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Since $\Gamma_{3}^{\prime\prime}$ again has an automorphism of order $2$, we
count such generators using a trick: if the vertices of
$\Gamma_{3}^{\prime\prime}$ were ordered, the generating series would be

$t^{3}\sum_{a,b,c,d\geq 0}t^{a+b+c+d}=\frac{t^{3}}{(1-t)^{4}}.$

Due to the automorphism, we counted almost all the generators twice, _except_
those fixed by the automorphism, for which $(a,b,c,d)=(a,b,b,a)$ and whose
generating series is

$t^{3}\sum_{a,b\geq 0}t^{2a+2b}=\frac{t^{3}}{(1-t^{2})^{2}}.$

Adding these two series, we count every generator twice, so we obtain the
correct count after dividing by two. Overall we get

$\displaystyle H_{\mathfrak{M}_{0}^{\leq 3}}$
$\displaystyle=H_{\mathfrak{M}_{0}^{\leq
2}}+\frac{t^{3}}{(1-t^{3})(1-t^{2})(1-t)}+\frac{1}{2}\left(\frac{t^{3}}{(1-t)^{4}}+\frac{t^{3}}{(1-t^{2})^{2}}\right)$
$\displaystyle=\frac{t^{6}+t^{5}+2t^{4}+t^{3}+1}{(1-t^{2})^{2}(1-t)(1-t^{3})}.$

However, expanding this series we obtain

$\frac{t^{6}+t^{5}+2t^{4}+t^{3}+1}{(1-t^{2})^{2}(1-t)(1-t^{3})}=1+t+3t^{2}+5t^{3}+10t^{4}+15t^{5}+26t^{6}+36t^{7}+\mathbf{54}t^{8}+\ldots$
(43)

Comparing with the expansion of the corresponding function $H^{\prime}$ in
Figure 4 we see that the coefficient of $t^{8}$ is $55$ for Fulghesu and $54$
for us. We used a modified version of the software package admcycles [DSv20]
for the open-source software SageMath [S+20] to verify the number $54$ above.

After revisiting Fulghesu’s proof, we think we can explain this discrepancy
from a relation that was missed in [Ful10a]. In the notation of this paper, we
claim that there is a relation

$r\cdot q-\gamma_{3}^{\prime\prime}\cdot
s+2u\cdot\gamma_{2}-\gamma_{3}^{\prime\prime}\cdot
q\cdot\kappa_{2}-s\cdot\gamma_{2}\cdot\kappa_{1}=0\in\mathrm{CH}^{8}(\mathfrak{M}_{0}^{\leq
3}).$ (44)

Here the classes $r,s,u,\gamma_{3}^{\prime\prime}$ are supported on the closed
stratum $\mathfrak{M}^{\Gamma_{3}^{\prime\prime}}\subset\mathfrak{M}_{0}^{\leq
3}$. The relation (44) follows from the description of the Chow ring
$\mathrm{CH}^{*}(\mathfrak{M}^{\Gamma_{3}^{\prime\prime}})$ and the formulas
for restrictions of classes
$q,\gamma_{3}^{\prime\prime},\gamma_{2},\kappa_{2},\kappa_{1}$ to
$\mathfrak{M}^{\Gamma_{3}^{\prime\prime}}$ computed in [Ful10a, Section 6.2].
On the other hand, using Macaulay2 we verified that the relation (44) is _not_
contained in the ideal of relations given in [Ful10a, Theorem 6.3]. Adding
this missing relation, we obtain the correct rank $54$ for
$\mathrm{CH}^{8}(\mathfrak{M}_{0}^{\leq 3})$.

Our numerical experiments indicated that there are further relations missing
in degrees $d>9$. So while the general proof strategy of [Ful10a] seems sound,
more care needed is needed in the final step of the computation.

