# Extensions of tautological rings and motivic structures in the cohomology of
$\overline{\mathcal{M}}_{g,n}$

Samir Canning  ,  Hannah Larson  and  Sam Payne

###### Abstract.

We study collections of subrings of $H^{*}(\overline{\mathcal{M}}_{g,n})$ that
are closed under the tautological operations that map cohomology classes on
moduli spaces of smaller dimension to those on moduli spaces of larger
dimension and contain the tautological subrings. Such extensions of
tautological rings are well-suited for inductive arguments and flexible enough
for a wide range of applications. In particular, we confirm predictions of
Chenevier and Lannes for the $\ell$-adic Galois representations and Hodge
structures that appear in $H^{k}(\overline{\mathcal{M}}_{g,n})$ for $k=13$,
$14$, and $15$. We also show that $H^{4}(\overline{\mathcal{M}}_{g,n})$ is
generated by tautological classes for all $g$ and $n$, confirming a prediction
of Arbarello and Cornalba from the 1990s. In order to establish the final
bases cases needed for the inductive proofs of our main results, we use
Mukai’s construction of canonically embedded pentagonal curves of genus 7 as
linear sections of an orthogonal Grassmannian and a decomposition of the
diagonal to show that the pure weight cohomolog of $\mathcal{M}_{7,n}$ is
generated by algebraic cycle classes, for $n\leq 3$.

S.C. was supported by a Hermann-Weyl-Instructorship from the
Forschungsinstitut für Mathematik at ETH Zürich. This research was partially
conducted during the period H.L. served as a Clay Research Fellow and as a
guest of the Forschungsinstitut für Mathematik at ETH Zürich. S.P. was
supported in part by NSF grants DMS–2001502 and DMS–2053261.

##  1\. Introduction

The moduli spaces of stable curves $\overline{\mathcal{M}}_{g,n}$ are smooth
and proper over the integers, and this implies strong restrictions on the
motivic structures, such as $\ell$-adic Galois representations, that can
appear in $H^{*}(\overline{\mathcal{M}}_{g,n})$. Widely believed conjectures
regarding analytic continuations and functional equations for $L$-functions
lead to precise predictions, by Chenevier and Lannes, about which such
structures can appear in degrees less than or equal to 22 [10, Theorem F].
These predictions are consistent with all previously known results on
$H^{*}(\overline{\mathcal{M}}_{g,n})$. Recent work inspired by these
predictions confirms their correctness in all degrees less than or equal to 12
[4, 8]. Here we introduce new methods to sytematically study the motivic
structures in $H^{k}(\overline{\mathcal{M}}_{g,n})$ for $k>12$ and confirm
these predictions in degrees $13$, $14$, and $15$.

Throughout, we write $H^{*}(X)$ for the rational singular cohomology of a
scheme or Deligne-Mumford stack $X$ endowed with its associated Hodge
structure or $\ell$-adic Galois representation and $H^{*}(X)^{\mathrm{ss}}$
for its semisimplification. Let $\mathsf{L}:=H^{2}(\mathbb{P}^{1})$ and
$\mathsf{S}_{12}:=H^{11}(\overline{\mathcal{M}}_{1,11})$.

######  Theorem 1.1.

For all $g$ and $n$, we have
$H^{13}(\overline{\mathcal{M}}_{g,n})^{\mathrm{ss}}\cong\bigoplus\mathsf{L}\mathsf{S}_{12}$
and
$H^{14}(\overline{\mathcal{M}}_{g,n})^{\mathrm{ss}}\cong\bigoplus\mathsf{L}^{7}$.
Moreover, for $g\geq 2$, we have
$H^{15}(\overline{\mathcal{M}}_{g,n})^{\mathrm{ss}}\cong\bigoplus\mathsf{L}^{2}\mathsf{S}_{12}$.

Theorem 1.1 confirms the predictions of Chenevier and Lannes for motivic
weights $k\leq 15$. Note that the Hodge structure on the cohomology of a
smooth and proper Deligne–Mumford stack such as $\overline{\mathcal{M}}_{g,n}$
is semi-simple, so the semi-simplification in Theorem 1.1 is relevant only
when considering $\ell$-adic Galois representations.

The proof of Theorem 1.1 uses the inductive structure of the boundary of the
moduli space and the maps induced by tautological morphisms between moduli
spaces, as do the proofs of the precursor results mentioned above. Recall that
the collection of tautological rings
$RH^{*}(\overline{\mathcal{M}}_{g,n})\subset
H^{*}(\overline{\mathcal{M}}_{g,n})$ is the smallest collection of subrings
that is closed under pushforward and pullback for the tautological morphisms
induced by gluing, forgetting, and permuting marked points. For many inductive
arguments, including those used here, it suffices to consider the operations
that produce cohomology classes on moduli spaces of larger dimension from
those on moduli spaces of smaller dimension.

