Now we summarize results from [Cha07]. For a hyperplane complement
$U\subset\mathbb{A}^{N}$, there is a finite index set $I$ and $n_{i}\geq 0$
such that

$\mathsf{M}(U)\cong\bigoplus_{i\in I}\mathbb{Q}(n_{i})[n_{i}]\,.$

As a corollary, $\mathrm{CH}^{*}(U,\bullet)$ is a finitely generated free
module over $\mathrm{CH}^{*}(k,\bullet)$ and

$\mathrm{CH}^{\ell}(U,1)_{\mathbb{Z}}=\begin{cases}H^{0}(U,\mathcal{O}_{U}^{\times})&\text{
, if $\ell=1$}\\\ 0&\text{ , otherwise.}\end{cases}$ (23)

There exists an isomorphism

$\mathrm{CH}^{1}(U,1)\cong\overline{\mathrm{CH}}^{1}(U,1)\oplus\mathrm{CH}^{1}(k,1)\,.$

######  Example 2.15.

Let $U=\textup{Spec}\,k[x,x^{-1}]$ be the complement of the origin in the
affine line. Then

$\mathrm{CH}^{1}(U,1)_{\mathbb{Z}}\cong\bigsqcup_{a\in\mathbb{Z}}k^{\times}\langle
x^{a}\rangle\cong\mathbb{Z}\oplus k^{\times}$

and the element $m\in\mathrm{CH}^{0}(k)_{\mathbb{Z}}=\mathbb{Z}$ acts by
$x^{a}\mapsto x^{ma}$ and
$\lambda\in\mathrm{CH}^{1}(k,1)_{\mathbb{Z}}=k^{\times}$ acts by
$x^{a}\mapsto\lambda x^{a}$. In fact $\mathrm{CH}^{*}(U,\bullet)$ is generated
by the fundamental class and $\langle x\rangle$ over
$\mathrm{CH}^{*}(k,\bullet)$.

######  Proposition 2.16.

Let $U\subset\mathbb{A}^{N}$ be a hyperplane complement as above. Then the
hCKP holds for quotient stacks.

###### Proof.

Let $Y$ be a quotient stack and hence $Y\times U$ is also a quotient stack.
$Y$ admits a vector bundle $E$ such that the vector bundle is represented by a
scheme off a locus of arbitrarily high codimension. Since $U$ is a scheme, the
pullback of $E$ to $Y\times U$ also satisfies the same property. For higher
Chow groups of quotient stacks, the homotopy invariance for vector bundle and
the extended localization sequence is proven in [Kri13]. Therefore we may
assume that $Y$ is a scheme. When $Y$ is a scheme, there exists isomorphisms

$\displaystyle\mathrm{CH}^{l}(Y\times U,1)$
$\displaystyle=\mathrm{Hom}(\mathsf{M}(Y\times U),\mathbb{Q}(l)[2l-1])$
$\displaystyle=\mathrm{Hom}(\mathsf{M}(Y)\otimes\mathsf{M}(U),\mathbb{Q}(l)[2l-1])$
$\displaystyle=\mathrm{Hom}(\bigoplus_{i\in
I}\mathsf{M}(Y)(n_{i})[n_{i}],\mathbb{Q}(l)[2l-1])$
$\displaystyle=\bigoplus_{i\in
I}\mathrm{Hom}(\mathsf{M}(Y)(n_{i})[n_{i}],\mathbb{Q}(l)[2l-1])$
$\displaystyle=\bigoplus_{i\in
I}\mathrm{Hom}(\mathsf{M}(Y),\mathbb{Q}(l-n_{i})[2l-n_{i}-1])$
$\displaystyle=\bigoplus_{i\in I}\mathrm{CH}^{l-n_{i}}(Y,1-n_{i})$
$\displaystyle=\bigoplus_{n_{i}\leq 1}\mathrm{CH}^{l-n_{i}}(Y,1-n_{i})$

where the fifth equality comes from the cancellation theorem. In the proof of
[Cha07, Proposition 1.1], the index $n_{i}=0$ corresponds to
$\mathrm{CH}^{0}(U,0)$ and the indices $n_{i}=1$ corresponds to generators of
$\overline{\mathrm{CH}}^{1}(U,1)$ over $\mathbb{Q}$. Therefore we get the
isomorphism. ∎

After identifying

$\mathcal{M}_{0,n}=\\{(x_{1},\ldots,x_{n})\in\mathbb{A}^{n-3}\colon x_{i}\neq
x_{j},\textup{for}\,i\neq j,\,x_{i}\neq 0,x_{j}\neq
1\\}\subset\mathbb{A}^{n-3}\,,$

Proposition 2.14 and 2.16 compute the higher Chow group of $\prod_{v\in
V(\Gamma)}\mathfrak{M}_{0,n(v)}^{\textup{sm}}$ for any prestable graph
$\Gamma$.

Now we revisit the CKP for the stack $\mathfrak{M}_{0,n}$. We recall the
definition of Bloch’s higher Chow groups [Blo86]. Let

$\Delta^{m}=\text{Spec}\left(k[t_{0},\ldots,t_{m}]/(t_{0}+\ldots+t_{m}-1)\right)$

be the algebraic $m$ simplex. For $0\leq i_{1}<\ldots<i_{a}\leq m$, the
equation $t_{i_{1}}=\ldots=t_{i_{a}}=0$ defines a face
$\Delta^{m-a}\subset\Delta^{n}$. Let $X$ be an equidimensional quasi-
projective scheme over $k$. Let $z^{i}(X,m)$ be the free abelian group
generated by all codimension $i$ subvarieties of $X\times\Delta^{m}$ which
intersect all faces $X\times\Delta^{l}$ properly for all $l<m$. Taking the
alternating sum of restriction maps to $i+1$ faces of $X\times\Delta^{i}$, we
get a chain complex $(z^{*}(X,m),\delta)$. The higher Chow group
$\mathrm{CH}^{i}(X,m)$ is the $i$-th cohomology of the complex $z^{*}(X,m)$.

When $m=1$, the proper intersection is equivalent to saying that cycles are
not contained in any of the (strict) faces. Let
$R=\Delta^{1}\setminus\\{[0],[1]\\}$. Then the group $z^{*}(X,1)$ is equal to
$z^{*}\left(X\times R\right)$ and the differential

$\ldots\xrightarrow{}z^{*}(X\times
R)\xrightarrow{\delta}z^{*}(X)\xrightarrow{}0$

is given by specialization maps. If $\sum a_{i}W_{i}$ is a cycle in $X\times
R$,

$\delta\left(\sum a_{i}W_{i}\right)=\sum a_{i}\overline{W_{i}}\cap
X\times[0]-\sum a_{i}\overline{W_{i}}\cap X\times[1]$ (24)

where $\overline{W_{i}}$ is the closure of $W_{i}$ in $X\times\Delta^{1}$.

######  Lemma 2.17.

Let $X_{1},X_{2}$ be algebraic stacks stratified by quotient stacks and
$Z_{1}\subset X_{1}$ and $Z_{2}\subset X_{2}$ be closed substacks with
complements

$i_{1}\colon U_{1}=X_{1}\setminus Z_{1}\hookrightarrow X_{1},\hskip
8.53581pti_{2}:U_{2}=X_{2}\setminus Z_{2}\hookrightarrow X_{2}$

where $U_{1},U_{2}$ are quotient stacks. Let $Z_{12}=X_{1}\times
X_{2}\setminus U_{1}\times U_{2}$. Denote by

$\displaystyle\partial_{1}$
$\displaystyle\colon\mathrm{CH}_{*}(U_{1},1)\to\mathrm{CH}_{*}(Z_{1})\,,$
$\displaystyle\partial_{2}$
$\displaystyle\colon\mathrm{CH}_{*}(U_{2},1)\to\mathrm{CH}_{*}(Z_{2})\,,$
$\displaystyle\partial$ $\displaystyle\colon\mathrm{CH}_{*}(U_{1}\times
U_{2},1)\to\mathrm{CH}_{*}(Z_{12})$

the boundary maps for the inclusions $U_{1}\subset X_{1}$, $U_{2}\subset
X_{2}$, $U_{1}\times U_{2}\subset X_{1}\times X_{2}$.

  1. (a)

For $\alpha\in\mathrm{CH}_{*}(U_{1},1)$ and $\beta\in\mathrm{CH}_{*}(U_{2})$,

$\partial(\alpha\otimes\beta)=\partial_{1}(\alpha)\otimes\overline{\beta}\,\text{
in }\mathrm{CH}_{*}(Z_{12})$

where $\overline{\beta}\in\mathrm{CH}_{*}(X_{2})$ is any extension of $\beta$.

  2. (b)

The following diagram commutes

${\begin{array}[]{r}\mathrm{CH}_{*}(U_{1},1)\otimes\mathrm{CH}_{*}(X_{2})\\\
\oplus\mathrm{CH}_{*}(X_{1})\otimes\mathrm{CH}_{*}(U_{2},1)\end{array}}$${\begin{array}[]{r}\mathrm{CH}_{*}(Z_{1})\otimes\mathrm{CH}_{*}(X_{2})\\\
\oplus\mathrm{CH}_{*}(X_{1})\otimes\mathrm{CH}_{*}(Z_{2})\end{array}}$${\mathrm{CH}_{*}(U_{1}\times
U_{2},1)}$${\mathrm{CH}_{*}(Z_{12})\,,}$$\scriptstyle{(\partial_{1}\otimes\mathrm{id})\oplus(\mathrm{id}\otimes\partial_{2})}$$\scriptstyle{(\mathrm{id}\otimes
i_{2}^{*})\oplus(i_{1}^{*}\otimes\mathrm{id})}$$\scriptstyle{\partial}$ (25)

where the arrow on the right is induced by the natural map

$Z_{1}\times X_{2}\sqcup X_{1}\times Z_{2}\to Z_{12}.$

###### Proof.

(a) We first prove that the right hand side is well-defined. For a different
choice of extension $\overline{\beta}^{\prime}$ of $\beta$, the difference
$\overline{\beta}-\overline{\beta}^{\prime}$ is a class supported on $Z_{2}$.
Therefore, the class
$\partial_{1}(\alpha)\otimes(\overline{\beta}-\overline{\beta}^{\prime})$ on
$Z_{12}$ is supported on $Z_{1}\times Z_{2}$. In particular, it a class pushed
forward from $\mathrm{CH}_{*}(X_{1}\times Z_{2})$. This class vanishes because
$\partial_{1}\alpha$ vanishes as a class in $\mathrm{CH}_{*}(X_{1})$.