###  2.6 Chow rings of open substacks of $\mathfrak{M}_{0,n}$ \- finite
generation and Hilbert series

In the previous section, we saw some explicit computations for Chow groups
$\mathrm{CH}^{*}(U)$ of of open substacks $U\subset\mathfrak{M}_{0,n}$ and
their Hilbert series

$H_{U}=\sum_{d\geq 0}\dim_{\mathbb{Q}}\mathrm{CH}^{d}(U)t^{d}.$

For $U=\mathfrak{M}_{0}^{\leq e}$ and $e=0,1,2$ we have that
$\mathrm{CH}^{*}(U)$ is a finitely generated graded algebra by the results of
[Ful10a]. But recall that any such algebra, having generators in degrees
$d_{1},\ldots,d_{r}$ has a Hilbert series which is the expansion (at $t=0$) of
a rational function $H(t)$ of the form

$H(t)=\frac{Q(t)}{\prod_{i=1}^{r}(1-t^{d_{i}})}\text{ for some
}Q(t)\in\mathbb{Z}[t]$

(see [Mat86, Theorem 13.2]). This explains the shape of the Hilbert functions
of $\mathfrak{M}_{0}^{\leq e}$ from Figure 4. We remark here, that all
functions $H(t)$ of the above form have poles only at roots of unity.

On the other hand, for the open substack
$\mathcal{T}\subset\mathfrak{M}_{0,3}$ studied by Oesinghaus, we saw
$\mathrm{CH}^{*}(\mathcal{T})\cong\text{QSym}$, where the algebra QSym had an
additive basis element $M_{J}$ in degree $d$ for each composition $J$ of $d$.
Since for $d\geq 1$ the number of compositions of $d$ is $2^{d-1}$, the
Hilbert series of the Chow ring of $\mathcal{T}$ is given by

$H_{\mathcal{T}}=1+\sum_{d\geq
0}2^{d-1}t^{d}=1+\frac{t}{1-2t}=\frac{1-t}{1-2t}.$

From this we can see two things:

  * •

The Chow ring $\mathrm{CH}^{*}(\mathcal{T})$ cannot be a finitely generated
algebra, since the function $H_{\mathcal{T}}$ has a pole at $1/2$, which is
not a root of unity.

  * •

On the other hand, we still have that $H_{\mathcal{T}}$ is the expansion of a
rational function, even though $\mathcal{T}$ is not even of finite type.

The above observations lead to the following two questions.

######  Question 2.44.

Is it true that for $U\subset\mathfrak{M}_{0,n}$ an open substack of finite
type, the Chow ring $\mathrm{CH}^{*}(U)$ is a finitely generated algebra?

######  Question 2.45.

Is it true that for $U\subset\mathfrak{M}_{0,n}$ any open substack which is a
union of strata $\mathfrak{M}^{\Gamma}$, the Hilbert series $H_{U}$ is the
expansion of a rational function at $t=0$?

For the first question, we note that by Theorem 1.2 we know that
$\mathrm{CH}^{*}(U)$ is additively generated by possibly infinitely many
decorated strata $[\Gamma,\alpha]$, supported on _finitely many_ prestable
graphs $\Gamma$. It is far from obvious whether we can obtain all of them
multiplicatively from a finite collection of $[\Gamma_{i},\alpha_{i}]$.

For the second question, we observe that it would be implied for all finite
type open substacks $U$ of $\mathfrak{M}_{0,n}$ assuming a positive answer to
the first question. Further evidence is provided by the results from [Oes19]:
as we saw above, the open substack $\mathcal{T}\subset\mathfrak{M}_{0,3}$ has
a rational generating series $H_{\mathcal{T}}$. In fact, as mentioned above
Oesinghaus computes the Chow ring for the the entire semistable locus in
$\mathfrak{M}_{0,2}$ and $\mathfrak{M}_{0,3}$ (see [Oes19, Corollary 2,3]) and
obtains

$\mathrm{CH}^{*}(\mathfrak{M}_{0,2}^{\textup{ss}})=\mathrm{QSym}\otimes_{\mathbb{Q}}\mathbb{Q}[\beta],\
\mathrm{CH}^{*}(\mathfrak{M}_{0,3}^{\textup{ss}})=\mathrm{QSym}\otimes_{\mathbb{Q}}\mathrm{QSym}\otimes_{\mathbb{Q}}\mathrm{QSym}.$

Since we know that the Hilbert series of $\mathrm{QSym}$ is $(1-t)/(1-2t)$ and
the Hilbert series of $\mathbb{Q}[t]$ is $1/(1-t)$ and that Hilbert series are
multiplicative under tensor products, we easily see that

$H_{\mathfrak{M}_{0,2}^{\textup{ss}}}=\frac{1}{1-2t},\
H_{\mathfrak{M}_{0,3}^{\textup{ss}}}=\frac{(1-t)^{3}}{(1-2t)^{3}}.$

So Question 2.45 has a positive answer for the non-finite type substacks of
semistable points in $\mathfrak{M}_{0,2}$ and $\mathfrak{M}_{0,3}$.