######  Definition 1.2.

A _semi-tautological extension_ (STE) is a collection of subrings
$S^{*}(\overline{\mathcal{M}}_{g,n})$ of $H^{*}(\overline{\mathcal{M}}_{g,n})$
that contains the tautological subrings $RH^{*}(\overline{\mathcal{M}}_{g,n})$
and is closed under pullback by forgetting and permuting marked points and
under pushforward for gluing marked points.

Examples of STEs include the trivial extension $RH^{*}$, the full cohomology
rings $H^{*}$, and the collection of subrings generated by algebraic cycle
classes. Not every STE is closed under the additional tautological operations
induced by push-forward for forgetting marked points and pullback for gluing
marked points. However, the main examples we study here are indeed closed
under all of the tautological operations. See Proposition 2.4.

Note that any intersection of STEs is an STE. An STE is _finitely generated_
if it is the smallest STE that contains a given finite subset (or,
equivalently, the union of finitely many $\mathbb{Q}$-vector subspaces) of
$\coprod_{g,n}H^{*}(\overline{\mathcal{M}}_{g,n})$. A finitely generated STE
is suitable for combinatorial study via algebraic operations on decorated
graphs. See [22, 31] for discussions of the graphical algebra underlying the
tautological ring, and [24, 25] for applications of such operadic methods,
with not necessarily tautological decorations, to the weight spectral sequence
for $\mathcal{M}_{g,n}$. Every STE that we consider is _motivic_ , meaning
that $S^{*}(\overline{\mathcal{M}}_{g,n})\subset
H^{*}(\overline{\mathcal{M}}_{g,n})$ is a sub-Hodge structure and its base
change to $\mathbb{Q}_{\ell}$ is preserved by the Galois action.

######  Theorem 1.3.

For any fixed degree $k$, the STE generated by

$\left\\{H^{k^{\prime}}(\overline{\mathcal{M}}_{g^{\prime},n^{\prime}}):k^{\prime}\leq
k,g^{\prime}<\tfrac{3}{2}k^{\prime}+1,\ n^{\prime}\leq k^{\prime},\
4g^{\prime}-4+n^{\prime}\geq k^{\prime}\right\\}$

contains $H^{k}(\overline{\mathcal{M}}_{g,n})$ for all $g$ and $n$.

In particular, there is a finitely generated STE that contains
$H^{k}(\overline{\mathcal{M}}_{g,n})$ for all $g$ and $n$.

######  Corollary 1.4.

For each $k$, there are only finitely many isomorphism classes of simple Hodge
structures (resp. $\ell$-adic Galois representations) in
$\bigoplus_{g,n}H^{k}(\overline{\mathcal{M}}_{g,n})^{\mathrm{ss}}$.

We developed the notion of STEs to study nontrivial extensions of tautological
rings, such as the STE generated by $H^{11}(\overline{\mathcal{M}}_{1,11})$,
but the same methods also yield new results on the tautological ring itself.
We apply the explicit bounds in Theorem 1.3, together with new tools and
results for small $g$, $n$, and $k$ to prove the following. At the level of
$\mathbb{Q}$-vector spaces, we identify $H_{k}(\overline{\mathcal{M}}_{g,n})$
with $H^{2d_{g,n}-k}(\overline{\mathcal{M}}_{g,n})$, where $d_{g,n}:=3g-3+n$
is the dimension of $\overline{\mathcal{M}}_{g,n}$. Similarly, when $S^{*}$ is
an STE we write $S_{k}(\overline{\mathcal{M}}_{g,n})$ for the
$\mathbb{Q}$-vector space $S^{2d_{g,n}-k}(\overline{\mathcal{M}}_{g,n})$.

######  Theorem 1.5.

The tautological ring $RH^{*}(\overline{\mathcal{M}}_{g,n})$ contains

  1. (1)

$H^{4}(\overline{\mathcal{M}}_{g,n}),$ for all $g$ and $n$,

  2. (2)

$H^{6}(\overline{\mathcal{M}}_{g,n}),$ for $g\geq 10$,

  3. (3)

$H_{k}(\overline{\mathcal{M}}_{g,n}),$ for even $k\leq 14$, for all $g$ and
$n$.

######  Theorem 1.6.

The STE generated by $H^{11}(\overline{\mathcal{M}}_{1,11})$ contains
$H_{13}(\overline{\mathcal{M}}_{g,n})$ for all $g$ and $n$.