We first prove the equality when $X_{1},X_{2}$ are schemes. The proof follows
from diagram chasing. Recall that the connecting homomorphism
$\partial\colon\mathrm{CH}_{*}(U_{1}\times U_{2},1)\to\mathrm{CH}_{*}(Z_{12})$
is defined using the following diagram

${z^{*}(Z_{12},1)}$${z^{*}(Z_{12})}$${0}$${z^{*}(X_{1}\times
X_{2},1)}$${z^{*}(X_{1}\times X_{2})}$${0}$${z^{*}(U_{1}\times
U_{2},1)}$${z^{*}(U_{1}\times
U_{2})}$${0\,.}$$\scriptstyle{j_{*}}$$\scriptstyle{\delta}$$\scriptstyle{i^{*}}$$\scriptstyle{\delta}$

For each class in $\mathrm{CH}_{*}(U_{1}\times U_{2},1)$ take a representative
in $z^{*}(U_{1}\times U_{2},1)$. By taking a preimage under $i^{*}$, applying
the map $\delta$ and taking a preimage under $j_{*}$, we get a class in
$\mathrm{CH}_{*}(Z_{12})$ which corresponds to the image of $\partial$. Fix a
representative of $\alpha$ in $z^{*}(U_{1}\times R)$ and $\beta$ in
$z^{*}(U_{2})$. Let $\overline{\alpha}$ be the closure of $\alpha$ in
$X_{1}\times R$ and $\overline{\beta}$ be the closure of $\beta$ in $X_{2}$.
Let $\widetilde{\alpha}$ be the closure of $\overline{\alpha}$ in
$X_{1}\times\mathbb{A}^{1}$. Then to compute $\partial(\alpha\times\beta)$ we
observe that
$\alpha\times\beta=i^{*}(\overline{\alpha}\times\overline{\beta})$. Applying
$\delta$, we have

$\displaystyle\delta(\overline{\alpha}\times\overline{\beta})$
$\displaystyle=\widetilde{\alpha}\times\overline{\beta}\cap
X_{1}\times[0]\times X_{2}-\widetilde{\alpha}\times\overline{\beta}\cap
X_{1}\times[1]\times X_{2}$ $\displaystyle=j_{*}\left(\widetilde{\alpha}\cap
X_{1}\times[0]-\widetilde{\alpha}\cap
X_{1}\times[1]\right)\times\overline{\beta}$
$\displaystyle=j_{*}(\partial(\alpha)\times\overline{\beta})$

and this proves the equality.

In general, let $U_{1}$ be a quotient stack by assumption. For a projective
morphism $S_{1}\to U_{1}$ from a reduced stack $S_{1}$, there exists a
projective morphism $T_{1}\to X_{1}$ such that $S_{1}\cong
U_{1}\times_{X_{1}}T_{1}$ ([Kre99, Corollary 2.3.2]). Let $E_{1}$ be a vector
bundle on $S_{1}$. By [Kre99, Proposition 2.3.3], there exists a projective
modification $T_{1}^{\prime}\to T_{1}$ and a vector bundle $E_{1}^{\prime}$
which restricts to $E_{1}$. We perform a similar construction for the quotient
stack $U_{2}$. The image of
$\mathrm{CH}_{*}(U_{1},1)\otimes\mathrm{CH}_{*}(U_{2})$ under the boundary map

$\partial\colon\mathrm{CH}_{*}(U_{1}\times U_{2},1)\to\mathrm{CH}_{*}(Z_{12})$

is defined by the limit of boundary maps for naive higher Chow groups of
$E_{1}\boxtimes E_{2}\subset E_{1}^{\prime}\boxtimes E_{2}^{\prime}$. The
corresponding computation is precisely equal to the case above. Therefore the
same formula holds for stacks $X_{1}$ and $X_{2}$.

(b) Let
$\alpha\otimes\beta\in\mathrm{CH}_{*}(U_{1},1)\otimes\mathrm{CH}_{*}(X_{2})$.
We take a natural extension $\beta$ of $i^{*}_{2}\beta$. Then by (a), we have

$\displaystyle\partial\circ(\mathrm{id}\otimes
i^{*}_{2})\big{(}\alpha\otimes\beta\big{)}$
$\displaystyle=\partial\big{(}\alpha\otimes i^{*}_{2}\beta\big{)}$
$\displaystyle=\partial_{1}(\alpha)\otimes\beta$
$\displaystyle=\partial_{1}\otimes\mathrm{id}\big{(}\alpha\otimes\beta\big{)}\,.$

The same computation holds for
$\mathrm{CH}_{*}(X_{1})\otimes\mathrm{CH}_{*}(U_{2},1)$ and we get the
commutativity of (25). ∎

######  Remark 2.18.

Applying Lemma 2.17 to $X_{1}=U_{1}=\mathrm{Spec}\,k$ (so that
$Z_{1}=\emptyset$) and $Z=Z_{2}\subseteq X=X_{2}$ with $U=X\setminus Z$, we
find that the composition

$\mathrm{CH}_{*}(k,1)\otimes\mathrm{CH}_{*}(U)\to\mathrm{CH}_{*}(U,1)\xrightarrow{\partial}\mathrm{CH}_{*}(Z)$

vanishes since for $\alpha\in\mathrm{CH}_{*}(k,1)$ and
$\beta\in\mathrm{CH}_{*}(U)$ we have
$\partial(\alpha\otimes\beta)=\partial_{1}(\alpha)\otimes\overline{\beta}=0$
as $\partial_{1}(\alpha)$ lives in
$\mathrm{CH}_{*}(Z_{1})=\mathrm{CH}_{*}(\emptyset)=0$. This implies that
$\partial$ factors through the indecomposable part
$\overline{\mathrm{CH}}_{*}(U,1)$ of $\mathrm{CH}_{*}(U,1)$.

To prove the CKP for $\mathfrak{M}_{0,n}$, we want to use that via the
boundary gluing morphisms, the space $\mathfrak{M}_{0,n}$ is stratified by
(finite quotients of) products of spaces
$\mathfrak{M}_{0,n_{i}}^{\textup{sm}}$, for which we know the CKP. The
following proposition tells us that indeed the CKP for such a stratified space
can be checked on the individual strata.

######  Proposition 2.19.

Let $X$ be an algebraic stack, locally of finite type over $k$ with a good
filtration and stratified by quotient stacks $X=\cup X_{i}$. Suppose each
stratum $X_{i}$ has the CKP and the hCKgP for quotient stacks. Then $X$ has
the CKP for quotient stacks.

###### Proof.

Since $X$ has a good filtration, the Chow groups of $X$ and $X\times Y$ of a
fixed degree can be computed on a sufficiently large finite-type open
substack. This allows us to reduce to the case where $X$ has finite type.

Now by assumption, there exists a nonempty open substack $U\subset X$ which is
a quotient stack and has the CKP. Let $Z=X\setminus U$ be the complement. For
a quotient stack $Y$ consider a commutative diagram

${\mathrm{CH}_{*}(U,1)\otimes\mathrm{CH}_{*}(Y)}$${\mathrm{CH}_{*}(Z)\otimes\mathrm{CH}_{*}(Y)}$${\mathrm{CH}_{*}(X)\otimes\mathrm{CH}_{*}(Y)}$${\mathrm{CH}_{*}(U)\otimes\mathrm{CH}_{*}(Y)}$${0}$${\mathrm{CH}_{*}(U\times
Y,1)}$${\mathrm{CH}_{*}(Z\times Y)}$${\mathrm{CH}_{*}(X\times
Y)}$${\mathrm{CH}_{*}(U\times
Y)}$${0}$$\scriptstyle{\gamma_{1}}$$\scriptstyle{\gamma_{2}}$$\scriptstyle{\gamma_{3}}$$\scriptstyle{\gamma_{4}}$
(26)

where the rows are exact by the excision sequence. Since $U$ has the CKP, the
arrow $\gamma_{4}$ is an isomorphism and by Noetherian induction, the same is
true for $\gamma_{2}$. We extend the domain of the map $\gamma_{1}$ by
inserting an extra component $\mathrm{CH}_{*}(U)\otimes\mathrm{CH}_{*}(Y,1)$.
Then the following diagram

${\begin{array}[]{r}\mathrm{CH}_{*}(U,1)\otimes\mathrm{CH}_{*}(Y)\\\
\oplus\mathrm{CH}_{*}(U)\otimes\mathrm{CH}_{*}(Y,1)\end{array}}$${\mathrm{CH}_{*}(Z)\otimes\mathrm{CH}_{*}(Y)}$${\mathrm{CH}_{*}(U\times
Y,1)}$${\mathrm{CH}_{*}(Z\times Y)}$$\scriptstyle{\gamma^{\prime}_{1}}$

commutes by applying Lemma 2.17 to $U\times Y\subset X\times Y$. Note that the
new factor $\mathrm{CH}_{*}(U)\otimes\mathrm{CH}_{*}(Y,1)$ maps to
$\mathrm{CH}_{*}(X)\otimes\mathrm{CH}_{*}(Y\setminus Y)=0$ under the top
arrow, and so in particular the top row of (26) remains exact after the
modification. Furthermore, the modified map $\gamma^{\prime}_{1}$ is
surjective because $U$ has the hCKgP for quotient stacks. Therefore
$\gamma_{3}$ is an isomorphism by applying the five lemma to the modified
version of (26). ∎

To apply this to the stratification of $\mathfrak{M}_{0,n}$ by prestable
graph, we need a small further technical lemma, due to the fact that the
strata of $\mathfrak{M}_{0,n}$ are quotients of products of
$\mathfrak{M}_{0,n_{i}}$ by finite groups.

######  Lemma 2.20.

Let $\mathfrak{M}$ be an algebraic stack of finite type over $k$ and
stratified by quotient stacks with an action of a finite group $G$. Then the
quotient map $\pi:\mathfrak{M}\to\mathfrak{M}/G$ induces an isomorphism

$\pi^{*}:\mathrm{CH}_{*}(\mathfrak{M}/G)\to\mathrm{CH}_{*}(\mathfrak{M})^{G}$
(27)

from the Chow group of the quotient $\mathfrak{M}/G$ to the $G$-invariant part
of the Chow group of $\mathfrak{M}$. On the other hand, the map

$\pi_{*}:\mathrm{CH}_{*}(\mathfrak{M})\to\mathrm{CH}_{*}(\mathfrak{M}/G)$ (28)

is a surjection.

###### Proof.

The map $\pi$ is representable and a principal $G$-bundle, hence in particular
it is finite and étale. Thus we can both pull back cycles and push forward
cycles under $\pi$. For $g\in G$ let $\sigma_{g}:\mathfrak{M}\to\mathfrak{M}$
be the action of $g$ on $\mathfrak{M}$. Then the relation
$\pi\circ\sigma_{g}\cong\pi$ shows that $\sigma_{g}^{*}$ acts as the identity
on the image of $\pi^{*}$ and thus $\pi^{*}$ has image in the $G$-invariant
part of $\mathrm{CH}_{*}(\mathfrak{M})$. The equality

$\pi_{*}\circ\pi^{*}=|G|\cdot\mathrm{id}:\mathrm{CH}_{*}(\mathfrak{M}/G)\to\mathrm{CH}_{*}(\mathfrak{M}/G)$

shows that $\pi^{*}$ is injective and that $\pi_{*}$ is surjective (since we
work with $\mathbb{Q}$-coefficients). On the other hand, we have

$\pi^{*}\circ\pi_{*}=\sum_{g\in
G}\sigma_{g}^{*}:\mathrm{CH}_{*}(\mathfrak{M})\to\mathrm{CH}_{*}(\mathfrak{M})$

thus restricted on the $G$-invariant part, we again have

$\pi^{*}\circ\pi_{*}|_{\mathrm{CH}_{*}(\mathfrak{M})^{G}}=|G|\cdot\mathrm{id}:\mathrm{CH}_{*}(\mathfrak{M})^{G}\to\mathrm{CH}_{*}(\mathfrak{M})^{G}\,,$

showing $\pi^{*}$ is surjective. ∎

######  Remark 2.21.

The above lemma is also true for the first higher Chow groups with
$\mathbb{Q}$-coefficients.