To finish this section, we want to record some numerical data about the Chow
groups of the full stacks $\mathfrak{M}_{0,n}$. Using Theorems 1.2 and 2.31
these groups have a completely combinatorial description. This has been
implemented in a modified version of the software package admcycles [DSv20],
which can enumerate prestable graphs, decorated strata in normal form and the
relations $\mathcal{R}_{\kappa,\psi},\mathcal{R}_{\mathrm{WDVV}}$ between
them. Thus, from linear algebra we can compute the ranks of Chow groups of
$\mathfrak{M}_{0,n}$ in many cases. We record the results in Figure 5.

d | n=0 | n=1 | n=2 | n=3 | n=4 | n=5 | n=6 | n=7 | n=8
---|---|---|---|---|---|---|---|---|---
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
1 | 1 | 2 | 3 | 4 | 6 | 11 | 23 | 50 | 108
2 | 3 | 5 | 9 | 16 | 33 | 80 | 215 | 621 | 1900
3 | 5 | 12 | 27 | 62 | 162 | 481 | 1572 |  |
4 | 13 | 32 | 84 | 235 | 739 | 2594 |  |  |
5 | 27 | 84 | 263 | 875 | 3219 |  |  |  |
6 | 70 | 234 | 837 | 3219 |  |  |  |  |
7 | 166 | 656 | 2683 |  |  |  |  |  |
8 | 438 | 1892 |  |  |  |  |  |  |
9 | 1135 |  |  |  |  |  |  |  |
10 | 3081 |  |  |  |  |  |  |  |
Figure 5: The rank of the Chow groups $\mathrm{CH}^{d}(\mathfrak{M}_{0,n})$

##  3 Comparison with the tautological ring of the moduli of stable curves

###  3.1 Injectivity of pullback by forgetful charts

Assume we are in the stable range $2g-2+n>0$ so that the moduli space
$\overline{\mathcal{M}}_{g,n}$ is nonempty. Since
$\overline{\mathcal{M}}_{g,n}\subset\mathfrak{M}_{g,n}$ is an open substack,
the Chow groups of $\mathfrak{M}_{g,n}$ determine those of
$\overline{\mathcal{M}}_{g,n}$: we have that
$\mathrm{CH}^{*}(\overline{\mathcal{M}}_{g,n})$ is the quotient of
$\mathrm{CH}^{*}(\mathfrak{M}_{g,n})$ by the span of classes supported on the
strictly unstable locus.

Restricting to the subrings of tautological classes, we note that the
tautological ring $\mathrm{R}^{*}(\overline{\mathcal{M}}_{g,n})$ of
$\overline{\mathcal{M}}_{g,n}$ is the subring of
$\mathrm{CH}^{*}(\overline{\mathcal{M}}_{g,n})$ given by the restriction of
$\mathrm{R}^{*}(\mathfrak{M}_{g,n})\subseteq\mathrm{CH}^{*}(\mathfrak{M}_{g,n})$
under the open embedding
$i\colon\overline{\mathcal{M}}_{g,n}\hookrightarrow\mathfrak{M}_{g,n}$. Thus
the tautological ring $\mathrm{R}^{*}(\mathfrak{M}_{g,n})$ determines
$\mathrm{R}^{*}(\overline{\mathcal{M}}_{g,n})$ since the composition

$\mathrm{R}^{*}(\overline{\mathcal{M}}_{g,n})\xrightarrow{\mathrm{st}^{*}}\mathrm{R}^{*}(\mathfrak{M}_{g,n})\xrightarrow{i^{*}}\mathrm{R}^{*}(\overline{\mathcal{M}}_{g,n})$

is the identity and thus
$\mathrm{R}^{*}(\overline{\mathcal{M}}_{g,n})\xrightarrow{\mathrm{st}^{*}}\mathrm{R}^{*}(\mathfrak{M}_{g,n})$
is injective.