######  Theorem 1.7.

The STE generated by $H^{11}(\overline{\mathcal{M}}_{1,11})$ and
$H^{15}(\overline{\mathcal{M}}_{1,15})$ contains
$H_{15}(\overline{\mathcal{M}}_{g,n})$ for all $g$ and $n$.

Theorem 1.5(1) confirms a prediction of Arbarello and Cornalba from the 1990s;
they proposed that their inductive method used to prove that
$H^{2}(\overline{\mathcal{M}}_{g,n})$ is tautological should also apply in
degree 4 [1, p. 1]. Shortly thereafter, Polito confirmed that
$H^{4}(\overline{\mathcal{M}}_{g,n})$ is tautological for $g\geq 8$ [32], but
the general case remained open until now.

Theorem 1.5(3) implies that
$H^{k}(\overline{\mathcal{M}}_{g,n})\cong\bigoplus\mathsf{L}^{k/2}$ for even
$k\leq 14$. The work of Chenevier and Lannes predicts that
$H^{k}(\overline{\mathcal{M}}_{g,n})$ should also be isomorphic to
$\bigoplus\mathsf{L}^{k/2}$ for $k=16,18,20$. The Hodge and Tate conjectures
then predict that these groups are generated by algebraic cycle classes.
However, generation by algebraic cycle classes is an open problem except in
the cases where $H^{k}(\overline{\mathcal{M}}_{g,n})$ is known to be generated
by tautological cycle classes.

By Theorems 1.6 and 1.7, $H_{13}(\overline{\mathcal{M}}_{g,n})$ and
$H_{15}(\overline{\mathcal{M}}_{g,n})$ lie in STEs generated by cohomology
from genus $1$ moduli spaces. On the other hand,
$H_{17}(\overline{\mathcal{M}}_{g,n})$ requires genus $2$ data because
$W_{17}H^{17}(\mathcal{M}_{2,14})\neq 0$ [13].

We note that any STE is top-heavy, in the sense that $\dim
S^{k}(\overline{\mathcal{M}}_{g,n})\leq\dim
S_{k}(\overline{\mathcal{M}}_{g,n})$ for
$k\leq\dim\overline{\mathcal{M}}_{g,n}$, cf. [30]. Moreover, the dimensions in
even and odd degrees are unimodal. This is because any STE contains $RH^{*}$
and hence contains an ample class. Multiplication by a suitable power of the
ample class gives an injection from $S^{k}(\overline{\mathcal{M}}_{g,n})$ to
$S_{k}(\overline{\mathcal{M}}_{g,n})$.

One could show that $H^{k}(\overline{\mathcal{M}}_{g,n})$ is generated by
tautological cycles in the remaining cases covered by Theorem 1.5(3) by
showing that the pairing on $RH^{k}(\overline{\mathcal{M}}_{g,n})\times
RH_{k}(\overline{\mathcal{M}}_{g,n})$ is perfect, e.g., using admcycles [11].
However, computational complexity prevents meaningful progress by brute force.
Note that Petersen and Tommasi showed that this pairing is not perfect in
general for $k\geq 22$ [30, 28]. Graber and Pandharipande had previously shown
that $H^{22}(\overline{\mathcal{M}}_{2,20})$ contains an algebraic cycle class
that is not tautological [17].

######  Conjecture 1.8.

The tautological ring $RH^{*}(\overline{\mathcal{M}}_{g,n})$ contains
$H^{k}(\overline{\mathcal{M}}_{g,n})$ for even $k\leq 20$.

The results above show that Conjecture 1.8 is true for $k\leq 4$, and for
$k=6$ and $g\geq 10$. For $k\leq 14$, the conjecture is true if and only if
the pairing $RH^{k}(\overline{\mathcal{M}}_{g,n})\times
RH_{k}(\overline{\mathcal{M}}_{g,n})$ is perfect. The examples of
Graber–Pandharipande and Petersen–Tommasi show that the conjectured bound of
$k\leq 20$ is the best possible. As further evidence for Conjecture 1.8, we
note that the Arbarello–Cornalba induction together with known base cases
implies the vanishing of $H^{16,0}(\overline{\mathcal{M}}_{g,n})$ and
$H^{18,0}(\overline{\mathcal{M}}_{g,n})$ for all $g$ and $n$, as recently
observed by Fontanari [15].