######  Corollary 2.22.

For all $n\geq 0$, the stacks $\mathfrak{M}_{0,n}$ have the CKP for quotient
stacks.

###### Proof.

Recall that for a prestable graph $\Gamma$ of genus $0$ with $n$ markings,
there exists the locally closed substack
$\mathfrak{M}^{\Gamma}\subset\mathfrak{M}_{0,n}$ of curves with dual graph
exactly $\Gamma$. By Proposition 2.19, it suffices to show that the stacks
$\mathfrak{M}^{\Gamma}$ have the CKP and the hCKgP for quotient stacks. Now
from [BS21, Proposition 2.4] we know that the restriction of the gluing map
$\xi_{\Gamma}$ induces an isomorphism

$\left(\prod_{v\in
V(\Gamma)}\mathfrak{M}_{0,n(v)}^{\textup{sm}}\right)/\mathrm{Aut}(\Gamma)\xrightarrow{\xi_{\Gamma}}\mathfrak{M}^{\Gamma}\,.$

The product of spaces $\mathfrak{M}_{0,n(v)}^{\textup{sm}}$ has the CKP by
Proposition 2.7 and the hCKgP for quotient stacks by Proposition 2.14 and
Proposition 2.16. From Lemma 2.20 and Remark 2.21 it follows that the quotient
of a space with the CKP (or hCKgP) under a finite group action still has the
CKP (or hCKgP), so by the above isomorphism all $\mathfrak{M}^{\Gamma}$ have
the CKP and hCKgP for quotient stacks. This finishes the proof. ∎

We proved the Chow-Künneth property of $\mathfrak{M}_{0,n}$ with respect to
quotient stacks. This assumption comes from technical assumptions in [Kre99].
For example, the extended excision sequence is only proven when the open
substack is a quotient stack. Such assumptions are not necessary for a
different cycle theory of algebraic stacks constructed in [Kha19]. Therefore,
the following remark could remove the technical assumptions in the above Chow-
Künneth property.

######  Remark 2.23.

Let $X$ be an algebraic stack, locally of finite type over $k$. Let
$\mathrm{H}_{*}^{\textup{BM}}$ be the rational motivic Borel–Moore homology
theory defined in [Kha19]. There exists a cycle class map

$\mathrm{cl}\colon\mathrm{CH}_{*}(X)_{\mathbb{Q}}\to\mathrm{H}_{*}^{\textup{BM}}(X)$

which is compatible with projective pushforward, Chern classes and lci
pullbacks. In [BP], we will show that the cycle class map $\mathrm{cl}$ is an
isomorphism when $X$ is stratified by quotient stacks.

###  2.4 Tautological relations

In this section, we formulate and prove a precise form of Theorem 1.4, see
Theorem 2.31. Recall that for a prestable graph $\Gamma$, a decoration
$\alpha$ is an element of $\mathrm{CH}^{*}(\mathfrak{M}_{\Gamma})$ given as a
product $\alpha=\prod_{v}\alpha_{v}$ where
$\alpha_{v}\in\mathrm{CH}^{*}(\mathfrak{M}_{0,n(v)})$ are monomials in
$\kappa$ and $\psi$-classes on the factors $\mathfrak{M}_{0,n(v)}$ of
$\mathfrak{M}_{\Gamma}$.

######  Definition 2.24.

Define the _strata space_ $\mathcal{S}_{g,n}$ to be the free
$\mathbb{Q}$-vector space with basis given by isomorphism classes of decorated
prestable graphs $[\Gamma,\alpha]$.

By definition, the image of the map

$\mathcal{S}_{g,n}\to\mathrm{CH}^{*}(\mathfrak{M}_{g,n})\,,\,\,\,[\Gamma,\alpha]\mapsto\xi_{\Gamma*}\alpha$

is the tautological ring $\mathrm{R}^{*}(\mathfrak{M}_{g,n})$141414From [BS21,
Corollary 3.7] we see that there is a $\mathbb{Q}$-algebra structure on
$\mathcal{S}_{g,n}$ which makes this map into a $\mathbb{Q}$-algebra
homomorphism, since products of decorated strata classes are given by explicit
combinations of further decorated strata classes..

For the proofs below, it is convenient to allow decorations $\alpha_{v}$ at
vertices of $\Gamma$ which are combinations of monomials in $\kappa$ and
$\psi$-classes as follows.

######  Definition 2.25.

Given a prestable graph $\Gamma$ in genus $0$ with $n$ markings, an element

$\alpha=\prod_{v\in V(\Gamma)}\alpha_{v}\in\prod_{v\in
V(\Gamma)}\mathrm{CH}^{*}(\mathfrak{M}_{0,n(v)})$

is said to be in _normal form_ if

  1. a)

for vertices $v\in V(\Gamma)$ with $n(v)=0$, we have
$\alpha_{v}=\kappa_{2}^{a}$ for some $a\geq 0$,

  2. b)

for vertices $v\in V(\Gamma)$ with $n(v)=1$, we have
$\alpha_{v}=\psi_{h}^{b}$, where $h$ is the unique half-edge at $v$ and $b\geq
0$,

  3. c)

for vertices $v\in V(\Gamma)$ with $n(v)=2$, we have
$\alpha_{v}=\psi_{h}^{c}+(-\psi_{h^{\prime}})^{c}$, where $h,h^{\prime}$ are
two half-edges at $v$ and $c\geq 0$,

  4. d)

for vertices $v\in V(\Gamma)$ with $n(v)\geq 3$, we have that $\alpha_{v}=1$
is trivial.

Note that because of the terms $\psi_{h}^{c}+(-\psi_{h^{\prime}})^{c}$ in case
c) above, the element $\alpha$ is not strictly speaking a decoration, since
the $\alpha_{v}$ are not monomials. However, given $\Gamma,\alpha$ as in
Definition 2.24, we write $[\Gamma,\alpha]$ for the element in
$\mathcal{S}_{0,n}$ obtained by expanding $\alpha$ in terms of monomial
decorations.

######  Definition 2.26.

For $g=0$ let $\mathcal{S}_{0,n}^{\textup{nf}}\subset\mathcal{S}_{0,n}$ be the
subspace additively generated by $[\Gamma,\alpha]$ for $\alpha$ in normal
form151515Note that at vertices $v\in V(\Gamma)$ with $n(v)=2$ we have a
choice of ordering of the two half-edges $h,h^{\prime}$, and the possible
decorations $\alpha_{v}=\psi_{h}^{c}+(-\psi_{h^{\prime}})^{c}$ differ by a
sign for $c$ odd. Still they generate the same subspace of $\mathcal{S}_{0,n}$
and the independence of this subspace from the choice of ordering of half-
edges will be important in a proof below..

######  Definition 2.27.

Let $R_{0}\in\mathcal{S}_{g_{0},n_{0}}$ be a tautological relation. Given
$g,n$, we say that _the set of relations in $\mathcal{S}_{g,n}$ generated by
$R_{0}$_ is the subspace of the $\mathbb{Q}$-vector space $\mathcal{S}_{g,n}$
generated by elements of $\mathcal{S}_{g,n}$ obtained by

  * •

choosing a prestable graph $\Gamma$ in genus $g$ with $n$ markings and a
vertex $v\in V(\Gamma)$ with $g(v)=g_{0},n(v)=n_{0}$,

  * •

choosing an identification of the $n_{0}$ half-edges incident to $v$ with the
markings $1,\ldots,n_{0}$ for $\mathcal{S}_{g_{0},n_{0}}$,

  * •

choosing decorations $\alpha_{w}\in\mathrm{CH}^{*}(\mathfrak{M}_{g(w),n(w)})$
for all vertices $w\in V(\Gamma)\setminus\\{v\\}$,

  * •

gluing the relation $R_{0}$ into the vertex $v_{0}$ of $\Gamma$, putting
decorations $\alpha_{w}$ in the other vertices and expanding as an element of
$\mathcal{S}_{g,n}$.

More generally, given any family
$(R_{0}^{i}\in\mathcal{S}_{g_{0}^{i},n_{0}^{i}})_{i\in I}$ of tautological
relations, we define the relations in $\mathcal{S}_{g,n}$ generated by this
family to be the sum of the spaces of relations generated by the $R_{0}^{i}$.

######  Example 2.28.

Let $n_{0}=4$ with markings labelled $\\{3,4,5,h\\}$ and let
$R_{0}\in\mathcal{S}_{0,n_{0}}$ be the WDVV relation

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}{}\pgfsys@moveto{2.35693pt}{0.0pt}\pgfsys@curveto{2.35693pt}{1.30171pt}{1.30171pt}{2.35693pt}{0.0pt}{2.35693pt}\pgfsys@curveto{-1.30171pt}{2.35693pt}{-2.35693pt}{1.30171pt}{-2.35693pt}{0.0pt}\pgfsys@curveto{-2.35693pt}{-1.30171pt}{-1.30171pt}{-2.35693pt}{0.0pt}{-2.35693pt}\pgfsys@curveto{1.30171pt}{-2.35693pt}{2.35693pt}{-1.30171pt}{2.35693pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.5}{0.0}{0.0}{0.5}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{30.8097pt}{0.0pt}\pgfsys@curveto{30.8097pt}{1.30171pt}{29.75447pt}{2.35693pt}{28.45276pt}{2.35693pt}\pgfsys@curveto{27.15105pt}{2.35693pt}{26.09583pt}{1.30171pt}{26.09583pt}{0.0pt}\pgfsys@curveto{26.09583pt}{-1.30171pt}{27.15105pt}{-2.35693pt}{28.45276pt}{-2.35693pt}\pgfsys@curveto{29.75447pt}{-2.35693pt}{30.8097pt}{-1.30171pt}{30.8097pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.5}{0.0}{0.0}{0.5}{28.45276pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{2.55693pt}{0.0pt}\pgfsys@lineto{25.89583pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.41684pt}{11.00417pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$h$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-2.08058pt}{1.48625pt}\pgfsys@lineto{-13.68384pt}{9.77495pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-2.08058pt}{-1.48625pt}\pgfsys@lineto{-13.30305pt}{-9.503pt}\pgfsys@stroke\pgfsys@invoke{
} \par{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{30.53334pt}{1.48625pt}\pgfsys@lineto{42.1366pt}{9.77495pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{30.53334pt}{-1.48625pt}\pgfsys@lineto{42.1366pt}{-9.77495pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}=\leavevmode\hbox
to80.33pt{\vbox to41.81pt{\pgfpicture\makeatletter\hbox{\hskip
26.13063pt\lower-21.0316pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{2.55693pt}{0.0pt}\pgfsys@lineto{25.89583pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-2.08058pt}{1.48625pt}\pgfsys@lineto{-13.68384pt}{9.77495pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-2.08058pt}{-1.48625pt}\pgfsys@lineto{-13.30305pt}{-9.503pt}\pgfsys@stroke\pgfsys@invoke{
} \par{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$5$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{30.53334pt}{1.48625pt}\pgfsys@lineto{42.1366pt}{9.77495pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{30.53334pt}{-1.48625pt}\pgfsys@lineto{42.1366pt}{-9.77495pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$