It is an interesting question whether the converse is true: do the Chow (or
tautological) rings of the moduli spaces of stable curves determine the Chow
(or tautological) ring of $\mathfrak{M}_{g,n}$? The following conjecture gives
a precise way in which this could be true.

######  Conjecture 3.1.

Let $(g,n)\neq(1,0)$, then for a fixed $d\geq 0$ there exists $m_{0}\geq 0$
such that for any $m\geq m_{0}$, the forgetful morphism

$F_{m}:\overline{\mathcal{M}}_{g,n+m}\to\mathfrak{M}_{g,n}$

satisfies that the pullback

$F_{m}^{*}:\mathrm{CH}^{d}(\mathfrak{M}_{g,n})\to\mathrm{CH}^{d}(\overline{\mathcal{M}}_{g,n+m})$

is injective.

We have seen in [BS21, Lemma 2.1] that (for $m$ sufficiently large) the image
of $F_{m}$ is open with complement of codimension
$\lfloor\frac{m}{2}\rfloor+1$. So for $m\geq 2d$, we have
$\mathrm{CH}^{d}(F_{m}(\overline{\mathcal{M}}_{g,n+m}))\cong\mathrm{CH}^{d}(\mathfrak{M}_{g,n})$,
so certainly the image of $F_{m}$ is sufficiently large to capture the Chow
group of codimension $d$ cycles. Still, it is not true that a surjective,
smooth morphism has injective pullback in Chow (if the fibres are not proper,
as is the case for $F_{m}$), so this does not suffice to prove the conjecture.

One aspect of the conjecture we can prove so far is the statement that if
$F_{m_{0}}^{*}$ is injective, it is true that for any $m\geq m_{0}$ the map
$F_{m}^{*}$ remains injective.

######  Proposition 3.2.

For $(g,n)\neq(1,0)$ and $0\leq m\leq m^{\prime}$ with $2g-2+n+m>0$ we have
$\ker F_{m^{\prime}}^{*}\subseteq\ker F_{m}^{*}$. In other words, the
subspaces $(\ker F_{\ell}^{*})_{\ell}$ form a _non-increasing_ sequence of
subspaces of $\mathrm{CH}^{d}(\mathfrak{M}_{g,n})$.

###### Proof.

It suffices to show the statement for $m^{\prime}=m+1$. Consider the following
_non-commutative_ diagram.

${\overline{\mathcal{M}}_{g,n+m+1}}$${\overline{\mathcal{M}}_{g,n+m}}$${\mathfrak{M}_{g,n}}$$\scriptstyle{\pi}$$\scriptstyle{F_{m+1}}$$\scriptstyle{F_{m}}$
(45)

Here $\pi$ is the usual map forgetting the marking $p_{n+m+1}$ and stabilizing
the curve. For this reason, the diagram is only commutative on the complement
of the locus

$Z=\left\\{(C,p_{1},\ldots,p_{n+m+1}):\begin{array}[]{c}p_{n+m+1}\text{
contained in}\\\ \text{rational component of $C$}\\\ \text{with $3$ special
points}\end{array}\right\\}\subseteq\overline{\mathcal{M}}_{g,n+m+1}\,.$

Let $i:Z\to\overline{\mathcal{M}}_{g,n+m+1}$ be the inclusion of $Z$ and let
$\alpha\in\mathrm{CH}^{*}(\mathfrak{M}_{g,n})$ be any class. By the
commutativity of the diagram (45) away from $Z$, we know that the class
$F_{m+1}^{*}\alpha-\pi^{*}F_{m}^{*}\alpha$ restricts to zero on the complement
of $Z$ and thus, by the usual excision sequence, there exists a class
$\beta\in\mathrm{CH}^{*}(Z)$ such that

$F_{m+1}^{*}\alpha-\pi^{*}F_{m}^{*}\alpha=i_{*}\beta\,.$

We want to transport this to an equality of classes on
$\overline{\mathcal{M}}_{g,n+m}$ by intersecting with $\psi_{n+m+1}$ and
pushing forward via $\pi$. But notice that $\psi_{n+m+1}|_{Z}=0$ since on $Z$
the component of $C$ containing $p_{n+m+1}$ is parametrized by
$\overline{\mathcal{M}}_{0,3}$ and thus the psi-class of $p_{n+m+1}$ vanishes
here. Thus