The inductive arguments used to study $H^{k}(\overline{\mathcal{M}}_{g,n})$
for all $g$ and $n$ rely on understanding base cases
$H^{k^{\prime}}(\overline{\mathcal{M}}_{g^{\prime},n^{\prime}})$ where
$g^{\prime}$ and $n^{\prime}$ are small relative to $k$. As $k$ grows, more
base cases are needed. With the exception of
$H^{11}(\overline{\mathcal{M}}_{1,11})$, all base cases required for previous
work with $k\leq 12$ have been pure Hodge–Tate [1, 4, 5]. Substantial work
went into establishing these base cases via point counts and other methods.
When $k\geq 13$, the problem becomes fundamentally more difficult, as an
increasing number of the required base cases are _not_ pure Hodge–Tate. In
particular, previous techniques for handling base cases do not apply.

The advances presented here depend on a new technique for controlling the pure
weight cohomology of $\mathcal{M}_{g,n}$. A space $X$ has the _Chow–Künneth
generation Property (CKgP)_ if the tensor product map on Chow groups
$A_{*}(X)\otimes A_{*}(Y)\to A_{*}(X\times Y)$ is surjective for all $Y$. If
$X$ is smooth, proper, and has the CKgP, then the cycle class map is an
isomorphism. However, in several of the base cases needed for our arguments,
the smooth and proper moduli space $\overline{\mathcal{M}}_{g,n}$ has odd
cohomology and hence does not have the CKgP. Nevertheless, we show that the
open moduli spaces $\mathcal{M}_{g,n}$ do have the CKgP for the relevant pairs
$(g,n)$. To apply this in the proof of our main results, the key new technical
statement is Lemma 4.3, which says that if $X$ is smooth and has the CKgP,
then $W_{k}H^{k}(X)$ is algebraic. This extension of the aforementioned result
on the cycle class map to spaces that are not necessarily proper is essential
for controlling the motivic structures that appear in
$H^{k}(\overline{\mathcal{M}}_{g,n})$ for $k\geq 13$. For example, using that
$\mathcal{M}_{3,n}$ has the CKgP for $n\leq 11$ [6, Theorem 1.4], we determine
the Hodge structures and Galois representations that appear in
$H^{*}(\overline{\mathcal{M}}_{3,n})$, for $n\leq 11$.

######  Theorem 1.9.

For $n\leq 11$, $H^{*}(\overline{\mathcal{M}}_{3,n})^{\mathrm{ss}}$ is a
polynomial in $\mathsf{L}$ and $\mathsf{S}_{12}$.

Bergström and Faber recently used point counting techniques to compute the
cohomology of $\overline{\mathcal{M}}_{3,n}$ as an
$\mathbb{S}_{n}$-equivariant Galois representation for $n\leq 14$. For $n\geq
9$, these computations are conditional on the assumption that the only
$\ell$-adic Galois representations appearing are those from the list of
Chenevier and Lannes [3]. Theorem 1.9 unconditionally confirms the
calculations of Bergström and Faber for $n=9$, $10$, and $11$.

In order to prove Theorems 1.1, 1.5(3), and 1.7, our inductive arguments
require several base cases beyond what was already in the literature. In
particular, we prove the following results in genus $7$, which are also of
independent interest.

######  Theorem 1.10.

For $n\leq 3$, the moduli space $\mathcal{M}_{7,n}$ has the CKgP and
$R^{*}(\mathcal{M}_{7,n})=A^{*}(\mathcal{M}_{7,n})$.

Here, $R^{*}$ denotes the subring of the Chow ring $A^{*}$ generated by
tautological cycle classes. Previous results proving that $\mathcal{M}_{g,n}$
has the CKgP and $R^{*}(\mathcal{M}_{g,n})=A^{*}(\mathcal{M}_{g,n})$ for small
$g$ and $n$ have primarily relied on corresponding results for Hurwitz spaces
with marked points [6]. Unfortunately, the numerics for degree $5$ covers
prevented this technique from working with marked points. Here, we take a new
approach to the pentagonal locus, using a modification of Mukai’s construction
[21] that includes markings.

Acknowledgements. We thank Jonas Bergström, Carel Faber, Rahul Pandharipande,
Dan Petersen, Kartik Prasanna, Burt Totaro, and Thomas Willwacher for helpful
conversations related to this work. We especially thank Dan Petersen for his
comments that led to an improved version of Lemma 3.1.

##  2\. Preliminaries

In this section, we establish notation and terminology that we will use
throughout the paper and discuss a few basic examples of STEs. We also recall
previously known facts about the cohomology groups of moduli spaces,
especially in genus 0, 1, and 2, that will be used in the base cases of our
inductive arguments.

###  2.1. Preliminaries on STEs

Recall that an STE is, by definition, closed under the tautological operations
given by pushing forward from the boundary or pulling back from moduli spaces
with fewer marked points. Let $\widetilde{\partial\mathcal{M}}_{g,n}$ denote
the normalization of the boundary. The sequence

$H^{k-2}(\widetilde{\partial\mathcal{M}}_{g,n})\to
H^{k}(\overline{\mathcal{M}}_{g,n})\to
W_{k}H^{k}(\mathcal{M}_{g,n})\rightarrow 0$

is right exact.