For a prestable graph

$\Gamma=\leavevmode\hbox to79.95pt{\vbox
to41.56pt{\pgfpicture\makeatletter\hbox{\hskip 25.74985pt\lower-20.7816pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{2.35693pt}{0.0pt}\pgfsys@curveto{2.35693pt}{1.30171pt}{1.30171pt}{2.35693pt}{0.0pt}{2.35693pt}\pgfsys@curveto{-1.30171pt}{2.35693pt}{-2.35693pt}{1.30171pt}{-2.35693pt}{0.0pt}\pgfsys@curveto{-2.35693pt}{-1.30171pt}{-1.30171pt}{-2.35693pt}{0.0pt}{-2.35693pt}\pgfsys@curveto{1.30171pt}{-2.35693pt}{2.35693pt}{-1.30171pt}{2.35693pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.5}{0.0}{0.0}{0.5}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{30.8097pt}{0.0pt}\pgfsys@curveto{30.8097pt}{1.30171pt}{29.75447pt}{2.35693pt}{28.45276pt}{2.35693pt}\pgfsys@curveto{27.15105pt}{2.35693pt}{26.09583pt}{1.30171pt}{26.09583pt}{0.0pt}\pgfsys@curveto{26.09583pt}{-1.30171pt}{27.15105pt}{-2.35693pt}{28.45276pt}{-2.35693pt}\pgfsys@curveto{29.75447pt}{-2.35693pt}{30.8097pt}{-1.30171pt}{30.8097pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{28.45276pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.5}{0.0}{0.0}{0.5}{28.45276pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{2.55693pt}{0.0pt}\pgfsys@lineto{25.89583pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.41684pt}{11.00417pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$w$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-2.08058pt}{1.48625pt}\pgfsys@lineto{-13.68384pt}{9.77495pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-2.08058pt}{-1.48625pt}\pgfsys@lineto{-13.68384pt}{-9.77495pt}\pgfsys@stroke\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{30.53334pt}{1.48625pt}\pgfsys@lineto{42.1366pt}{9.77495pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{30.53334pt}{-1.48625pt}\pgfsys@lineto{42.1366pt}{-9.77495pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{31.00969pt}{0.0pt}\pgfsys@lineto{42.1366pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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14.22636pt\text{ in $\mathcal{S}_{0,5}$}$

and a decoration $\alpha_{w}=\kappa_{3}$, the corresponding relation is

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{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{2.55693pt}{0.0pt}\pgfsys@lineto{25.89583pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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######  Definition 2.29.

Consider the family $\mathcal{R}_{\kappa,\psi}^{0}$ of relations obtained by
multiplying the relations of $\kappa$ and $\psi$-classes from Lemmas 2.2, 2.3
and 2.4 with an arbitrary monomial in $\kappa$ and $\psi$-classes. Define the
space $\mathcal{R}_{\kappa,\psi}\subset\mathcal{S}_{0,n}$ as the space of
relations generated by $\mathcal{R}_{\kappa,\psi}^{0}$.

Define $\mathcal{R}_{\mathrm{WDVV}}\subset\mathcal{S}_{0,n}^{\textup{nf}}$ as
the space of relations obtained by gluing some WDVV relation into a decorated
prestable graph $[\Gamma,\alpha]$ in normal form at a vertex $v$ with
$n(v)\geq 4$. In other words, it is the space of relations generated by the
WDVV relation as in Definition 2.27 where we restrict to
$\Gamma,(\alpha_{w})_{w\neq v}$ such that $[\Gamma,\alpha]$ (with
$\alpha_{v}=1$) is in normal form161616Again we relax the condition of the
$\alpha_{w}$ being monomials and allow $\alpha_{w}$ of the form
$\psi_{h}^{c}+(-\psi_{h^{\prime}})^{c}$ at vertices of valence $2$..

######  Remark 2.30.

Let us comment on the role of the sets of relations appearing above. The
relations $\mathcal{R}_{\kappa,\psi}^{0}$ allow to write any monomial $\alpha$
in $\kappa$ and $\psi$-classes on $\mathfrak{M}_{0,n}$ as a sum
$\alpha=\alpha_{0}+\beta$ of

  * •

a (possibly zero) monomial term $\alpha_{0}$ in $\kappa$ and $\psi$-classes
such that the trivial prestable graph with decoration $\alpha_{0}$ is in
normal form (implying that $\alpha_{0}$ restricts to a basis element of
$\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{\textup{sm}})$ as computed in Lemma 2.1),

  * •

a sum $\beta$ of generators $[\Gamma_{i},\alpha_{i}]$ supported in the
boundary (i.e. with $\Gamma_{i}$ nontrivial).

The relations $\mathcal{R}_{\kappa,\psi}$ allow to do the above at each of the
vertices of a decorated stratum class $[\Gamma,\alpha]$ and by a recursive
procedure allow to write $[\Gamma,\alpha]$ as a sum of decorated strata
classes in normal form. The relations in $\mathcal{R}_{\mathrm{WDVV}}$ then
encode the remaining freedom to express relations among these classes in
normal form generated by the $\mathrm{WDVV}$ relation. The following theorem
and the course of its proof make precise the statement that these processes
describe tautological relations on $\mathfrak{M}_{0,n}$.

######  Theorem 2.31.

The kernel of the surjection
$\mathcal{S}_{0,n}\to\mathrm{CH}^{*}(\mathfrak{M}_{0,n})$ is given by
$\mathcal{R}_{\kappa,\psi}+\mathcal{R}_{\mathrm{WDVV}}$. In particular, we
have

$\mathrm{CH}^{*}(\mathfrak{M}_{0,n})=\mathcal{S}_{0,n}/(\mathcal{R}_{\kappa,\psi}+\mathcal{R}_{\mathrm{WDVV}})\,.$

We split the proof of the above theorem into two parts.

######  Proposition 2.32.

The map
$\mathcal{S}_{0,n}^{\textup{nf}}\hookrightarrow\mathcal{S}_{0,n}\to\mathcal{S}_{0,n}/\mathcal{R}_{\kappa,\psi}$
is surjective.

###### Proof.

The statement says that we can use $\kappa,\psi$ relations on each vertex to
express any decorated stratum class as a linear combination of stratum classes
in normal form. This follows from Lemma 2.2, 2.3 and 2.4 as described in
Remark 2.30. ∎

######  Theorem 2.33.

The kernel of the surjection
$\mathcal{S}_{0,n}^{\textup{nf}}\to\mathrm{CH}^{*}(\mathfrak{M}_{0,n})$ is
given by $\mathcal{R}_{\mathrm{WDVV}}$.

The proof is separated into several steps. The overall strategy is to stratify
$\mathfrak{M}_{0,n}$ by the number of edges of the prestable graph $\Gamma$
and use an excision sequence argument. For $p\geq 0$ we denote by
$\mathfrak{M}_{0,n}^{\geq p}$ the closed substack of $\mathfrak{M}_{0,n}$ of
curves with at least $p$ nodes. Similarly, we denote by
$\mathfrak{M}_{0,n}^{=p}$ the open substack of $\mathfrak{M}_{0,n}^{\geq p}$
of curves with exactly $p$ nodes. It is clear that

$\mathfrak{M}_{0,n}^{\geq
p}\setminus\mathfrak{M}_{0,n}^{=p}=\mathfrak{M}_{0,n}^{\geq p+1}$

and also

$\mathfrak{M}_{0,n}^{=p}=\coprod_{\Gamma\in\mathcal{G}_{p}}\mathfrak{M}^{\Gamma}\,,$

where $\mathcal{G}_{p}$ is the set of prestable graphs of genus $0$ with $n$
markings having exactly $p$ edges. For the strata space $\mathcal{S}_{0,n}$,
consider the decomposition

$\mathcal{S}_{0,n}=\bigoplus_{p\geq 0}\mathcal{S}_{0,n}^{p}$

according to the number $p$ of edges of graph $\Gamma$.171717This
decomposition is not equal to the standard decomposition of
$\mathcal{S}_{0,n}$ via degree of a class. This descends to decompositions

$\mathcal{S}_{0,n}^{\textup{nf}}=\bigoplus_{p\geq
0}\mathcal{S}_{0,n}^{\textup{nf,$p$}},\
\mathcal{R}_{\mathrm{WDVV}}=\bigoplus_{p\geq
0}\mathcal{R}_{\mathrm{WDVV}}^{p}$

for $\mathcal{S}_{0,n}^{\textup{nf}}$ and $\mathcal{R}_{\mathrm{WDVV}}$. We
note that $\mathcal{R}_{\mathrm{WDVV}}^{p}$ is exactly the space of relations
obtained by taking a prestable graph $\Gamma$ with $p-1$ edges, a decoration
$\alpha$ on $\Gamma$ in normal form and inserting a $\mathrm{WDVV}$ relation
at a vertex $v_{0}\in V(\Gamma)$ with $n(v_{0})\geq 4$.

From Proposition [BS21, Proposition 2.4] and Lemma 2.20 it follows that

$\displaystyle\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{=p})$
$\displaystyle=\bigoplus_{\Gamma\in\mathcal{G}_{p}}\mathrm{CH}^{*}(\mathfrak{M}^{\Gamma})=\bigoplus_{\Gamma\in\mathcal{G}_{p}}\mathrm{CH}^{*}(\mathfrak{M}_{\Gamma}^{\textup{sm}})^{\mathrm{Aut}(\Gamma)}\,,$
(29) $\displaystyle\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{=p},1)$
$\displaystyle=\bigoplus_{\Gamma\in\mathcal{G}_{p}}\mathrm{CH}^{*}(\mathfrak{M}^{\Gamma},1)=\bigoplus_{\Gamma\in\mathcal{G}_{p}}\mathrm{CH}^{*}(\mathfrak{M}_{\Gamma}^{\textup{sm}},1)^{\mathrm{Aut}(\Gamma)}\,.$
(30)

Note that we have a natural map
$\mathcal{S}_{0,n}^{\textup{nf,$p$}}\to\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{\geq
p})$.

######  Lemma 2.34.

The composition
$\mathcal{S}_{0,n}^{\textup{nf,$p$}}\to\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{\geq
p})\to\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{=p})$ is an isomorphism.

