write down the localization formula.
It will be the most convenient to write it in a power series form. We have

$\nu_{*}^{vir}\varphi(\mathbf{a})=\left[\sum_{\Gamma,\zeta,d}\frac{1}{\operatorname{Aut}(\Gamma)}\xi_{\Gamma*}\varepsilon_{*}\left(\prod_{v}\mathrm{Vertex}^{1}_{v}\prod_{e}\mathrm{Edge}^{1}_{e}\prod_{i=n^{\prime}+1}^{n}\mathrm{Point}_{i}^{1}\right)\right]_{x^{d}\mathbf{p}^{\mathbf{a}}},$

where $\nu_{*}^{vir}$ again denotes push-forward using $\nu$ after capping
with the virtual fundamental class and the vertex, edge and point series still
need to be defined.

The vertex series $\mathrm{Vertex}^{1}_{v}$ is given by

$\mathrm{Vertex}^{1}_{v}=(\zeta(v)s)^{g(v)-1}\exp(-T_{1})T_{2}\in\bigoplus_{d=0}^{\infty}A_{\mathbb{C}^{*}}^{*}(\overline{M}_{g(v),(\mathbf{w}(v),\varepsilon^{d})})\otimes\mathbb{Q}[[x,\mathbf{p},s,s^{-1}]]$

with

$\displaystyle T_{1}=$ $\displaystyle\sum_{n^{\prime}\geq i\in
p^{-1}(v)}s_{i}p_{i}+\frac{1}{2}\zeta(v)\sum_{n^{\prime}\geq i\in
p^{-1}(v)}sp_{i}$ $\displaystyle T_{2}=$
$\displaystyle\sum_{j=0}^{\infty}\sum_{d=0}^{\infty}(\zeta(v)s)^{d-j}c_{j}(\mathbb{F}_{d})\frac{x^{d}}{(\zeta(v)s)^{2d}d!}.$

For the edge series $\mathrm{Edge}^{1}_{e}$ we have

$\displaystyle\mathrm{Edge}^{1}_{e}=$
$\displaystyle\sum_{d=1}^{\infty}x^{d}\sum_{\Gamma_{e}^{d}=(V_{e},E_{e})}\frac{1}{\operatorname{Aut}(\Gamma_{e}^{d})}$
$\displaystyle\frac{1}{\omega_{\Gamma_{e}^{d},1}-\psi_{1}}\frac{1}{\omega_{\Gamma_{e}^{d},2}-\psi_{2}}\prod_{f\text{
edge}}\operatorname{Cont}(f)\prod_{v\text{ vertex}}\operatorname{Cont}(v),$

where $\Gamma_{e}^{d}$ is a stable quotient localization graph of
$\overline{Q}_{0,2}(\mathbb{P}^{1},d)$ 777which is here a chain similar to the
first in Figure 1 with color of the two vertices determined by $\zeta$ at the
two vertices adjacent to $e$. The variables $\zeta_{i}$, $d_{i}$, $\psi_{i}$
for $i\in\\{1,2\\}$ denote the color, the weight and the $\psi$ classes of the
two marked points corresponding to the edge respectively. The contributions
$\operatorname{Cont}(f)$ and $\operatorname{Cont}(v)$ are contributions to the
calculation of the integrals

$\int_{\overline{Q}_{0,2}(\mathbb{P}^{1},d)}\operatorname{ev}_{1}^{*}(\phi_{1})\operatorname{ev}_{2}^{*}(\phi_{2})$

as in Section 5. By the stable quotients Gromov-Witten comparison we can
replace $\overline{Q}_{0,2}(\mathbb{P}^{1},d)$ everywhere in this paragraph by
$\overline{M}_{0,2}(\mathbb{P}^{1},d)$ and in then $\mathrm{Edge}^{1}_{e}$ is
very similar to $E^{ij}(t_{1},t_{2},x)$ from Section 5.

Finally the point series $\mathrm{Point}_{i}^{1}$ is similarly given by

$\displaystyle\mathrm{Point}_{i}^{1}=$
$\displaystyle\left(\frac{s}{2}\zeta_{i}\right)^{a_{i}}+$
$\displaystyle\sum_{d=1}^{\infty}x^{d}\sum_{\Gamma_{i}^{d}=(V_{i},E_{i})}\frac{1}{\operatorname{Aut}(\Gamma_{i}^{d})}\frac{1}{\omega_{\Gamma_{i}^{d}}-\psi_{1}}\prod_{f\text{
edge}}\operatorname{Cont}(f)\prod_{v\text{ vertex}}\operatorname{Cont}(v),$

where $\zeta_{i}$ is $\zeta$ at the vertex with $i$, $\Gamma_{i}^{d}$ is a
stable quotient localization graph of $\overline{Q}_{0,2}(\mathbb{P}^{1},d)$
with color of the the first vertex determined by $\zeta_{i}$. Here the
contributions $\operatorname{Cont}(f)$ and $\operatorname{Cont}(v)$ are
contributions to the calculation of the integrals