Let $\pi_{i}\colon\mathcal{M}_{g,n}\to\mathcal{M}_{g,n-1}$ be the tautological
morphism forgetting the $i$th marking and let

$\Phi^{k}_{g,n}:=\pi_{1}^{*}W_{k}H^{k}(\mathcal{M}_{g,n-1})+\ldots+\pi_{n}^{*}W_{k}H^{k}(\mathcal{M}_{g,n-1})\subset
W_{k}H^{k}(\mathcal{M}_{g,n}).$

The following lemma is a cohomological analogue of the “filling criteria” in
[6, Section 4].

We consider the partial order in which $(g^{\prime},n^{\prime})\prec(g,n)$ if
$g^{\prime}\leq g$, $2g^{\prime}+n^{\prime}\leq 2g+n$, and
$(g^{\prime},n^{\prime})\neq(g,n)$. The moduli space
$\overline{\mathcal{M}}_{g,n}$ is stratified according to the topological
types of stable curves, and each stratum of the boundary is a finite quotient
of a product of moduli spaces $\mathcal{M}_{g^{\prime},n^{\prime}}$ such that
$(g^{\prime},n^{\prime})\prec(g,n)$. Recall that we write $d_{g,n}:=3g-3+n$.

######  Lemma 2.1.

Let $S^{*}$ be an STE and let $2g-2+n>0$. If the canonical map

(2.1) $S^{k^{\prime}}(\overline{\mathcal{M}}_{g^{\prime},n^{\prime}})\to
W_{k^{\prime}}H^{k^{\prime}}(\mathcal{M}_{g^{\prime},n^{\prime}})/\big{(}\Phi^{k^{\prime}}_{g^{\prime},n^{\prime}}+RH^{k^{\prime}}(\mathcal{M}_{g^{\prime},n^{\prime}})\big{)}$

is surjective for $(g^{\prime},n^{\prime},k^{\prime})=(g,n,k)$ and all
$(g^{\prime},n^{\prime},k^{\prime})$ satisfying

(2.2) $(g^{\prime},n^{\prime})\prec(g,n),\qquad
2d_{g^{\prime},n^{\prime}}-k^{\prime}\leq 2d_{g,n}-k\qquad\text{and}\qquad
k^{\prime}\leq k-2$

then
$S^{k}(\overline{\mathcal{M}}_{g,n})=H^{k}(\overline{\mathcal{M}}_{g,n})$.

###### Proof.

The proof is by induction on $g$ and $n$. Consider the diagram

(2.3)
${S^{k-2}(\widetilde{\partial\mathcal{M}_{g,n}})}$${S^{k}(\overline{\mathcal{M}}_{g,n})}$${H^{k-2}(\widetilde{\partial\mathcal{M}_{g,n}})}$${H^{k}(\overline{\mathcal{M}}_{g,n})}$${W_{k}H^{k}(\mathcal{M}_{g,n})}$${0.}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\scriptstyle{\phi}$

Here, we extend $S^{*}$ to $\widetilde{\partial\mathcal{M}_{g,n}}$ in the
natural way, by summing over components, using the Künneth formula, and taking
invariants under automorphisms of the dual graph. We claim that $\alpha$ is an
isomorphism. By the Künneth formula, the domain of $\alpha$ is a sum of tensor
products of $S^{\ell}(\overline{\mathcal{M}}_{\gamma,\nu})$ with $\ell\leq
k-2$ and $(\gamma,\nu)\prec(g,n)$. Furthermore, by considering dimensions of
the cycles involved, we must also have $2d_{\gamma,\nu}-\ell\leq 2d_{g,n}-k$.
This is because we must have $k-2-\ell\leq 2(d_{g,n}-1-d_{\gamma,\nu})$, so
that the degree of the other Künneth component does not exceed its real
dimension. Now, suppose we are given $(g^{\prime},n^{\prime},k^{\prime})$ that
satisfy

$(g^{\prime},n^{\prime})\prec(\gamma,\nu),\qquad
2d_{g^{\prime},n^{\prime}}-k^{\prime}\leq
2d_{\gamma,\nu}-\ell\qquad\text{and}\qquad k^{\prime}\leq\ell-2.$

Then $(g^{\prime},n^{\prime},k^{\prime})$ satisfies (2.2), so
$S^{\ell}(\overline{\mathcal{M}}_{\gamma,\nu})=H^{\ell}(\overline{\mathcal{M}}_{\gamma,\nu})$
by induction. This proves the claim.