$\psi_{n+m+1}\cdot
F_{m+1}^{*}\alpha-\psi_{n+m+1}\cdot\pi^{*}F_{m}^{*}\alpha=(i_{*}\beta)\psi_{n+m+1}=i_{*}(i^{*}\psi_{n+m+1}\cdot\beta)=0\,.$
(46)

Pushing forward by $\pi$ and using
$\pi_{*}\psi_{n+m+1}=(2g-2+n+m)\cdot[\overline{\mathcal{M}}_{g,n+m}]$ we
obtain

$\pi_{*}\left(\psi_{n+m+1}\cdot F_{m+1}^{*}\alpha\right)=(2g-2+n+m)\cdot
F_{m}^{*}\alpha\,.$

Thus, since $2g-2+n+m\geq 2g-2+n>0$, any class $\alpha$ with
$F_{m+1}^{*}\alpha=0$ also satisfies $F_{m}^{*}\alpha=0$, finishing the proof.
∎

Again, for $g=0$ we can give some numerical evidence for the above conjecture.
In Figure 6 we compare ranks of the Chow groups
$\mathrm{CH}^{d}(\mathfrak{M}_{0,n})$ to (lower bounds on) the ranks of
$F_{m}^{*}(\mathrm{CH}^{d}(\mathfrak{M}_{0,n}))$. We see that the bounds for
$F_{m}^{*}(\mathrm{CH}^{d}(\mathfrak{M}_{0,n}))$ increase monotonically in $m$
(as predicted by Proposition 3.2) and, in all cases which we could handle
computationally, stabilize at the rank of
$\mathrm{CH}^{d}(\mathfrak{M}_{0,n})$, implying that the corresponding
pullbacks $F_{m}^{*}$ are indeed injective.

$(n,d)$ | $\mathrm{CH}^{d}(\mathfrak{M}_{0,n})$ | $F_{m}^{*}(\mathrm{CH}^{d}(\mathfrak{M}_{0,n}))$
---|---|---
|  | m=0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
(0,0) | 1 |  |  |  | 1 | 1 | 1 | 1 | 1 | 1 | 1
(0,1) | 1 |  |  |  |  | 1 | 1 | 1 | 1 | 1 | 1
(0,2) | 3 |  |  |  |  |  | 1 | 2 | 3 | 3 | 3
(0,3) | 5 |  |  |  |  |  |  | 1 | 2 | 4 | 5
(0,4) | 13 |  |  |  |  |  |  |  | 1 | 2 | 7
(1,0) | 1 |  |  | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1
(1,1) | 2 |  |  |  | 1 | 2 | 2 | 2 | 2 | 2 | 2
(1,2) | 5 |  |  |  |  | 1 | 3 | 5 | 5 | 5 | 5
(1,3) | 12 |  |  |  |  |  | 1 | 4 | 7 | 12 | 12
(2,0) | 1 |  | 1 | 1 | 1 | 1 | 1 | 1 | 1 |  |
(2,1) | 3 |  |  | 1 | 3 | 3 | 3 | 3 | 3 |  |
(2,2) | 9 |  |  |  | 1 | 5 | 9 | 9 | 9 |  |
(2,3) | 27 |  |  |  |  | 1 | 7 | 11 | 27 |  |
(3,0) | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |  |  |
(3,1) | 4 |  | 1 | 4 | 4 | 4 | 4 | 4 |  |  |
(3,2) | 16 |  |  | 1 | 5 | 15 | 16 | 16 |  |  |
(3,3) | 62 |  |  |  | 1 | 5 | 16 | 62 |  |  |
Figure 6: The ranks of the Chow groups $\mathrm{CH}^{d}(\mathfrak{M}_{0,n})$
compared to (lower bounds on) the ranks of
$F_{m}^{*}(\mathrm{CH}^{d}(\mathfrak{M}_{0,n}))$; in many cases it was not
feasible to obtain the precise rank of
$F_{m}^{*}(\mathrm{CH}^{d}(\mathfrak{M}_{0,n}))$, but a lower bound could be
achieved by computing the rank of the intersection pairing of
$F_{m}^{*}(\mathrm{CH}^{d}(\mathfrak{M}_{0,n}))$ with a selection of
tautological classes on $\overline{\mathcal{M}}_{0,n+m}$