###### Proof.

First we note that $\mathcal{S}_{0,n}^{\textup{nf,$p$}}$ decomposes into a
direct sum of subspaces $\mathcal{S}_{0,n}^{\textup{nf,$\Gamma$}}$ indexed by
prestable graphs $\Gamma\in\mathcal{G}_{p}$, according to the underlying
prestable graph of the generators. The analogous decomposition of
$\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{=p})$ is given by (29). Now for two non-
isomorphic prestable graphs $\Gamma$ and $\Gamma^{\prime}$ with the same
number $p$ of edges, the induced map
$\mathcal{S}_{0,n}^{\textup{nf,$\Gamma$}}\to\mathrm{CH}^{*}(\mathfrak{M}^{\Gamma^{\prime}})$
vanishes. Indeed, the locally closed substack $\mathfrak{M}^{\Gamma^{\prime}}$
is disjoint from the image of the gluing map $\xi_{\Gamma}$ and all generators
of $\mathcal{S}_{0,n}^{\textup{nf,$\Gamma$}}$ are pushforwards under
$\xi_{\Gamma}$. Thus we are reduced to showing that
$\mathcal{S}_{0,n}^{\textup{nf,$\Gamma$}}\to\mathrm{CH}^{*}(\mathfrak{M}^{\Gamma})$
is an isomorphism. The image of a generator $[\Gamma,\alpha]$ under this map
is obtained by pushing forward
$\alpha\in\mathrm{CH}^{*}(\mathfrak{M}_{\Gamma})$ to $\mathfrak{M}_{0,n}^{=p}$
under $\xi_{\Gamma}$ and restricting to the open subset
$\mathfrak{M}^{\Gamma}$. From the cartesian diagram

${\mathfrak{M}_{\Gamma}}$${\mathfrak{M}_{0,n}^{\geq
p}}$${\mathfrak{M}_{\Gamma}^{\textup{sm}}}$${\mathfrak{M}^{\Gamma}=\mathfrak{M}^{\textup{sm}}_{\Gamma}/\mathrm{Aut}(\Gamma)}$$\scriptstyle{\xi_{\Gamma}}$$\scriptstyle{\xi_{\Gamma}^{\textup{sm}}}$

in which the vertical arrows are open embeddings, it follows that this is
equivalent to first restricting $\alpha$ to
$\mathfrak{M}_{\Gamma}^{\textup{sm}}$ and then pushing forward to
$\mathfrak{M}^{\Gamma}$. As we saw in (29), we can identify
$\mathrm{CH}^{*}(\mathfrak{M}^{\Gamma})$ with the
$\mathrm{Aut}(\Gamma)$-invariant part of
$\mathrm{CH}^{*}(\mathfrak{M}_{\Gamma}^{\textup{sm}})$ via pullback under
$\xi_{\Gamma}^{\textup{sm}}$. But clearly

$(\xi_{\Gamma}^{\textup{sm}})^{*}[\Gamma,\alpha]=(\xi_{\Gamma}^{\textup{sm}})^{*}(\xi_{\Gamma}^{\textup{sm}})_{*}\alpha=\sum_{\sigma\in\mathrm{Aut}(\Gamma)}\sigma^{*}\alpha\,,$
(31)

where the automorphisms $\sigma$ act on $\mathfrak{M}_{\Gamma}^{\textup{sm}}$
by permuting the factors.

Now by Proposition 2.7 we have

$\mathrm{CH}^{*}(\mathfrak{M}_{\Gamma}^{\textup{sm}})=\bigotimes_{v\in
V(\Gamma)}\mathrm{CH}^{*}(\mathfrak{M}_{g(v),n(v)}^{\textup{sm}})\,.$

Thus it follows from Lemma 2.1 that the set of all possible $\alpha$ such that
$[\Gamma,\alpha]$ is in normal form is a basis of
$\mathrm{CH}^{*}(\mathfrak{M}_{\Gamma}^{\textup{sm}})$ and the action of
automorphisms $\sigma$ of $\Gamma$ acts by permuting these basis elements.
Hence a basis of the $\mathrm{Aut}(\Gamma)$-invariant part of
$\mathrm{CH}^{*}(\mathfrak{M}_{\Gamma}^{\textup{sm}})$ is given by the sums of
orbits of these basis elements (with the dimension of
$\mathrm{CH}^{*}(\mathfrak{M}_{\Gamma}^{\textup{sm}})^{\mathrm{Aut}(\Gamma)}$
being the number of such orbits). Then the fact that we chose the basis of
$\mathcal{S}_{0,n}^{\textup{nf,$\Gamma$}}$ to be the set of $[\Gamma,\alpha]$
_up to isomorphism_ implies via (31) that they exactly map to this basis of
$\mathrm{CH}^{*}(\mathfrak{M}_{\Gamma}^{\textup{sm}})^{\mathrm{Aut}(\Gamma)}$.
∎

Next we realize the WDVV relation as the image of the connecting homomorphism
$\partial$ of the excision sequence

$\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{=p},1)\xrightarrow{\partial}\mathrm{CH}^{*-1}(\mathfrak{M}_{0,n}^{\geq
p+1})\to\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{\geq
p})\to\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{=p})\to 0\,.$ (32)

By[BS21, Proposition 2.4], the stack $\mathfrak{M}^{=p}_{0,n}$ is a quotient
stack and hence (32) is exact by [Kre99, Proposition 4.2.1].

Before we study the map $\partial$ in the sequence (32), we consider an easier
situation: we show that in the setting of the moduli spaces
$\overline{\mathcal{M}}_{0,n}$ of stable curves, we can explicitly compute the
connecting homomorphism $\partial$, see Proposition 2.36 below. In the proof,
we will need the following technical lemma about the connecting homomorphisms
of excision sequences.

######  Lemma 2.35.

Let $X$ be an equidimensional scheme and let

$Z^{\prime}\xrightarrow{j^{\prime}}Z\xrightarrow{j}X$

be two closed immersions. Consider the open embedding

$U=X\setminus Z\xrightarrow{i}U^{\prime}=X\setminus Z^{\prime}.$

Then we have a commutative diagram

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{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-8.21515pt}{-24.26387pt}\pgfsys@lineto{17.08041pt}{-24.26387pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.2804pt}{-24.26387pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{2.77446pt}{-21.9111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{$\scriptstyle{\partial}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$ (33)

where $\partial$ and $\partial^{\prime}$ are the connecting homomorphisms for
the inclusions of $U$ and $U^{\prime}$ in $X$.

###### Proof.

Elements of $\mathrm{CH}_{n}(U^{\prime},1)$ are represented by cycles $\sum
a_{i}W_{i}$ on $U^{\prime}\times\Delta^{1}$ with the $W_{i}$ of dimension
$n+1$ intersecting the faces of $U^{\prime}\times\partial\Delta^{1}$ properly.
On the one hand, to evaluate the connecting homomorphism $\partial^{\prime}$,
we form the closures $\overline{W_{i}}$ in $X\times\Delta^{1}$ and take
alternating intersections with faces. This is a sum of cycles of dimension $n$
supported on $Z^{\prime}\times\partial\Delta^{1}$ and via $j^{\prime}_{*}$ we
regard it as a sum of cycles on $Z\times\partial\Delta^{1}$.

On the other hand, to evaluate $\partial\circ i^{*}$ we first restrict all
$W_{i}$ to $U\times\Delta^{1}$, take the closure $\overline{W_{i}\cap
U\times\Delta^{1}}$ and take alternating intersection with faces. But the only
way that this closure can be different from $\overline{W_{i}}$ is when $W_{i}$
has generic point in $Z\times\Delta^{1}$. But then it defines an element of
$z_{n}(Z,1)$ and thus it maps to zero in $\mathrm{CH}_{*}(Z)$. ∎

######  Proposition 2.36.

For $n\geq 4$, the image of the connecting homomorphism $\partial$ of

$\mathrm{CH}^{1}(\mathcal{M}_{0,n},1)\xrightarrow{\partial}\mathrm{CH}^{0}(\overline{\mathcal{M}}_{0,n}^{\geq
1})\xrightarrow{\iota_{*}}\mathrm{CH}^{1}(\overline{\mathcal{M}}_{0,n})\to
0=\mathrm{CH}^{1}(\mathcal{M}_{0,n})\,.$

is the set of WDVV relations, where we identify
$\mathrm{CH}^{0}(\overline{\mathcal{M}}_{0,n}^{\geq 1})$ as the
$\mathbb{Q}$-vector space with basis given by boundary divisors of
$\overline{\mathcal{M}}_{0,n}$.

###### Proof.

First, we prove this proposition when $n=4$. Identify
$\overline{\mathcal{M}}_{0,4}\cong\mathbb{P}^{1}$ and
$\mathcal{M}_{0,4}\cong\mathbb{A}^{1}-\\{0,1\\}$. Then
$\mathrm{CH}^{*}(\mathcal{M}_{0,4},\bullet)$ is a
$\mathrm{CH}^{*}(k,\bullet)$-algebra generated by two elements $f_{0}$ and
$f_{1}$ corresponding to two points in $\mathbb{A}^{1}$ ([Cha07]). Fix a (non-
canonical) isomorphism $\Delta^{1}\cong\mathbb{A}^{1}$ and set the two faces
as $0$ and $1$. Consider a line $L_{0}$ through $(0,0)$ and $(1,1)$ in
$\mathbb{P}^{1}\times\Delta^{1}$ restricted to
$(\mathbb{P}^{1}-\\{0,1,\infty\\})\times\Delta^{1}$ as illustrated in Figure
3.

$(\mathbb{P}^{1}-\\{0,1,\infty\\})\times\Delta^{1}$$L_{0}$$\Delta^{1}$$0$$1$
Figure 3: The line $L_{0}$ in
$(\mathbb{P}^{1}-\\{0,1,\infty\\})\times\Delta^{1}$

Then $f_{0}=[L_{0}]$ and

$\partial(L_{0})=[0]-[1]\in\mathrm{CH}^{0}(\overline{\mathcal{M}}_{0,4}\setminus\mathcal{M}_{0,4})\,.$

This is one of the WDVV relations on $\overline{\mathcal{M}}_{0,4}$ after
identifying $[0],[1]$ and $[\infty]$ with three boundary strata in (4). The
second one is obtained from the generator $f_{1}$ in an analogous way,
finishing the proof for $n=4$.

For the case of general $n\geq 4$, the space $\mathcal{M}_{0,n}$ is a
hyperplane complement with hyperplanes associated to pairs of points that
collide and there is a correspondence between generators of
$\mathrm{CH}^{1}(\mathcal{M}_{0,n},1)$ and hyperplanes. On the one hand, the
action of the symmetric group $S_{n}$ on $\mathcal{M}_{0,n}$ is transitive on
the hyperplanes (and thus on the generators). On the other hand, we can obtain
one of the hyperplanes as the pullback of a boundary point in
$\mathcal{M}_{0.4}$ under the forgetful morphism
$\pi:\overline{\mathcal{M}}_{0,n}\to\overline{\mathcal{M}}_{0,4}$ and thus,
via the action of $S_{n}$, any hyperplane can be obtained under a suitable
forgetful morphism to $\overline{\mathcal{M}}_{0,4}$ (varying the subset of
four points to remember).

Note that the morphism $\pi$ is flat and that we have an open embedding
$i:\mathcal{M}_{0,n}\to\pi^{-1}(\mathcal{M}_{0,4})$ and a closed embedding
$j^{\prime}:\pi^{-1}(\partial\overline{\mathcal{M}}_{0,4})\to\partial\overline{\mathcal{M}}_{0,n}$.
Combining the compatibility of the connecting homomorphism $\partial$ with
flat pullback and Lemma 2.35 above, we obtain a commutative diagram

${\mathrm{CH}^{1}(\mathcal{M}_{0,n},1)}$${\mathrm{CH}^{0}(\partial\overline{\mathcal{M}}_{0,n})}$${\mathrm{CH}^{1}(\pi^{-1}(\mathcal{M}_{0,4}),1)}$${\mathrm{CH}^{0}(\pi^{-1}(\partial\overline{\mathcal{M}}_{0,4}))}$${\mathrm{CH}^{1}(\mathcal{M}_{0,4},1)}$${\mathrm{CH}^{0}(\partial\overline{\mathcal{M}}_{0,4})}$$\scriptstyle{\partial}$$\scriptstyle{\partial}$$\scriptstyle{i^{*}}$$\scriptstyle{j^{\prime}_{*}}$$\scriptstyle{\partial}$$\scriptstyle{\pi^{*}}$$\scriptstyle{\pi^{*}}$

Therefore, since (under a suitable permutation of the markings) every
generator of $\mathrm{CH}^{1}(\mathcal{M}_{0,n},1)$ can be obtained as the
image of one of the generators of $\mathrm{CH}^{1}(\mathcal{M}_{0,4},1)$, the
image of

$\mathrm{CH}^{1}(\mathcal{M}_{0,n},1)\to\mathrm{CH}^{0}(\partial\overline{\mathcal{M}}_{0,n})$

is generated by WDVV relations on
$\mathrm{CH}^{0}(\partial\overline{\mathcal{M}}_{0,4})$ pulled back via $\pi$.
∎

We extend the above computation to $\mathfrak{M}_{0,n}$.