Next, by induction we have
$H^{k}(\overline{\mathcal{M}}_{g,n-1})=S^{k}(\overline{\mathcal{M}}_{g,n-1})$,
so the image of

$\pi_{i}^{*}\colon H^{k}(\overline{\mathcal{M}}_{g,n-1})\to
H^{k}(\overline{\mathcal{M}}_{g,n})$

is contained in $S^{k}(\overline{\mathcal{M}}_{g,n})$ for all $i$. Hence,
$\Phi^{k}_{g,n}$ is contained in the image of $\phi$. The tautological classes
$RH^{k}(\mathcal{M}_{g,n})\subseteq W_{k}H^{k}(\mathcal{M}_{g,n})$ are also
contained in the image of $\phi$ by definition. Thus, the surjectivity of

$S^{k}(\overline{\mathcal{M}}_{g,n})\to
W_{k}H^{k}(\mathcal{M}_{g,n})/\left(\Phi^{k}_{g,n}+RH^{k}(\mathcal{M}_{g,n})\right)$

implies that $\phi$ is surjective. Hence, $\beta$ is also surjective, as
desired. ∎

In order to apply Lemma 2.1, we need results that help us understand
generators for
$W_{k}H^{k}(\mathcal{M}_{g,n})/\left(\Phi^{k}_{g,n}+RH^{k}(\mathcal{M}_{g,n})\right)$.
This is the topic of Sections 3 and 4.

###  2.2. Pure cohomology in genus 1 and 2

On $\mathcal{M}_{0,n}$, the only nonvanishing pure cohomology is
$W_{0}H^{0}(\mathcal{M}_{0,n})$. Below, we present the complete classification
of pure cohomology in genus $1$, which is due to Getzler [16]. In genus $2$,
we present a classification in low cohomological degree, following Petersen
[27].

####  2.2.1. Genus 1

The following statement appeared in [23, Proposition 7]; the proof there is
omitted. Here, we include the proof (and corrected statement), which was
explained to us by Dan Petersen. Let $\mathsf{S}_{k+1}$ denote the weight $k$
structure associated to the space of cusp forms of weight $k+1$ for
$\operatorname{SL}_{2}(\mathbb{Z})$. By the Eichler–Shimura correspondence, we
have
$\mathsf{S}_{k+1}=W_{k}H^{1}(\mathcal{M}_{1,1},\mathbb{V}^{\otimes{k-1}})$. We
will see that the latter is identified with $W_{k}H^{k}(\mathcal{M}_{1,k})$.
More generally, we have the following. Given a partition $\lambda$ of $n$, let
$V_{\lambda}$ be the associated Specht module representation of
$\mathbb{S}_{n}$.

######  Proposition 2.2.

For $n\geq k,$ a basis for $W_{k}H^{k}(\mathcal{M}_{1,n})$ is given by the
${n-1\choose k-1}$ pullbacks from $W_{k}H^{k}(\mathcal{M}_{1,A})$ where $A$
runs over all subsets of $\\{1,\ldots,n\\}$ of size $k$ such that $1\in A$.
Consequently, there is an $\mathbb{S}_{n}$-equivariant isomorphism

$W_{k}H^{k}(\mathcal{M}_{1,n})\cong\mathsf{S}_{k+1}\otimes V_{n-k+1,1^{k-1}}.$

For $n<k$, we have $W_{k}H^{k}(\mathcal{M}_{1,n})=0$.

###### Proof.

Let $\pi\colon E\rightarrow\mathcal{M}_{1,1}$ denote the universal elliptic
curve, and $\sigma\colon\mathcal{M}_{1,1}\to E$ the section. Associated to the
open embedding $\mathcal{M}_{1,n}\hookrightarrow E^{n-1}$, we have a right
exact sequence

(2.4) $\bigoplus W_{k-2}H^{k-2}(E^{n-2})\rightarrow
W_{k}H^{k}(E^{n-1})\rightarrow W_{k}H^{k}(\mathcal{M}_{1,n})\rightarrow 0.$

Since $f\colon E^{n-1}\rightarrow\mathcal{M}_{1,1}$ is smooth and proper,
using the Leray spectral sequence, we have

(2.5)
$H^{k}(E^{n-1})=\bigoplus_{p+q=k}H^{p}(\mathcal{M}_{1,1},R^{q}f_{*}\mathbb{Q}).$

By the Künneth formula, we have

$R^{q}f_{*}\mathbb{Q}=\bigoplus_{i_{2}+\ldots+i_{n}=q}R^{i_{2}}\pi_{*}\mathbb{Q}\otimes\cdots\otimes
R^{i_{n}}\pi_{*}\mathbb{Q}.$

Let $\mathbb{V}:=R^{1}\pi_{*}\mathbb{Q}$ and note that
$R^{0}\pi_{*}\mathbb{Q}=\mathbb{Q}$ and
$R^{2}\pi_{*}\mathbb{Q}=\mathbb{Q}(-1)$.