$\int_{\overline{Q}_{0,2}(\mathbb{P}^{1},d)}\operatorname{ev}_{1}^{*}(\phi_{1})\operatorname{ev}_{2}^{*}\left(\phi_{2}\left(\frac{[0]+[\infty]}{2}\right)^{a_{i}}\right).$

The first summand corresponds to the case when the length of the chain is $0$.
Its form comes from the identity $[0]-[\infty]=s$ and the fact that $[0]$
vanishes when at a vertex at $\infty$ and vice versa.

####  6.2.1. Pushing forward

Next we will study the $\varepsilon$-push-forward in the formula for
$\nu_{*}^{vir}\varphi(\mathbf{a})$. For this we want to replace the $\psi$
classes in the edge and point terms by $\psi$ classes pulled back via
$\varepsilon$, since the other $\psi$ classes are already pull-backs. For a
fixed localization graph $\varepsilon$ is a composition of local maps, one for
each vertex in the graph, of the form
$\overline{M}_{g(v),(w(v),\varepsilon^{d(v)})}\to\overline{M}_{g(v),w(v)}$,
and one for each edge and marked point which contracts products of factors of
the form $\overline{M}_{0,2|d}$.

Let us first look at the push-forward local to a vertex. For this we only need
to look at the product of the vertex term with the factors of the form

$\frac{1}{\omega_{i}-\psi_{i}}$

from the adjacent edge and point series. Let us simplify the notation for this
local discussion. With $d$ we denote the degree at this vertex, with
$n=n^{\prime}+n^{\prime\prime}$ the number of marked points with weight
different or equal to $1$ respectively, and with $\mathbf{w}$ the weights at
the vertex. We will index the marked points by the set
$\\{1,\dotsc,n\\}\sqcup\\{1,\dotsc,d\\}$. Hopefully the non-empty intersection
of these sets will not cause any confusion.

The basic pull-back formula is that

$\psi_{i}=\varepsilon^{*}(\psi_{i})+\Delta_{i1}$

in the case that $d=1$. Here $\Delta_{i1}$ is the boundary divisor of curves
who have one irreducible component of genus $0$ containing only $i$ and the
weight $\varepsilon$ point. This generalizes to the formula

$\psi_{i}^{k}=\varepsilon^{*}(\psi_{i}^{k})+\varepsilon^{*}(\psi_{i}^{k-1})\sum_{\emptyset\neq
S\subset\\{1,\dotsc,d\\}}\Delta_{iS},$

where $\Delta_{iS}$ is the boundary divisor of curves who have one irreducible
component containing only $i$ and the weight $\varepsilon$ points indexed by
$S$. Thus

$\frac{1}{\omega_{i}-\psi_{i}}=\frac{1}{\omega_{i}-\varepsilon^{*}(\psi_{i})}\left(1+\omega_{i}^{-1}\sum_{\emptyset\neq
S\subset\\{1,\dotsc,d\\}}\Delta_{iS}\right).$

Modulo factors pulled back via $\varepsilon$ the most general classes we will
need to push forward are

$\prod_{i=1}^{n^{\prime}}D_{iS_{i}}\prod_{i=n^{\prime}+1}^{n}\Delta_{iT_{i}}M$

for subsets $S_{i},T_{i}\subset\\{1,\dotsc,d\\}$ and a monomial $M$ in $\psi$
and diagonal classes of the $d$ points of weight $\varepsilon$. In order that
this class is not zero the $S_{i}$ and $T_{i}$ must be pairwise disjoint.
Moreover for each $i\in\\{n^{\prime}+1,\dots,n\\}$ the monomial $M$ must
contain the diagonal corresponding to $T_{i}$ as a connected factor.

Therefore for a monomial $M=\prod_{i\in P}M_{i}$ corresponding to a set
partition $P\vdash\\{1,\dotsc,d\\}$ we obtain