######  Corollary 2.37.

For $n\geq 4$, the image of the connecting homomorphism $\partial$ of

$\mathrm{CH}^{\ell+1}(\mathcal{M}_{0,n},1)\xrightarrow{\partial}\mathrm{CH}^{\ell}(\mathfrak{M}_{0,n}^{\geq
1})\xrightarrow{\iota_{*}}\mathrm{CH}^{\ell+1}(\mathfrak{M}_{0,n})\to 0\,$

is the set of WDVV relations for $\ell=0$ and is zero for $\ell>0$.

###### Proof.

By (23), we have $\mathrm{CH}^{\ell+1}(\mathcal{M}_{0,n},1)=0$ when $\ell>0$
and hence $\partial$ is trivial in this range.

For the statement in degree $\ell=0$ we in fact ignore the definition of
$\partial$ and the machinery of higher Chow groups and simply use that here
the image of $\partial$ is given by the kernel of the map

$\mathrm{CH}^{0}(\mathfrak{M}_{0,n}^{\geq
1})\xrightarrow{\iota_{*}}\mathrm{CH}^{1}(\mathfrak{M}_{0,n}),$ (34)

in other words by linear combinations of boundary divisors adding to zero in
$\mathrm{CH}^{1}(\mathfrak{M}_{0,n})$. Given such a relation, restricting to
the open substack $\overline{\mathcal{M}}_{0,n}$ simply kills all unstable
boundary divisors and by Proposition 2.36 (or classical theory) the result is
a combination of WDVV relations. After subtracting those from the original
relation, we obtain a combination of unstable boundary divisors forming a
relation. The proof is finished if we can show that this must be the trivial
linear combination, i.e. that the unstable boundary divisors are linearly
independent.

There are exactly $n+1$ strictly prestable graphs with one edge. Let
$\Gamma_{0}$ be the prestable graph with a vertex of valence $1$ and
$\Gamma_{i}$ be the semistable graph with the $i$-th leg on the semistable
vertex. Suppose there is a linear relation

$R=a_{0}[\Gamma_{0}]+a_{1}[\Gamma_{1}]+\ldots+a_{n}[\Gamma_{n}]=0,\,\,a_{i}\in\mathbb{Q}$

in $\mathrm{CH}^{1}(\mathfrak{M}_{0,n})$, then we want to show that all
$a_{i}=0$.

To see this we can simply construct test curves
$\sigma_{i}:\mathbb{P}^{1}\to\mathfrak{M}_{0,n}$ intersecting precisely the
divisor $[\Gamma_{i}]$ and none of the others. To obtain $\sigma_{i}$, start
with the trivial family $\mathbb{P}^{1}\times\mathbb{P}^{1}\to\mathbb{P}^{1}$
and a tuple of $n$ disjoint constant sections $p_{1},\ldots,p_{n}$. Let
$\sigma_{0}$ be defined by the family of prestable curves obtained by blowing
up a point on $\mathbb{P}^{1}\times\mathbb{P}^{1}$ away from any of the
sections. For $1\leq i\leq n$ we similarly obtain $\sigma_{i}$ by blowing up a
point on the image of $p_{i}$ and taking the strict transform of the old
section $p_{i}$. Then we have $0=\sigma_{i}^{*}R=a_{i}$ for all $i$, finishing
the proof. ∎

######  Remark 2.38.

As a consequence of the above result, the map

$\iota_{*}:\mathrm{CH}^{\ell}(\mathfrak{M}_{0,n}^{\geq
1})\to\mathrm{CH}^{\ell+1}(\mathfrak{M}_{0,n})$

is an isomorphism in degree $\ell\geq 1$. Restricting to the locus of stable
curves, the same proof implies that

$\iota_{*}:\mathrm{CH}^{\ell}(\partial\overline{\mathcal{M}}_{0,n})\to\mathrm{CH}^{\ell+1}(\overline{\mathcal{M}}_{0,n})$

is an isomorphism for $\ell\geq 1$.

The surjectivity of $\iota_{*}$ comes from the excision sequence that we
discussed. The injectivity can be explained from the results of Kontsevich and
Manin ([KM94]). Indeed, by Proposition [BS21, Proposition B.19], the vector
space $\mathrm{CH}^{\ell}(\partial\overline{\mathcal{M}}_{0,n})$ is generated
by boundary strata of $\overline{\mathcal{M}}_{0,n}$ with at least one edge.
To show injectivity of $\iota_{*}$, it is enough to show that any relation
among boundary strata in $\mathrm{CH}^{\ell+1}(\overline{\mathcal{M}}_{0,n})$
is a pushforward of a relation holding already in the Chow group of
$\partial\overline{\mathcal{M}}_{0,n}$. By [KM94, Theorem 7.3] the set of
relations between boundary strata in
$\mathrm{CH}^{\ell+1}(\overline{\mathcal{M}}_{0,n})$ is spanned by the
relations obtained from gluing the WDVV relation into a vertex $v_{0}$ of a
stable graph $\Gamma$ with at least $\ell$ edges. When $\ell\geq 1$, this
relation is a pushforward of a class

$\prod_{v\neq
v_{0}}\left[\overline{\mathcal{M}}_{0,n(v)}\right]\times\textup{WDVV}\in\mathrm{CH}^{1}(\overline{\mathcal{M}}_{\Gamma})$

where $\textup{WDVV}\in\mathrm{CH}^{1}(\overline{\mathcal{M}}_{0,n(v_{0})})$
is the WDVV relation corresponding to the choice of four half edges at
$v_{0}$. Under the gluing map, this class is a relation on
$\partial\overline{\mathcal{M}}_{0,n}$. Therefore we get the injectivity of
$\iota_{*}$.

This corollary is enough to compute the connecting homomorphism in arbitrary
degree.

######  Proposition 2.39.

The image of
$\partial\colon\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{=p},1)\to\mathrm{CH}^{*-1}(\mathfrak{M}_{0,n}^{\geq
p+1})$ in (32) is equal to the image of the composition

$\mathcal{R}_{\mathrm{WDVV}}^{p+1}\to\mathcal{S}_{0,n}^{\textup{nf,$p+1$}}\to\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{\geq
p+1})\,.$

Thus we can write

$\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{\geq
p+1})/\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{=p},1)=\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{\geq
p+1})/\mathcal{R}_{\mathrm{WDVV}}^{p+1}\,.$ (35)

###### Proof.

From the functoriality of higher Chow groups, it follows that we have a
commutative diagram

${\bigoplus_{\Gamma}\mathrm{CH}^{*}(\mathfrak{M}_{\Gamma}^{\textup{sm}},1)}$${\bigoplus_{\Gamma}\mathrm{CH}^{*}(\mathfrak{M}^{\Gamma},1)}$${\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{=p},1)}$${\bigoplus_{\Gamma}\mathrm{CH}^{*-1}(\mathfrak{M}_{\Gamma}\setminus\mathfrak{M}_{\Gamma}^{\textup{sm}})}$${\mathrm{CH}^{*-1}(\mathfrak{M}_{0,n}^{\geq
p+1})}$$\scriptstyle{\bigoplus(\xi_{\Gamma}^{\textup{sm}})_{*}}$$\scriptstyle{\bigoplus\partial_{\Gamma}}$$\scriptstyle{\partial}$$\scriptstyle{\sum(\xi_{\Gamma})_{*}}$
(36)

where the sums run over prestable graphs with exactly $p$ edges. By Remark
2.21, the maps $(\xi_{\Gamma}^{\textup{sm}})_{*}$ are surjective. Thus the
image of $\partial$ is given by the sum of the images of the maps
$(\xi_{\Gamma})_{*}\circ\partial_{\Gamma}$.

From Remark 2.18 we know that $\partial_{\Gamma}$ vanishes on the image of the
map

$\mathrm{CH}_{*}(k,1)\otimes\mathrm{CH}_{*}(\mathfrak{M}_{\Gamma}^{\textup{sm}})\to\mathrm{CH}_{*}(\mathfrak{M}_{\Gamma}^{\textup{sm}},1),$
(37)

and thus factors through its cokernel. On the other hand, it follows from
Propositions 2.14 and 2.16 that the cokernel of (37) is generated by classes
coming from the direct sum

$\bigoplus_{v\in V(\Gamma),n(v)\geq
4}\Big{(}\mathrm{CH}^{1}(\mathfrak{M}^{\textup{sm}}_{0,n(v)},1)\otimes\bigotimes_{v^{\prime}\in
V(\Gamma),v\neq
v^{\prime}}\mathrm{CH}^{*}(\mathfrak{M}^{\textup{sm}}_{0,n(v^{\prime})})\Big{)}\,.$

For an element $\alpha_{v}\otimes\bigotimes_{v\neq
v^{\prime}}\alpha_{v^{\prime}}$ in
$\mathrm{CH}^{*}(\mathfrak{M}_{\Gamma}^{\textup{sm}},1)$, choose an extension
$\overline{\alpha}_{v^{\prime}}$ of each $\alpha_{v^{\prime}}$ from
$\mathfrak{M}_{0,n(v^{\prime})}^{\textup{sm}}$ to
$\mathfrak{M}_{0,n(v^{\prime})}$. By Lemma 2.17 (a), the boundary map
$\partial_{\Gamma}$ has the form

$\partial_{\Gamma}\Big{(}\alpha_{v}\otimes\bigotimes_{v\neq
v^{\prime}}\alpha_{v^{\prime}}\Big{)}=\partial(\alpha_{v})\otimes\bigotimes_{v\neq
v^{\prime}}\overline{\alpha}_{v^{\prime}}\,.$

By Corollary 2.37, the elements $\partial(\alpha_{v})$ at vertices $v$ with
$n(v)\geq 4$ are precisely the $\mathrm{WDVV}$ relations on
$\mathfrak{M}_{0,n(v)}$, whereas the classes $\overline{\alpha}_{v^{\prime}}$
at other vertices $v^{\prime}$ are exactly the types of decorations allowed in
decorated strata classes in normal form. After pushing forward via
$\xi_{\Gamma}$ this is precisely our definition of the relations
$\mathcal{R}_{\mathrm{WDVV}}^{p+1}$. ∎

###### Proof of Theorem 2.33.

Recall that by Lemma 2.34, the composition

$\mathcal{S}_{0,n}^{\textup{nf,$p$}}\to\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{\geq
p})\to\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{=p})$

is an isomorphism. Thus in the diagram

${\mathcal{S}_{0,n}^{\textup{nf,$p$}}}$${\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{=p},1)}$${\mathrm{CH}^{*-1}(\mathfrak{M}_{0,n}^{\geq
p+1})}$${\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{\geq
p})}$${\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{=p})}$${0}$$\scriptstyle{\cong}$$\scriptstyle{\partial}$

we obtain a canonical splitting of the excision exact sequence (32) and thus
we have