Fix some $p$ and $(i_{2},\ldots,i_{n})$ with $p+i_{2}+\ldots+i_{n}=k$. If
$i_{j}=2$, then we claim that

(2.6)
$W_{k}H^{p}(\mathcal{M}_{1,1},R^{i_{2}}\pi_{*}\mathbb{Q}\otimes\cdots\otimes
R^{i_{n}}\pi_{*}\mathbb{Q})\subset W_{k}H^{k}(E^{n-1})$

lies in the image of of the first map in (2.4). More precisely, let
$\alpha_{j}\colon E^{n-2}\rightarrow E^{n-1}$, be the locus where the $j$th
entry in $E^{n-1}$ agrees with the section $\sigma\colon\mathcal{M}_{1,1}\to
E$. In other words, $\alpha_{j}$ is defined by the fiber diagram

${E^{n-2}}$${E^{n-1}}$${\mathcal{M}_{1,1}}$${E.}$$\scriptstyle{\alpha_{j}}$$\scriptstyle{\mathrm{pr}_{j}}$$\scriptstyle{\sigma}$

If $i_{j}=2,$ we have $p+i_{2}+\ldots+i_{j-1}+i_{j+1}+\ldots+i_{n}=k-2$.
Therefore, using the Leray spectral sequence for
$E^{n-2}\to\mathcal{M}_{1,1}$, there is a corresponding term

(2.7)
$W_{k}H^{p}(\mathcal{M}_{1,1},R^{i_{2}}\pi_{*}\mathbb{Q}\otimes\cdots\otimes
R^{i_{j-1}}\pi_{*}\mathbb{Q}\otimes
R^{i_{j+1}}\pi_{*}\mathbb{Q}\otimes\cdots\otimes
R^{i_{n}}\pi_{*}\mathbb{Q})\subset W_{k}H^{k-2}(E^{n-2}).$

Then the pushforward $\alpha_{j*}\colon H^{k-2}(E^{n-2})\to H^{k}(E^{n-1})$
sends the subspace on the left of (2.7) isomorphically onto the subspace on
the left of (2.6), which proves the claim

It follows that $W_{k}H^{k}(\mathcal{M}_{1,n})$ is generated by the terms
$W_{k}H^{p}(\mathcal{M}_{1,1},R^{i_{2}}\pi_{*}\mathbb{Q}\otimes\cdots\otimes
R^{i_{n}}\pi_{*}\mathbb{Q})$ in $W_{k}H^{k}(E^{n-1})$ where all $i_{j}\leq 1$.
By [26, Section 2], we have $W_{k}H^{p}(\mathcal{M}_{1,1},\mathbb{V}^{\otimes
q})=0$ unless $p=1$ and $q=k-1$, in which case
$W_{k}H^{1}(\mathcal{M}_{1,1},\mathbb{V}^{\otimes k-1})=\mathsf{S}_{k+1}$ by
Eichler–Shimura.

There are ${n-1\choose k-1}$ terms of the form
$H^{1}(\mathcal{M}_{1,1},\mathbb{V}^{\otimes k-1})$ in (2.5), coming from
choosing which $k-1$ of the $n-1$ indices have $i_{j}=1$. Each of these terms
is pulled back along the projection map $E^{n-1}\to E^{k-1}$, which remembers
the $k-1$ factors for which $i_{j}=1$. Let $A$ be the collection of indices
$j$ such that $i_{j}=1$ together with $1$. There is a commutative diagram

${W_{k}H^{k}(E^{n-1})}$${W_{k}H^{k}(\mathcal{M}_{1,n})}$${W_{k}H^{k}(E^{k-1})}$${W_{k}H^{k}(\mathcal{M}_{1,A}).}$

It follows that $W_{k}H^{k}(\mathcal{M}_{1,n})$ is generated by the pullbacks
from $W_{k}H^{k}(\mathcal{M}_{1,A})$ as $A$ ranges over all subsets of size
$k$ containing $1$. Finally, we note that there can be no relations among
these ${n-1\choose k-1}$ copies of
$W_{k}H^{1}(\mathcal{M}_{1,1},\mathbb{V}^{\otimes k-1})=\mathsf{S}_{k+1}$
since the image of the left-hand map of (2.4) lies in the subspace of type
$\mathsf{L}^{i}\mathsf{S}_{k+1-2i}$ for $i\geq 1$.