$\displaystyle\varepsilon_{*}\left(\prod_{i=n^{\prime}+1}^{n}\left(1+\sum_{\emptyset\neq
S\subset\\{1,\dotsc,d\\}}\omega_{i}^{-1}\Delta_{iS}\right)\exp\left(\sum_{i=1}^{n^{\prime}}s_{i}p_{i}\right)M\right)=$
$\displaystyle\prod_{i\in
P}\left(\delta_{i}\sum_{j=n^{\prime}+1}^{n}\omega_{j}^{-1}+\varepsilon_{i*}\left(\exp\left(\sum_{j=1}^{n^{\prime}}s_{j}p_{j}\right)M_{i}\right)\right),$

where here $\delta_{i}$ is one if $M_{i}$ is a diagonal class and zero
otherwise and the $\varepsilon_{i}$ are forgetful maps
$\varepsilon_{i}:\overline{M}_{g,(\mathbf{w},\varepsilon^{|i|})}\to\overline{M}_{g,\mathbf{w}}$
888To think of $m_{i}$ as living on
$\overline{M}_{g,(\mathbf{w},\varepsilon^{|i|})}$ one needs to choose a
bijection $i\to\\{1,\dotsc,|i|\\}$ but the $\varepsilon_{i}$-push-forward is
independent of that choice.. Notice that the push-forward is a product where
each factor is dependent only on one factor of $M$.

This allows us to use the arguments from Section 4.2.2 to calculate the
necessary $\varepsilon$-push-forwards modulo classes pulled back via
$\varepsilon$. We obtain

(12) $\displaystyle\begin{aligned}
\sum_{d=0}^{\infty}\varepsilon_{*}\sum_{i=n^{\prime}+1}^{n}\left(1+\sum_{\emptyset\neq
S\subset\\{1,\dotsc,d\\}}\omega_{i}^{-1}\Delta_{iS}\right)\exp\left(\sum_{i=1}^{n^{\prime}}s_{i}p_{i}\right)\\\
\sum_{j=0}^{\infty}(\zeta s)^{d-j}c_{j}(-\mathbb{B}_{d})\frac{x^{d}}{(\zeta
s)^{2d}d!}=\end{aligned}$ $\displaystyle\begin{aligned}
\prod_{i=n^{\prime}+1}^{n}&\exp\left(\frac{u^{\zeta}}{\omega_{i}}-\log(\Phi^{\prime})\left(\frac{\psi_{i}}{\zeta(v)s},\frac{x}{\zeta(v)s}\right)\right)\\\
&\exp\left(\left\\{\exp(-pD)\log(\Phi)\left(\frac{t}{\zeta(v)s},\frac{x}{\zeta(v)s}\right)\right\\}_{\Delta}\right)\Big{|}_{t=1},\end{aligned}$

where $u^{\zeta}$ is as in (10) and $p=p_{1}+\dotsb+p_{n^{\prime}}$. Here we
rather artificially introduced a variable $t$ to make use of the bracket
notation. Notice that in the first factor of this formula the factor
corresponding to some $i\in\\{n^{\prime}+1,\dotsc,n\\}$ depends only on
$\omega_{i}$, which dependends on the degree splitting of the chain, whereas
the second factor is independent of the degree splittings.

Now we can again step back from the vertex and look at the global picture.
With (12) we can give a new formula for $\nu_{*}^{vir}\varphi(\mathbf{a})$

$\nu_{*}^{vir}\varphi(\mathbf{a})=\left[\sum_{\Gamma,\zeta}\frac{1}{\operatorname{Aut}(\Gamma)}\xi_{\Gamma*}\left(\prod_{v}\mathrm{Vertex}^{2}_{v}\prod_{e}\mathrm{Edge}^{2}_{e}\prod_{i=1}^{n}\mathrm{Point}_{i}^{2}\right)\right]_{x^{d}\mathbf{p}^{\mathbf{a}}},$

with new vertex, edge and point series. The term $\mathrm{Vertex}^{2}_{v}$
already has the form as in the stable quotient relations

$\displaystyle\mathrm{Vertex}^{2}_{v}=(\zeta(v)s)^{g(v)-1}$
$\displaystyle\exp\left(-\frac{1}{2}\zeta(v)sp_{(v)}+\left\\{\exp(p_{(v)}sD)\log(\Phi)\left(\frac{t}{\zeta(v)s},\frac{x}{(\zeta(v)s)^{2}}\right)\right\\}_{\Delta^{(v)}}\Big{|}_{t=1}+V^{\prime\zeta(v)}\right),$

where

$V^{\prime\zeta}=\sum_{j=0}^{\infty}(\zeta s)^{-j}c_{j}(\mathbb{E}^{*})\in
A^{*}(\overline{M}_{g(v),\mathbf{w}_{v}})[[x,s,s^{-1}]].$