###  3.2 The divisor group of $\mathfrak{M}_{g,n}$

As for the moduli space of stable curves, the group of divisor classes on
$\mathfrak{M}_{g,n}$ can be fully understood in terms of tautological classes
and relations. For $g=0$ we already saw that all divisor classes are
tautological and we explicitly described the relations, so below we can
restrict to $g\geq 1$. As before, we also want to exclude the case $g=1,n=0$
since $\mathfrak{M}_{1,0}$ does not have a stratification by quotient stacks.

Thus we can restrict to the range $2g-2+n>0$, where the space
$\overline{\mathcal{M}}_{g,n}$ is nonempty. Then we have an exact sequence

${\mathrm{CH}_{*}(\overline{\mathcal{M}}_{g,n},1)}$${\mathrm{CH}_{*}(\mathfrak{M}_{g,n}^{\textup{us}})}$${\mathrm{CH}_{*}(\mathfrak{M}_{g,n})}$${\mathrm{CH}_{*}(\overline{\mathcal{M}}_{g,n})}$${0}$$\scriptstyle{\mathrm{st}^{*}}$
(47)

where $\mathfrak{M}_{g,n}^{\textup{us}}$ is the _unstable locus_ of
$\mathfrak{M}_{g,n}$, i.e. the complement of the open substack
$\overline{\mathcal{M}}_{g,n}\subset\mathfrak{M}_{g,n}$. Using this sequence,
we can completely understand $\mathrm{CH}^{1}(\mathfrak{M}_{g,n})$ from the
explicit description of $\mathrm{CH}^{1}(\overline{\mathcal{M}}_{g,n})$ in
[AC98, Theorem 2.2].

######  Proposition 3.3.

For $(g,n)\neq(1,0)$ we have
$\mathrm{R}^{1}(\mathfrak{M}_{g,n})=\mathrm{CH}^{1}(\mathfrak{M}_{g,n})$.
Furthermore, for $2g-2+n>0$, all tautological relations in
$\mathrm{R}^{1}(\mathfrak{M}_{g,n})$ are pulled backed from relations in
$\mathrm{R}^{1}(\overline{\mathcal{M}}_{g,n})$ via the stabilization morphism.

###### Proof.

As discussed before, for the statement that all divisor classes are
tautological we can restrict to the stable range $2g-2+n>0$, since the case of
$g=0$ was treated before. For the moduli spaces of stable curves it holds that
$\mathrm{R}^{1}(\overline{\mathcal{M}}_{g,n})$ and
$\mathrm{CH}^{1}(\overline{\mathcal{M}}_{g,n})$ coincide by [AC98]. Since the
locus $\mathfrak{M}^{\textup{us}}_{g,n}$ is a union of boundary divisors,
whose fundamental classes are pushforwards of appropriate gluing maps, the
image of the pushforward map
$\mathrm{CH}^{0}(\mathfrak{M}^{\textup{us}}_{g,n})\to\mathrm{CH}^{1}(\mathfrak{M}_{g,n})$
is contained in $\mathrm{R}^{1}(\mathfrak{M}_{g,n})$. Therefore the excision
sequence (47) gives the conclusion.

To compute the set of relations, we observe that

$\mathrm{CH}^{1}(\overline{\mathcal{M}}_{g,n},1)\cong
H^{0}(\overline{\mathcal{M}}_{g,n},\mathcal{O}^{\times}_{\overline{\mathcal{M}}_{g,n}})=k^{\times}$

because $\overline{\mathcal{M}}_{g,n}$ is smooth and projective over $k$.
Therefore, in the degree $1$ part of the sequence (47), the image of the
connecting homomorphism is trivial. Thus, since the pullback $\mathrm{st}^{*}$
by the stabilization morphism defines a splitting of (47) on the right, we
have that

$\mathrm{CH}^{1}(\mathfrak{M}_{g,n})=\bigoplus_{\begin{subarray}{c}\Gamma\text{
unstable}\\\
|E(\Gamma)|=1\end{subarray}}\mathbb{Q}\cdot[\Gamma]\oplus\mathrm{CH}^{1}(\overline{\mathcal{M}}_{g,n}).$