$\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{\geq
p})=\mathcal{S}_{0,n}^{\textup{nf,$p$}}\oplus\mathrm{CH}^{*-1}(\mathfrak{M}_{0,n}^{\geq
p+1})/\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{=p},1)\,.$ (38)

Combining (38) and equation (35) from Proposition 2.39, we see

$\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{\geq
p})=\mathcal{S}_{0,n}^{\textup{nf,$p$}}\oplus\mathrm{CH}^{*-1}(\mathfrak{M}_{0,n}^{\geq
p+1})/\mathcal{R}_{\mathrm{WDVV}}^{p+1}\,.$ (39)

Applying (39) for $p=0,1,2,\ldots$ we obtain

$\displaystyle\mathrm{CH}^{*}(\mathfrak{M}_{0,n})$
$\displaystyle=\mathrm{CH}^{*}(\mathfrak{M}_{0,n}^{\geq 0})$
$\displaystyle=\mathcal{S}_{0,n}^{\textup{nf,$0$}}\oplus\mathrm{CH}^{*-1}(\mathfrak{M}_{0,n}^{\geq
1})/\mathcal{R}_{\mathrm{WDVV}}^{1}$
$\displaystyle=\mathcal{S}_{0,n}^{\textup{nf,$0$}}\oplus\left(\mathcal{S}_{0,n}^{\textup{nf,$1$}}/\mathcal{R}_{\mathrm{WDVV}}^{1}\right)\oplus\mathrm{CH}^{*-2}(\mathfrak{M}_{0,n}^{\geq
2})/\mathcal{R}_{\mathrm{WDVV}}^{2}$ $\displaystyle=\ldots$
$\displaystyle=\bigoplus_{p\geq
0}\mathcal{S}_{0,n}^{\textup{nf,$p$}}/\mathcal{R}_{\mathrm{WDVV}}^{p}$
$\displaystyle=\mathcal{S}_{0,n}^{\textup{nf}}/\mathcal{R}_{\mathrm{WDVV}}\,,$

finishing the proof. ∎

Finally we end the proof of the main theorem.

###### Proof of Theorem 2.31.

We know that the kernel of
$\mathcal{S}_{0,n}\to\mathrm{CH}^{*}(\mathfrak{M}_{0,n})$ contains
$\mathcal{R}_{\kappa,\psi}$. We define

$\mathcal{R}_{\text{res}}=\ker\big{(}\mathcal{S}_{0,n}/\mathcal{R}_{\kappa,\psi}\to\mathrm{CH}^{*}(\mathfrak{M}_{0,n})\big{)}\,.$

Likewise, by Theorem 2.33 we know that the kernel of
$\mathcal{S}_{0,n}^{\textup{nf}}\to\mathrm{CH}^{*}(\mathfrak{M}_{0,n})$ is
equal to $\mathcal{R}_{\mathrm{WDVV}}$. We then obtain a diagram of morphisms
with exact rows

${0}$${\mathcal{R}_{\mathrm{WDVV}}}$${\mathcal{S}_{0,n}^{\textup{nf}}}$${\mathrm{CH}^{*}(\mathfrak{M}_{0,n})}$${0}$${0}$${\mathcal{R}_{\text{res}}}$${\mathcal{S}_{0,n}/\mathcal{R}_{\kappa,\psi}}$${\mathrm{CH}^{*}(\mathfrak{M}_{0,n})}$${0}$
(40)

where the arrow
$\mathcal{S}_{0,n}^{\textup{nf}}\to\mathcal{S}_{0,n}/\mathcal{R}_{\kappa,\psi}$
is surjective by Proposition 2.32. A short diagram chase shows that
$\mathcal{R}_{\mathrm{WDVV}}$ factors through $\mathcal{R}_{\text{res}}$ via
the dashed arrow. By the four-lemma, the map
$\mathcal{R}_{\mathrm{WDVV}}\to\mathcal{R}_{\text{res}}$ is surjective. This
clearly implies the statement of the theorem. ∎

###  2.5 Relation to previous works

Let us start by pointing out several results in Gromov–Witten theory, studying
intersection numbers on moduli spaces of stable maps, which can be seen as
coming from results about the tautological ring of $\mathfrak{M}_{g,n}$.

######  Example 2.40.

In [LP04], degree one relations on the moduli space of stable maps to a
projective space $\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{N},d)$ are used to
reduce two pointed genus $0$ potentials to one pointed genus $0$ potentials.
Theorem 1.(2) from [LP04] can be obtained by the pullback of the relation in
Lemma 2.2 along the forgetful morphism

$\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{N},d)\to\mathfrak{M}_{0,2}\,.$

Similarly, Theorem 1.(1) of [LP04] can be obtained from Lemma 2.3 on
$\mathfrak{M}_{0,3}$. The relevant computations are given explicitly in
[Bae20].

From Theorem 2.31 we see that any universal relation in the Gromov–Witten
theory of genus $0$ obtained from tautological relations on
$\mathfrak{M}_{0,n}$ must follow either from the WDVV relation (4) or the
relation (8) between $\psi$ and boundary classes on $\mathfrak{M}_{0,2}$.

Apart from applications to Gromov–Witten theory, there are several results in
the literature which compute the Chow groups of some strict open subloci of
$\mathfrak{M}_{0,n}$.

######  Example 2.41.

Restricting to the locus
$\overline{\mathcal{M}}_{0,n}\subset\mathfrak{M}_{0,n}$ of stable curves,
Theorem 2.31 specializes to the classical result in [Kee92, KM94] that all
relations between undecorated strata of $\overline{\mathcal{M}}_{0,n}$ are
additively generated by the WDVV relations.

######  Example 2.42.

In [Oes19], Oesinghaus computes the Chow ring (with integer coefficients) of
the open substack $\mathcal{T}$ of $\mathfrak{M}_{0,3}$ of curves with
prestable graph of the form

$1$$2$$3$

where we denote by $\Gamma_{k}$ the graph of the shape above with $k$ edges
(for $k\geq 0$). Oesinghaus shows that the Chow ring
$\mathrm{CH}^{*}(\mathcal{T})$ is given by the ring QSym of quasi-symmetric
functions on the index set $\mathbb{Z}_{>0}$. QSym can be seen as the subring
of $\mathbb{Q}[\alpha_{1},\alpha_{2},\ldots]$ with additive basis given by

$M_{J}=\sum_{i_{1}<\ldots<i_{k}}\alpha_{i_{1}}^{j_{1}}\cdots\alpha_{i_{k}}^{j_{k}}\text{
for }k\geq 1,J=(j_{1},\ldots,j_{k})\in\mathbb{Z}_{\geq 1}^{k}.$ (41)

Under the isomorphism $\mathrm{CH}^{*}(\mathcal{T})\cong\text{QSym}$, the
element $M_{J}$ is a basis element of degree $\sum_{\ell}j_{\ell}$ in the Chow
group of $\mathcal{T}$. As we explain in [BS21, Example 4.3], the cycle
$M_{J}$ corresponds to the tautological class supported on the stratum
$\mathfrak{M}^{\Gamma_{k}}$ given by

$(-\psi-\psi^{\prime})^{j_{1}-1}$$(-\psi-\psi^{\prime})^{j_{\ell}-1}$$(-\psi-\psi^{\prime})^{j_{k}-1}$$1$$2$$3$
(42)

Using the correspondence, we can verify several of the results of our paper in
this particular example. Indeed, one can use Theorem 2.33 to verify that the
classes (42) form a basis of $\mathrm{CH}^{*}(\mathcal{T})$. For this, one
observes that decorated strata in normal form generically supported on
$\mathcal{T}$ must have underlying graph $\Gamma_{k}$ for some $k$, with
trivial decoration on the valence $3$ vertex and decorations
$(-\psi_{h}-\psi_{h^{\prime}})^{c_{\ell}}$ on the valence $2$ vertices. Since
every term appearing in a WDVV relation has at least two vertices of valence
at least $3$, all these relations restrict to zero on $\mathcal{T}$ and thus
the above generators form a basis by Theorem 2.33. Note that the form of these
generators in normal form is not quite the same as the one shown in (42), but
a small combinatorial argument shows that the two bases can be converted to
each other by using the relation (8) between $\psi$-classes and the boundary
divisor on $\mathfrak{M}_{0,2}$.

Note that [Oes19] also computes the Chow group of the semistable loci
$\mathfrak{M}_{0,2}^{ss}$ and $\mathfrak{M}_{0,3}^{ss}$. By a straightforward
generalization of the discussion above, a correspondence of the generators in
[Oes19] to the tautological generators on these spaces, as well as a
comparison of relations can be established.

######  Example 2.43.

In a series of paper [Ful10b, Ful10c, Ful10a], Fulghesu presented a
computation of the Chow ring of the open substack $\mathfrak{M}_{0}^{\leq
3}\subset\mathfrak{M}_{0}$ of rational curves with at most $3$ nodes, as an
explicit algebra with $10$ generators and $11$ relations. Some of the
generators are given by $\kappa$-classes, some are classes of strata and
others are decorated classes supported on strata.

Establishing a precise correspondence to the generators and relations
discussed in our paper is challenging due to the complexity of the involved
combinatorics. However, as a nontrivial check of our results we can compare
the dimensions $\dim\mathrm{CH}^{d}(\mathfrak{M}_{0}^{\leq 3})$ of the graded
pieces of the Chow ring. Given any open substack $U\subseteq\mathfrak{M}_{0}$,
we package the ranks of the Chow groups of $U$ in the generating function

$H_{U}=\sum_{d\geq 0}\dim\mathrm{CH}^{d}(U)t^{d},$

which is the _Hilbert series_ of the graded ring $\mathrm{CH}^{*}(U)$.

In [Ful10a], Fulghesu computes the Chow rings of the open substacks
$U=\mathfrak{M}_{0}^{\leq e}$ for $e=0,1,2,3$ in terms of generators and
relations. Using the software Macaulay2 [GS] we can compute181818The output of
the relevant computation can be found here. the Hilbert functions $H_{U}^{F}$
of the graded algebras given by Fulghesu. We list them in Figure 4.