Furthermore the edge and point series are given by

$\displaystyle\mathrm{Edge^{2}}_{e}=$
$\displaystyle{\Phi^{\prime}}^{-1}\left(\frac{\psi_{1}}{\zeta_{1}s},\frac{x}{\zeta_{1}s}\right){\Phi^{\prime}}^{-1}\left(\frac{\psi_{2}}{\zeta_{2}s},\frac{x}{\zeta_{2}s}\right)$
$\displaystyle\sum_{d=1}^{\infty}x^{d}\sum_{\Gamma_{e}^{d}=(V_{e},E_{e})}$
$\displaystyle\frac{1}{\operatorname{Aut}(\Gamma_{e}^{d})}\frac{\zeta_{1}se^{u^{\zeta_{1}}/\omega_{\Gamma_{e}^{d},1}}}{\omega_{\Gamma_{e}^{d},1}-\psi_{1}}\frac{\zeta_{2}se^{u^{\zeta_{2}}/\omega_{\Gamma_{e}^{d},2}}}{\omega_{\Gamma_{e}^{d},2}-\psi_{2}}\prod_{f\text{
edge}}\operatorname{Cont}(f)\prod_{v\text{ vertex}}\operatorname{Cont}(v)$

and

$\displaystyle\mathrm{Point}_{i}^{2}=$
$\displaystyle{\Phi^{\prime}}^{-1}\left(\frac{\psi_{i}}{\zeta_{i}s},\frac{x}{\zeta_{i}s}\right)$
$\displaystyle\sum_{d=0}^{\infty}x^{d}$
$\displaystyle\sum_{\Gamma_{i}^{d}=(V_{i},E_{i})}\frac{1}{\operatorname{Aut}(\Gamma_{i}^{d})}\frac{\zeta_{i}se^{u^{\zeta_{i}}/\omega_{\Gamma_{i}^{d}}}}{\omega_{\Gamma_{i}^{d}}-\psi_{i}}\prod_{f\text{
edge}}\operatorname{Cont}(f)\prod_{v\text{ vertex}}\operatorname{Cont}(v).$

Using Mumford’s formula (pulled back from $\overline{M}_{g}$ to
$\overline{M}_{g,\mathbf{w}}$) we find

$\displaystyle V^{\prime\zeta}=$ $\displaystyle-\sum_{i\geq
1}\frac{B_{2i}}{2i\cdot(2i-1)}\kappa_{2i-1}(\zeta s)^{1-2i}$
$\displaystyle-\sum_{i\geq
1}\frac{B_{2i}}{2i\cdot(2i-1)}s^{2}\sum_{\Delta\in\mathcal{DG}}\frac{1}{\operatorname{Aut}(\Delta)}\xi_{\Delta,*}\left(\frac{\psi_{a}^{2i-1}+\psi_{b}^{2i-1}}{\psi_{a}+\psi_{b}}\right)(\zeta
s)^{1-2i}.$

Notice that the edge series $\mathrm{Edge}_{e}^{2}$ modulo a slight change in
notation and the ${\Phi^{\prime}}^{-1}$ factors is the series
$E^{ij}(\psi_{1},\psi_{2},x)$ from Section 5, where $i$ and $j$ correspond to
the color of the vertices $e$ connects. Similarly we identify the point term
$\mathrm{Point}_{i}^{2}$ up to the ${\Phi^{\prime}}^{-1}$ factor with

$2^{-a_{i}}\left(s^{a_{i}}P^{i0}(\psi_{i},x)+(-s)^{a_{i}}P^{i\infty}(\psi_{i},x)\right).$

We finally obtain the relations by taking the $s^{c}$ part of
$\nu_{*}^{vir}\varphi(\mathbf{a})$ for $c<0$. So we replace everywhere $s$ by
$t^{-1}$, $x$ by $xt^{2}$ and $p_{(v)}$ by $p_{(v)}t^{-1}$. After dividing out
a common factor of $t^{e}$ for

$e=\sum_{v}(-g(v)+1)+2d+|a|=-g+1+|E|+2d+|a|$

and introducing variables $p_{i}$ for the $a^{\prime\prime}_{i}$ we arrive at
the stable quotient relations of Section 6.1.

###  6.3. Evaluation of the relations

####  6.3.1. Minor simplification

With the same proof as in Section 4.3.2 the stable quotient relations are
implied from the stable quotient relations in the case that the
$n^{\prime}$-tuple $\mathbf{a}^{\prime}$ is only $\\{0,1\\}$-valued. The same
holds trivially for $\mathbf{a}^{\prime\prime}$ because of the form of the
point term.

Furthermore the relations in the case that a point is of weight $1$ and a
point is of weight slightly smaller than $1$ are the same. This is because a
point $i$ of weight slightly smaller than $1$ is not allowed to meet any other
point, therefore the contribution of that point, which is solely in the vertex
contribution, is

$\exp\left(-\frac{1}{2}\zeta(p(i))p_{i}+p_{i}D\gamma(t\zeta(p(i))\psi_{i},x)\right)=1+p_{i}\zeta(p(i))\delta(t\zeta(p(i))\psi_{i},x)$

modulo $(p_{i}^{2})$. This is the point term after suitably renaming $p_{i}$.