Thus all relations between decorated strata classes in codimension $1$ are
pulled back from $\overline{\mathcal{M}}_{g,n}$. ∎

###  3.3 Zero cycles on $\mathfrak{M}_{g,n}$

After treating the case of codimension $1$ cycles in the previous section, we
want to make some remarks about cycles of _dimension_ $0$. For the moduli
spaces of stable curves, these exhibit many interesting properties:

  * •

In [GV01], Graber and Vakil showed that the group
$\mathrm{R}_{0}(\overline{\mathcal{M}}_{g,n})$ of tautological zero cycles on
$\overline{\mathcal{M}}_{g,n}$ is always isomorphic to $\mathbb{Q}$, even
though the full Chow group $\mathrm{CH}_{0}(\overline{\mathcal{M}}_{g,n})$ can
be infinite-dimensional (e.g. for $(g,n)=(1,11)$, see [GV01, Remark 1.1]).

  * •

In [PS20], Pandharipande and the second author presented geometric conditions
on stable curves $(C,p_{1},\ldots,p_{n})$ ensuring that the zero cycle
$[(C,p_{1},\ldots,p_{n})]$ in $\overline{\mathcal{M}}_{g,n}$ is tautological.

We want to note here, that for the moduli stacks of prestable curves, the
behaviour of tautological zero cycles becomes more complicated:

  * •

For $\mathfrak{M}_{0,n}$ with $n=0,1,2$, we have
$\mathrm{R}_{0}(\mathfrak{M}_{0,n})=\mathrm{CH}_{0}(\mathfrak{M}_{0,n})=0$ for
dimension reasons.

  * •

As visible from Figure 5, the group $\mathrm{R}_{0}(\mathfrak{M}_{0,n})$ is no
longer one-dimensional for $n\geq 4$. Indeed, looking at the example of $n=4$
we note that the boundary divisor of curves with one component having no
marked points is a nonvanishing zero cycle (since it pulls back to an
effective boundary divisor under the forgetful map
$F_{2}:\overline{\mathcal{M}}_{0,6}\to\mathfrak{M}_{0,4}$), but it restricts
to zero on $\overline{\mathcal{M}}_{0,4}\subset\mathfrak{M}_{0,4}$ and is thus
linearly independent of the generator of
$\mathrm{R}_{0}(\overline{\mathcal{M}}_{0,4})$.

This indicates that for the moduli stacks of prestable curves, the group of
zero cycles plays less of a special role than for the moduli spaces of stable
curves.

##  Appendix A Gysin pullback for higher Chow groups

In [DJK18], Déglise, Jin and Khan generalized Gysin pullback along a regular
imbedding to motivic homotopy theories. We summerize the construction in the
language of higher Chow groups. For a moment, let $X$ be a quasi-projective
scheme over $k$ and we consider higher Chow groups with
$\mathbb{Z}$-coefficients. For simplicity, we write
$\mathbb{G}_{m}X=X\times\mathbb{G}_{m}$. Let $[t]$ be a generator of

$\mathrm{CH}_{0}(\mathbb{G}_{m},1)\cong\left(k[t,t^{-1}]\right)^{\times}$

and let

$\gamma_{t}\colon\mathrm{CH}_{*}(X,m)\to\mathrm{CH}_{*}(\mathbb{G}_{m}X,m+1)\,,\;\alpha\mapsto\alpha\times[t]$

be the morphism defined by the exterior product. Let $i\colon Z\to X$ be a
regular imbedding of codimension $r$ and let $q:N_{Z}X\to Z$ be the normal
bundle. Let $D_{Z}X$ be the Fulton–MacPherson’s deformation space defined by

$D_{Z}X=\textup{Bl}_{Z\times 0}(X\times\mathbb{A}^{1})-\textup{Bl}_{Z\times
0}(X\times 0)$

which fits into the cartesian diagram

${N_{Z}X}$${D_{Z}X}$${\mathbb{G}_{m}X}$${\\{0\\}}$${\mathbb{A}^{1}}$${\mathbb{G}_{m}\,.}$