$U$ | $H_{U}^{F}$ |
---|---|---
$\mathfrak{M}_{0}^{\leq 0}$ | $\frac{1}{1-t^{2}}$ | $=1+t^{2}+t^{4}+\ldots$
$\mathfrak{M}_{0}^{\leq 1}$ | $\frac{1}{(1-t^{2})(1-t)}$ | $=1+t+2t^{2}+2t^{3}+3t^{4}+\ldots$
$\mathfrak{M}_{0}^{\leq 2}$ | $\frac{t^{4}+1}{(1-t^{2})^{2}(1-t)}$ | $=1+t+3t^{2}+3t^{3}+7t^{4}+7t^{5}+13t^{6}+13t^{7}+21t^{8}+\ldots$
$\mathfrak{M}_{0}^{\leq 3}$ | $H^{\prime}$ | $=1+t+3t^{2}+5t^{3}+10t^{4}+15t^{5}+26t^{6}+36t^{7}+\mathbf{55}t^{8}+\ldots$
Figure 4: The Hilbert series of the Chow rings of open substacks $U$ of
$\mathfrak{M}_{0}$, as computed by Fulghesu; for space reasons we don’t write
the full formula for the rational function $H^{\prime}$, only giving the
expansion

On the other hand, since for stable graphs with at most three edges no WDVV
relations can appear, Theorem 2.33 implies that the Chow group
$\mathrm{CH}^{*}(\mathfrak{M}_{0}^{\leq e})$ is equal to the subspace of the
strata algebra $\mathcal{S}_{0,0}$ spanned by decorated strata in normal form
with at most $e$ nodes (for $e\leq 3$). By some small combinatorial arguments,
this allows us to compute the Hilbert functions $H_{U}$ of the spaces
$U=\mathfrak{M}_{0}^{\leq e}$:

$\mathbf{e=0}$ For $U=\mathfrak{M}_{0}^{\leq 0}$ the only generators in normal
form are the classes $\kappa_{2}^{a}$, existing in every even degree $d=2a$,
so that the generating function is given by

$H_{\mathfrak{M}_{0}^{\leq 0}}=1+t^{2}+t^{4}+\ldots=\frac{1}{1-t^{2}},$

recovering the formula from Figure 4.

$\mathbf{e=1}$ On $U=\mathfrak{M}_{0}^{\leq 0}$ we get additional generators

$[\Gamma_{1},\psi_{h_{1}}^{a}\psi_{h_{2}}^{b}]\text{ for
}\Gamma_{1}=\leavevmode\hbox to45.35pt{\vbox
to18.43pt{\pgfpicture\makeatletter\hbox{\hskip 2.75693pt\lower-2.75693pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{2.35693pt}{0.0pt}\pgfsys@curveto{2.35693pt}{1.30171pt}{1.30171pt}{2.35693pt}{0.0pt}{2.35693pt}\pgfsys@curveto{-1.30171pt}{2.35693pt}{-2.35693pt}{1.30171pt}{-2.35693pt}{0.0pt}\pgfsys@curveto{-2.35693pt}{-1.30171pt}{-1.30171pt}{-2.35693pt}{0.0pt}{-2.35693pt}\pgfsys@curveto{1.30171pt}{-2.35693pt}{2.35693pt}{-1.30171pt}{2.35693pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.5}{0.0}{0.0}{0.5}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{42.19061pt}{0.0pt}\pgfsys@curveto{42.19061pt}{1.30171pt}{41.13539pt}{2.35693pt}{39.83368pt}{2.35693pt}\pgfsys@curveto{38.53197pt}{2.35693pt}{37.47675pt}{1.30171pt}{37.47675pt}{0.0pt}\pgfsys@curveto{37.47675pt}{-1.30171pt}{38.53197pt}{-2.35693pt}{39.83368pt}{-2.35693pt}\pgfsys@curveto{41.13539pt}{-2.35693pt}{42.19061pt}{-1.30171pt}{42.19061pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.5}{0.0}{0.0}{0.5}{39.83368pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{2.55692pt}{0.0pt}\pgfsys@lineto{37.27664pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.67763pt}{5.3966pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$h_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.59447pt}{5.3966pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$h_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$

Since the automorphism group of $\Gamma_{1}$ exchanges $h_{1},h_{2}$, the
numbers $a,b$ above are only unique up to ordering. We get a canonical
representative by requiring $a\leq b$. Overall, we obtain the generating
series

$\displaystyle H_{\mathfrak{M}_{0}^{\leq 1}}=H_{\mathfrak{M}_{0}^{\leq
0}}+\sum_{0\leq a\leq b}t^{a+b+1}$
$\displaystyle=\frac{1}{1-t^{2}}+t\sum_{a\geq 0}\sum_{c\geq 0}t^{a+(a+c)}$
$\displaystyle=\frac{1}{1-t^{2}}+t\left(\sum_{a\geq
0}t^{2a}\right)\left(\sum_{c\geq 0}t^{c}\right)$
$\displaystyle=\frac{1}{1-t^{2}}+t\frac{1}{1-t^{2}}\frac{1}{1-t}=\frac{1}{(1-t^{2})(1-t)},$

where we used the substitution $b=a+c$. Again we recover the formula from
Figure 4.

$\mathbf{e=2}$ The additional generators for $U=\mathfrak{M}_{0}^{\leq 2}$ are
given by

$[\Gamma_{2},\psi_{h_{1}}^{a}(\psi_{h_{2}}^{b}+(-\psi_{h_{3}})^{b})\psi_{h_{4}}^{c}]\text{
for }\Gamma_{2}=\leavevmode\hbox to85.18pt{\vbox
to18.43pt{\pgfpicture\makeatletter\hbox{\hskip 2.75693pt\lower-2.75693pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{2.35693pt}{0.0pt}\pgfsys@curveto{2.35693pt}{1.30171pt}{1.30171pt}{2.35693pt}{0.0pt}{2.35693pt}\pgfsys@curveto{-1.30171pt}{2.35693pt}{-2.35693pt}{1.30171pt}{-2.35693pt}{0.0pt}\pgfsys@curveto{-2.35693pt}{-1.30171pt}{-1.30171pt}{-2.35693pt}{0.0pt}{-2.35693pt}\pgfsys@curveto{1.30171pt}{-2.35693pt}{2.35693pt}{-1.30171pt}{2.35693pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.5}{0.0}{0.0}{0.5}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{42.19061pt}{0.0pt}\pgfsys@curveto{42.19061pt}{1.30171pt}{41.13539pt}{2.35693pt}{39.83368pt}{2.35693pt}\pgfsys@curveto{38.53197pt}{2.35693pt}{37.47675pt}{1.30171pt}{37.47675pt}{0.0pt}\pgfsys@curveto{37.47675pt}{-1.30171pt}{38.53197pt}{-2.35693pt}{39.83368pt}{-2.35693pt}\pgfsys@curveto{41.13539pt}{-2.35693pt}{42.19061pt}{-1.30171pt}{42.19061pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{39.83368pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.5}{0.0}{0.0}{0.5}{39.83368pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{82.0243pt}{0.0pt}\pgfsys@curveto{82.0243pt}{1.30171pt}{80.96909pt}{2.35693pt}{79.66737pt}{2.35693pt}\pgfsys@curveto{78.36566pt}{2.35693pt}{77.31044pt}{1.30171pt}{77.31044pt}{0.0pt}\pgfsys@curveto{77.31044pt}{-1.30171pt}{78.36566pt}{-2.35693pt}{79.66737pt}{-2.35693pt}\pgfsys@curveto{80.96909pt}{-2.35693pt}{82.0243pt}{-1.30171pt}{82.0243pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{79.66737pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.5}{0.0}{0.0}{0.5}{79.66737pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{2.55692pt}{0.0pt}\pgfsys@lineto{37.27664pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{42.39049pt}{0.0pt}\pgfsys@lineto{77.11021pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.67763pt}{5.3966pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$h_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.59447pt}{5.3966pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$h_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{45.5113pt}{5.3966pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$h_{3}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{65.42816pt}{5.3966pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$h_{4}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$

The automorphism group of $\Gamma_{2}$ exchanges $h_{1},h_{4}$ and
$h_{2},h_{3}$, so $a,c$ are only well-defined up to ordering. Moreover, for
$a=c$ and $b=2\ell+1$ odd, this symmetry implies

$[\Gamma_{2},\psi_{h_{1}}^{a}(\psi_{h_{2}}^{b}+(-\psi_{h_{3}})^{b})\psi_{h_{4}}^{a}]=-[\Gamma_{2},\psi_{h_{1}}^{a}(\psi_{h_{2}}^{b}+(-\psi_{h_{3}})^{b})\psi_{h_{4}}^{a}],$

so the corresponding generator vanishes. Overall, the numbers of basis
elements supported on $\Gamma_{2}$ have generating series

$t^{2}\cdot\big{(}\underbrace{\sum_{0\leq a\leq c}\sum_{b\geq
0}t^{a+b+c}}_{=\frac{1}{(1-t^{2})(1-t)^{2}}}-\underbrace{\sum_{a\geq
0}\sum_{\ell\geq
0}t^{2a+2\ell+1}}_{=\frac{t}{(1-t^{2})^{2}}}\big{)}=\frac{t^{2}(t^{2}+1)}{(1-t)(1-t^{2})^{2}}\,.$

Adding this to the generating series for $\mathfrak{M}_{0}^{\leq 1}$ we obtain
the formula

$H_{\mathfrak{M}_{0}^{\leq 2}}=H_{\mathfrak{M}_{0}^{\leq
1}}+\frac{t^{2}(t^{2}+1)}{(1-t)(1-t^{2})^{2}}=\frac{t^{4}+1}{(1-t^{2})^{2}(1-t)},$

again obtaining the same formula as in Figure 4.

$\mathbf{e=3}$ For the full locus $U=\mathfrak{M}_{0}^{\leq 3}$ a discrepancy
between our results and Fulghesu’s computations appears. There are two new
types of generators appearing: firstly we have

$[\Gamma_{3}^{\prime},\psi_{h_{1}}^{a}\psi_{h_{2}}^{b}\psi_{h_{3}}^{c}]\text{
for }\Gamma_{3}^{\prime}=\leavevmode\hbox to65.26pt{\vbox
to74.51pt{\pgfpicture\makeatletter\hbox{\hskip 22.67377pt\lower-37.254pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{}\pgfsys@moveto{2.35693pt}{0.0pt}\pgfsys@curveto{2.35693pt}{1.30171pt}{1.30171pt}{2.35693pt}{0.0pt}{2.35693pt}\pgfsys@curveto{-1.30171pt}{2.35693pt}{-2.35693pt}{1.30171pt}{-2.35693pt}{0.0pt}\pgfsys@curveto{-2.35693pt}{-1.30171pt}{-1.30171pt}{-2.35693pt}{0.0pt}{-2.35693pt}\pgfsys@curveto{1.30171pt}{-2.35693pt}{2.35693pt}{-1.30171pt}{2.35693pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.5}{0.0}{0.0}{0.5}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{2.55692pt}{0.0pt}\pgfsys@lineto{37.27664pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-1.27844pt}{2.21436pt}\pgfsys@lineto{-18.63832pt}{32.2826pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-1.27844pt}{-2.21436pt}\pgfsys@lineto{-18.63832pt}{-32.2826pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{25.59447pt}{5.3966pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$h_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.25589pt}{25.31346pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$h_{2}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-10.25589pt}{-30.45345pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$h_{3}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$

giving a contribution of

$t^{3}\sum_{0\leq a\leq b\leq
c}t^{a+b+c}=\frac{t^{3}}{(1-t^{3})(1-t^{2})(1-t)}$

to the generating series. The second type of generator is

$[\Gamma_{3}^{\prime\prime},\psi_{h_{1}}^{a}(\psi_{h_{2}}^{b}+(-\psi_{h_{3}})^{b})((-\psi_{h_{4}})^{c}+\psi_{h_{5}}^{c})\psi_{h_{6}}^{d}]\text{
for }\Gamma_{3}^{\prime\prime}=\leavevmode\hbox to125.01pt{\vbox
to18.43pt{\pgfpicture\makeatletter\hbox{\hskip 2.75693pt\lower-2.75693pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{2.55692pt}{0.0pt}\pgfsys@lineto{37.27664pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{82.22406pt}{0.0pt}\pgfsys@lineto{116.94377pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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