Therefore we can from now on treat points of weight $1$ the same way as points
of weight slightly less than $1$.

####  6.3.2. Edge terms

The factor $\exp(V^{\zeta})$ of the vertex contribution to the stable quotient
relations contains intersections of classes supported on divisor classes of
$\overline{M}_{g(v),\mathbf{w}_{v}}$. We want to reformulate the relations
such that the vertex term only contains $\kappa$, diagonal and $\psi$ classes
corresponding to the markings. Some excess intersection calculations will be
necessary to deal with $\exp(V^{\zeta})$.

######  Proposition 4.

The set of stable quotient relations is equivalent to the following set of
relations: Under the conditions of Proposition 3 it holds

$\displaystyle 0=\Bigg{[}\sum_{\begin{subarray}{c}\Gamma\in\mathcal{G}\\\
\zeta:\Gamma\to\\{\pm
1\\}\end{subarray}}\frac{1}{\operatorname{Aut}(\Gamma)}\xi_{\Gamma*}\Big{(}\prod_{v}\mathrm{Vertex^{4}}_{v}^{\zeta(v)}$
$\displaystyle\prod_{e}\mathrm{Edge^{4}}_{e}^{\zeta(v_{1}),\zeta(v_{2})}\Big{)}\Bigg{]}_{t^{r-|E|}x^{d}\mathbf{p}^{\mathbf{a}}},$

with

$\mathrm{Vertex^{4}}_{v}^{\zeta}=\zeta^{g(v)-1}\exp\left(\frac{1}{2}\zeta
p_{(v)}+\\{\exp(p_{(v)}D)\gamma(t\zeta,x)\\}_{\Delta^{(v)}}\right),$

where $p_{(v)}=\sum_{i\in p^{-1}(v)}p_{i}$, and

$t(\psi_{1}+\psi_{2})\mathrm{Edge^{4}}_{e}^{\zeta_{1},\zeta_{2}}=\frac{\zeta_{1}+\zeta_{2}}{2}\exp(-\gamma^{\prime}(t\zeta_{1}\psi_{1})-\gamma^{\prime}(t\zeta_{2}\psi_{2}))+\zeta_{1}\delta(t\zeta_{1}\psi_{1})+\zeta_{2}\delta(t\zeta_{2}\psi_{2}),$

where $\gamma^{\prime}$ is defined in the same way from $\Phi^{\prime}$ as
$\gamma$ is from $\Phi$:

$\gamma^{\prime}=\sum_{i\geq
1}\frac{B_{2i}}{2i\cdot(2i-1)}t^{2i-1}+\log(\Phi^{\prime})$

######  Remark 3.

As in Section 4.3.2 we can also write

$\mathrm{Vertex^{4}}_{v}^{\zeta}=\zeta^{g(v)-1}\exp\left(-\\{\gamma\\}_{\kappa^{(v)}}^{\zeta}+\sum_{i=1}^{\infty}\frac{\zeta^{i}}{i!}\\{p_{(v)}^{i}D^{i-1}\delta\\}_{\Delta^{(v)}}^{\zeta}\right).$

The power $\zeta^{i}$ appears because of the $t$ in
$D=tx\frac{\mathrm{d}}{\mathrm{d}x}$.

The proof of Proposition 4 depends on the following lemma.

######  Lemma 7.

For a polynomial $f$ in two variables we have

$\displaystyle\exp\left(\sum_{\Delta\in\mathcal{DG}}\frac{1}{|\operatorname{Aut}(\Delta)|}\xi_{\Delta,*}(f(\psi_{a},\psi_{b}))\right)=$
$\displaystyle\sum_{\Gamma\in\mathcal{G}}\frac{1}{|\operatorname{Aut}(\Gamma)|}\xi_{\Gamma,*}\left(\prod_{e}\frac{\exp(-f(\psi_{1}^{(e)},\psi_{2}^{(e)})(\psi_{1}^{(e)}+\psi_{2}^{(e)}))-1}{-(\psi_{1}^{(e)}+\psi_{2}^{(e)})}\right),$

where $\psi_{i}^{(e)}$ are the two cotangent line classes belonging to edge
$e$.

###### Proof.

Formally expanding the left hand side using the intersection formulas, for
example described in [5, Appendix A], we can write it as a sum over stable
graphs $(\Gamma,E)$. Let us look at the term corresponding to a given graph
$\Gamma$. By contracting all but one edge of $\Gamma$ one obtains a divisor
graph. This process gives a map $e_{\Gamma}:E\to\mathcal{DG}$. Counting the
preimages of $e_{\Gamma}$ gives a map
$m_{\Gamma}:\mathcal{DG}\to\mathbb{N}_{0}$. In the formal expansion of the
exponential on the left hand side each term also corresponds to a function
$\sigma:\mathcal{DG}\to\mathbb{N}_{0}$.