By [Blo86, Blo94] we have the localization sequence

$\ldots\to\mathrm{CH}_{d}(\mathbb{G}_{m}X,m+1)\xrightarrow{\partial}\mathrm{CH}_{d}(N_{Z}X,m)\to\mathrm{CH}_{d}(D_{Z}X,m)\to\ldots\,.$
(48)

######  Definition A.1.

For a regular imbedding $i\colon Z\to X$, we define

$i^{*}\colon\mathrm{CH}_{d}(X,m)\to\mathrm{CH}_{d-r}(Z,m)$

as a composition of following morphisms

$\mathrm{CH}_{d}(X,m)\xrightarrow{\gamma_{t}}\mathrm{CH}_{d}(\mathbb{G}_{m}X,m+1)\xrightarrow{\partial}\mathrm{CH}_{d}(N_{Z}X,m)\xrightarrow[\cong]{(q^{*})^{-1}}\mathrm{CH}_{d-r}(Z,m)$
(49)

where $\partial$ is the boundary map in (48) and the flat pullback $q^{*}$ is
an isomorphism by [Blo86].

This definition extends the Gysin pullback for Chow groups in the sense that
it coincides with the Gysin pullback defined in [Ful98] when $m=0$. This
construction extends to all lci morphism and satisfies functoriality,
transverse base change and excess intersection formula, see [DJK18].

Given a line bundle $q:L\to X$ with the zero section $0:X\to L$, the action of
the first Chern class on higher Chow groups can be defined by

$c_{1}(L)\cap\colon\mathrm{CH}_{d}(X,m)\xrightarrow{0_{*}}\mathrm{CH}_{d}(L,m)\xrightarrow[\cong]{(q^{*})^{-1}}\mathrm{CH}_{d-1}(X,m)\,.$

We want to note two basic compatibilities of this operation: firstly, given a
proper morphism $f:X^{\prime}\to X$ and the line bundle $L\to X$, a short
computation shows the projection formula

$f_{*}\left(c_{1}(f^{*}L)\cap\alpha\right)=c_{1}(L)\cap f_{*}\alpha\text{ for
}\alpha\in\mathrm{CH}_{*}(X^{\prime},m)\,.$ (50)

Secondly, intersecting higher Chow cycles with a Cartier divisor has the same
formula as for ordinary Chow groups.

######  Lemma A.2.

Let $i\colon D\to X$ be an effective divisor and let $q\colon\mathcal{O}(D)\to
X$ be the associated line bundle. Then

$i_{*}i^{*}\alpha=c_{1}(\mathcal{O}(D))\cap\alpha\,,\,\alpha\in\mathrm{CH}_{d}(X,m)\,.$
(51)

###### Proof.

Consider the following cartesian diagram

${D}$${X}$${X}$${\mathcal{O}(D)}$$\scriptstyle{i}$$\scriptstyle{i}$$\scriptstyle{s}$$\scriptstyle{0}$

where $s\colon X\to\mathcal{O}(D)$ be the regular section defining $D$ and $0$
is the zero section. Recall that the action of the first Chern class can be
defined by

$c_{1}(\mathcal{O}(D))\cap-\colon\mathrm{CH}_{d}(X,m)\xrightarrow{0_{*}}\mathrm{CH}_{d}(\mathcal{O}(D),m)\xrightarrow[\cong]{(q^{*})^{-1}}\mathrm{CH}_{d-1}(X,m)\,.$

By the transverse base change formula [DJK18, Proposition 2.4.2],

$i_{*}i^{*}\alpha=s^{*}0_{*}\alpha\,.$

Now we can conclude the result because $s^{*}$ is an inverse of $q^{*}$. ∎

The above construction can be extended to global quotient stacks without any
difficulty. Let $X$ be an equidimensional quasi-projective scheme with a
linearized $G$-action. Applying the Borel construction developed in [EG98]
yields the definition of higher Chow groups for $[X/G]$, see [Kri13]. For an
arbitrary algebraic stack, the authors do not know whether a direct
generalization of [Ful98] is possible. Relying on the recent development of
motivic homotopy theories, Khan used the six-operator formalism ([Kha16]) to
construct motivic Borel–Moore homology for derived algebraic stacks in
[Kha19].

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