A term contributes to a graph $\Gamma$ if and only if $m_{\Gamma}\leq\sigma$,
of the $|\sigma|$ intersections $|m_{\Gamma}|=|E|$ are transversal and the
others are excess. In addition, a contributing term of the left hand side
determines a partition $\mathbf{p}$ indexed by $E$ of $\sigma=\sum_{e\in
E}p_{e}$ such that $p_{e}(\Delta)=0$ unless $e_{\Gamma}(e)=\Delta$.

With this we can explicitly write down $|\operatorname{Aut}(\Gamma)|$ times
the $\Gamma$-contribution as

$\displaystyle\xi_{\Gamma,*}\sum_{\sigma\geq m_{\Gamma}}\sum_{\mathbf{p}}$
$\displaystyle\prod_{\Delta\in\mathcal{DG}}\frac{1}{\sigma(\Delta)!}\binom{\sigma(\Delta)}{\mathbf{p}(\Delta)}$
$\displaystyle\prod_{e}f(\psi_{1}^{(e)},\psi_{2}^{(e)})^{p_{e}(e_{\Gamma}(e))}(-(\psi_{1}^{(e)}+\psi_{2}^{(e)}))^{p_{e}(e_{\Gamma}(e))-1}$
$\displaystyle=$ $\displaystyle\xi_{\Gamma,*}\sum_{\sigma\geq
m_{\Gamma}}\sum_{\mathbf{p}}$
$\displaystyle\prod_{e}\frac{1}{(p_{e}(e_{\Gamma}(e)))!}f(\psi_{1}^{(e)},\psi_{2}^{(e)})^{p_{e}(e_{\Gamma}(e))}(-(\psi_{1}^{(e)}+\psi_{2}^{(e)}))^{p_{e}(e_{\Gamma}(e))-1}$
$\displaystyle=$ $\displaystyle\xi_{\Gamma,*}$
$\displaystyle\prod_{e}\sum_{i=1}^{\infty}\frac{1}{i!}f(\psi_{1}^{(e)},\psi_{2}^{(e)})^{i}(-(\psi_{1}^{(e)}+\psi_{2}^{(e)}))^{i-1}$
$\displaystyle=$ $\displaystyle\xi_{\Gamma,*}$
$\displaystyle\prod_{e}\frac{\exp(-f(\psi_{1}^{(e)},\psi_{2}^{(e)})(\psi_{1}^{(e)}+\psi_{2}^{(e)}))-1}{-(\psi_{1}^{(e)}+\psi_{2}^{(e)})}.$

Here the factor $(\sigma(\Delta)!)^{-1}$ comes from the exponential and

$\prod_{\Delta\in\mathcal{DG}}\binom{\sigma(\Delta)}{\mathbf{p}(\Delta)}$

comes from the choice of which intersections are excess.

Summing the contributions for all $\Gamma$ finishes the proof. ∎

###### Proof of the proposition.

We will apply the lemma in the case that

$f(x_{1},x_{2})=-\sum_{i\geq
1}\frac{B_{2i}}{2i(2i-1)}\frac{x_{1}^{2i-1}+x_{2}^{2i-1}}{x_{1}+x_{2}},$

but now we also need to take care of the coloring of the vertices.

For each graph $\Gamma\in\mathcal{G}$ with a coloring $\zeta:\Gamma\to\\{\pm
1\\}$ we can construct a new graph $\Gamma_{red}$, its _reduction_ , by
contracting all edges of $\Gamma$ connecting two vertices of the same color.
The induced coloring on $\Gamma_{red}$ satisfies the property that neighboring
vertices are differently colored. Let us call such a graph reduced. Having the
same reduction also defines an equivalence relation on $\mathcal{G}$.

The idea is now to apply lemma 7 to each vertex of each graph $\Gamma$. In
this way one gets terms at each specialization $\Gamma^{\prime}$ of $\Gamma$
in the same equivalence class of $\Gamma$.

Let us collect all the different contributions at a graph $\Gamma^{\prime}$
coming from graphs $\Gamma$. Recall the pull-back formula for the $\kappa$
classes

$p_{v*}(\xi_{\Gamma}^{*}\kappa_{i})=\kappa_{i}+\sum_{e}\psi_{e}^{i},$

where $p_{v}$ denotes the projection map to the factor corresponding to each
vertex $v$ and the sum is over all outgoing edges at $v$. This implies that
the contributions at $\Gamma^{\prime}$ all have the same vertex contribution
up a sign and a factor

$\exp(-\gamma^{\prime}(t\zeta_{1}\psi_{1}^{(e)})-\gamma^{\prime}(t\zeta_{2}\psi_{2}^{(e)}))$

for each edge of $\Gamma$ not in $\Gamma^{\prime}$ 999$\gamma^{\prime}$
appears here instead of $\gamma$ because $\kappa_{-1}=0$ while $\psi^{-1}$ is
not defined.. The the edge terms corresponding to common edges do exactly
coincide. The different factors split into a product over the connected
components of the graph obtained by removing the edges which need to be
contracted to obtain $\Gamma^{\prime}_{red}$.

So let us look at just one connected component
$\Gamma^{\prime\prime}\subset\Gamma^{\prime}\setminus\Gamma^{\prime}_{red}$.
We have to sum over the possibilities $E\subseteq E(\Gamma^{\prime\prime})$ of
contracting edges in $\Gamma^{\prime\prime}$. We thus have the contribution

$\displaystyle\sum_{E(\Gamma^{\prime\prime})=E\coprod F}\zeta^{|E|}\prod_{e\in
F}\mathrm{Edge^{3}}_{e}^{\zeta,\zeta}$ $\displaystyle\prod_{e\in
E}\frac{\exp(-f(t\zeta\psi_{1}^{(e)},t\zeta\psi_{2}^{(e)})(t\zeta\psi_{1}^{(e)}+t\zeta\psi_{2}^{(e)}))-1}{-(t\zeta\psi_{1}^{(e)}+t\zeta\psi_{2}^{(e)})}\exp(-\gamma^{\prime}(t\zeta\psi_{1}^{(e)})-\gamma^{\prime}(t\zeta\psi_{2}^{(e)}))$
$\displaystyle=$ $\displaystyle\prod_{e\in
E(\Gamma^{\prime\prime})}\Big{(}\mathrm{Edge^{3}}_{e}^{\zeta,\zeta}+$
$\displaystyle\zeta\frac{\exp(-f(t\zeta\psi_{1}^{(e)},t\zeta\psi_{2}^{(e)})(t\zeta\psi_{1}^{(e)}+t\zeta\psi_{2}^{(e)}))-1}{-(t\zeta\psi_{1}^{(e)}+t\zeta\psi_{2}^{(e)})}\exp(-\gamma^{\prime}(t\zeta\psi_{1}^{(e)})-\gamma^{\prime}(t\zeta\psi_{2}^{(e)}))\Big{)}$
$\displaystyle=$ $\displaystyle\prod_{e\in
E(\Gamma^{\prime\prime})}\mathrm{Edge^{4}}_{e}^{\zeta,\zeta}.$

Because of

$\mathrm{Edge^{3}}_{e}^{\zeta,-\zeta}=\mathrm{Edge^{4}}_{e}^{\zeta,-\zeta}$

we can replace $\mathrm{Edge^{3}}$ by $\mathrm{Edge^{4}}$ also for the edges
connecting differently colored vertices. ∎

####  6.3.3. Variable transformations

Using the results of Section 4.3.3 we can give a new formulation of the stable
quotient relations.

We have

$\displaystyle 0=\Bigg{[}\sum_{\begin{subarray}{c}\Gamma\in\mathcal{G}\\\
\zeta:\Gamma\to\\{\pm
1\\}\end{subarray}}\frac{1}{\operatorname{Aut}(\Gamma)}\xi_{\Gamma*}\Big{(}(1+4y)^{e_{\Gamma}}\prod_{v}\mathrm{Vertex^{5}}_{v}^{\zeta(v)}\prod_{e}\mathrm{Edge^{5}}_{e}^{\zeta(v_{1}),\zeta(v_{2})}\Big{)}\Bigg{]}_{u^{r-|E|}y^{d}\mathbf{p}^{\mathbf{a}}},$

with

$\mathrm{Vertex^{5}}_{v}^{\zeta}=\zeta^{g(v)-1}\exp\left(-\left\\{\sum_{k=1}^{\infty}\sum_{j=0}^{k}c_{k,j}u^{k}y^{j}\right\\}_{\kappa}^{\zeta}+\sum_{i=1}^{\infty}\frac{\zeta^{i}}{i!}\\{p_{(v)}^{i}\delta_{i}\\}_{\Delta}^{\zeta}\right),$
$\displaystyle
u(\psi_{1}+\psi_{2})\mathrm{Edge^{5}}_{e}^{\zeta_{1},\zeta_{2}}=\frac{\zeta_{1}+\zeta_{2}}{2}\exp\left(-\sum_{k=1}^{\infty}\sum_{j=0}^{k}c_{k,j}(u\zeta_{1}\psi_{1})^{k}+(u\zeta_{2}\psi_{2})^{k})y^{j}\right)$
$\displaystyle+\zeta_{1}\delta_{1}(u\zeta_{1}\psi_{1})+\zeta_{2}\delta_{1}(u\zeta_{2}\psi_{2})$

and the exponent

$e_{\Gamma}=\frac{r+2d-2}{2}-\frac{\kappa_{0}}{4}-\frac{|\mathbf{a}|}{2}=\frac{r-g+2d-1-|\mathbf{a}|}{2}$

under the condition of Proposition 3 on $r$.

We can assume that $e_{\Gamma}$ is integral because otherwise the relation is
zero since the term corresponding to a coloring $\zeta$ and the opposite
coloring $-\zeta$ exactly cancel each other in this case.

Next we look as in Section 4.3.4 at the extremal coefficients of this series
and obtain the FZ relations of Proposition 1.

###  6.4. Final remarks

Let $\mathcal{R}(g,\mathbf{w},r;S)$ denote the relation on
$\overline{M}_{g,\mathbf{w}}$ of Proposition 1 in codimension $r$
corresponding to $S\subset\\{1,\dotsc,n\\}$ viewed as a class in the formal
strata algebra, i.e. the formal $\mathbb{Q}$-algebra generated by the symbols

$\xi_{\Gamma*}\left(\prod_{v}M_{v}\right),$

where $\Gamma$ is a stable graph of $\overline{M}_{g,\mathbf{w}}$ and the
$M_{v}$ are formal monomials in $\kappa$, $\psi$ and diagonal classes, modulo
the relations given by the formal multiplication rules for boundary strata
described in [5, Appendix A] and the relations between diagonal and $\psi$
classes from 4.2.1. One can describe formal analogs of the push-forwards and
pull-backs along the forgetful, gluing and weight reduction maps.

By the way we have constructed the stable quotient relations, for
$\mathbf{w}^{\prime}\leq\mathbf{w}$ the push-forward of
$\mathcal{R}(g,\mathbf{w},r;S)$ via the weight reduction map is
$\mathcal{R}(g,\mathbf{w}^{\prime},r;S)$. Therefore the relations of
Proposition 1 are (up to a constant factor) the push-forward of a subset of
Pixton’s generalized FZ relations.

As mentioned in the introduction more relations than in Proposition 1 can be
obtained by taking for a partition $\sigma$ with no part equal to $2\pmod{3}$
the class

$\mathcal{R}(g,(\mathbf{w},1^{\ell(\sigma)}),r-|\lfloor\sigma/3\rfloor|;\bar{S})\prod_{i}\psi_{n+i}^{\lfloor\sigma_{i}/3\rfloor+1}$

in $A^{r+\ell(\sigma)}(\overline{M}_{g,(\mathbf{w},1^{\ell(\sigma)})})$, where
$\bar{S}$ equals $S$ on the first $n$ markings and is given by the remainders
when dividing the parts of $\sigma$ by $3$ on the other markings, and pushing
this class forward to $\overline{M}_{g,\mathbf{w}}$ under the forgetful map.
For explicitly calculating this push-forward it is better to use the usual
$\kappa$ classes $\tilde{\kappa}_{i}=\pi_{*}(c_{1}(\omega_{\pi}(D))^{i+1})$,
which are related to the $\kappa$ classes we have used in this article by
$\tilde{\kappa}_{i}=\kappa_{i}+\sum_{j=1}^{n}\psi_{j}^{i}$, in order to use
Faber’s formula for the push-forward of monomials in cotangent line classes
[4]. Let us call these relations $\mathcal{R}(g,\mathbf{w},r;\sigma,S)$.

As in [7] even more generally one can look at the $\mathbb{Q}$-vector space
$\mathcal{R}_{g,\mathbf{w}}$ generated by the relations obtained by choosing a
boundary stratum corresponding to a dual graph $\Gamma$, taking a FZ relation
$\mathcal{R}(g_{i},\mathbf{w}_{i},r;\sigma,S)M_{i}$ for any $r$, $S$, $\sigma$
and monomial $M_{i}$ in the diagonal and cotangent line classes on one of the
components, arbitrary tautological classes on the other components and pushing
this forward along $\xi_{\Gamma}$. Because of the compatibility with the
birational weight reduction maps [7, Proposition 1] implies that the system
$\mathcal{R}_{g,\mathbf{w}}$ of $\mathbb{Q}$-vector spaces cannot be
tautologically enlarged, i.e. it is closed under formal push-forward and pull-
back along forgetful and gluing maps as well as multiplication with arbitrary
tautological classes.

As in [6] we have thrown away many of the stable quotient relations: We looked
only at the extremal relations in Sections 4.3.4 and 6.3.3. However one should
expect that these additional relations can also be expressed in terms of FZ
relations.

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Departement Mathematik
ETH Zürich
<EMAIL_ADDRESS>