We have now shown that $W_{k}H^{k}(\mathcal{M}_{1,n})=\mathsf{S}_{k+1}\otimes
U$ for some ${n-1\choose k-1}$-dimensional vector space $U$. From the
discussion above, it is not difficult to identify $U$ as an
$\mathbb{S}_{n}$-representation. When $n=k$, the $\mathbb{S}_{k}$ action on
$W_{k}H^{k}(\mathcal{M}_{1,k})$ is the sign representation. To identify $U$
for $n>k$, let $\mathbb{S}_{n-1}\subset\mathbb{S}_{n}$ be the subgroup that
fixes $1$. Since $W_{k}H^{k}(\mathcal{M}_{1,n})$ is freely generated by the
pullbacks from $W_{k}H^{k}(\mathcal{M}_{1,A})$ as $A$ runs over subsets of
size $k$ containing $1$, we have

$\mathrm{Res}^{\mathbb{S}_{n}}_{\mathbb{S}_{n-1}}U=\mathrm{Ind}_{\mathbb{S}_{k-1}\times\mathbb{S}_{n-k}}^{\mathbb{S}_{n-1}}(\mathrm{sgn}\boxtimes\mathbf{1}).$

By the Pieri rule, we have

(2.8)
$\mathrm{Ind}_{\mathbb{S}_{k-1}\times\mathbb{S}_{n-k}}^{\mathbb{S}_{n-1}}(\mathrm{sgn}\boxtimes\mathbf{1})=\mathrm{Ind}_{\mathbb{S}_{k-1}\times\mathbb{S}_{n-k}}^{\mathbb{S}_{n-1}}(V_{1^{k-1}}\boxtimes
V_{n-k})=V_{n-k+1,1^{k-2}}\oplus V_{n-k,1^{k-1}}.$

By the branching rule, $V_{n-k+1,1^{k-1}}$ is the unique $\mathbb{S}_{n}$
representation whose restriction to $\mathbb{S}_{n-1}$ is the representation
in (2.8). ∎

Since $\mathsf{S}_{14}=0$, we have the following.

######  Corollary 2.3.

$H^{13}(\overline{\mathcal{M}}_{1,n})$ lies in the STE generated by
$H^{11}(\overline{\mathcal{M}}_{1,11})$ for all $n$.

Let $S^{*}_{\omega}$ denote the STE generated by
$H^{11}(\overline{\mathcal{M}}_{1,11})$. Note that $S^{*}_{\omega}$ is the STE
generated by the class $\omega\in H^{11,0}(\overline{\mathcal{M}}_{1,11})$
associated to the weight $12$ cusp form for
$\operatorname{SL}_{2}(\mathbb{Z})$, i.e., it is the smallest STE whose
complexification contains $\omega$. In [8], we showed that $S^{*}_{\omega}$
contains $H^{11}(\overline{\mathcal{M}}_{g,n})$, and hence
$H_{11}(\overline{\mathcal{M}}_{g,n})$, for all $g$ and $n$. In this paper, we
show that it also contains $H_{13}(\overline{\mathcal{M}}_{g,n})$ for all $g$
and $n$ (Theorem 1.6). In contrast with the system of tautological rings, an
arbitrary STE need not be closed under pushforward along maps forgetting
marked points or pullback to the boundary. Nevertheless, we have the following
result for $S_{\omega}^{*}(\overline{\mathcal{M}}_{g,n})$.

######  Proposition 2.4.

The STE $S_{\omega}^{*}$ is closed under the tautological operations induced
by pushforward for forgetting marked points and pullback for gluing marked
points.

###### Proof.

By [26], all even cohomology in genus $1$ is represented by boundary strata.
Therefore, any product of two odd degree classes can be written as a sum of
boundary strata. It follows that every class in
$S_{\omega}^{*}(\overline{\mathcal{M}}_{g,n})$ can be represented as a linear
combination of decorated graphs all of whose nontautological decorations are
$H^{11}$-classes on genus 1 vertices.

The pushforward of $H^{11}(\overline{\mathcal{M}}_{1,n})$ along the maps
forgetting marked points is zero. Since the tautological rings are closed
under pushforward, it follows that $S_{\omega}^{*}$ is closed under
pushforward.

By [8, Lemma 2.2], the image of $H^{11}(\overline{\mathcal{M}}_{1,n})$ under
pullback to the boundary lies in $S^{*}_{\omega}$. By the excess intersection
formula, and using the fact that $\psi$-classes in genus 1 are boundary
classes, it follows that the pullback of any such decorated graph in
$S_{\omega}^{*}$ to any boundary stratum is a sum of decorated graphs of the
same form. In particular, $S_{\omega}^{*}$ is closed under pullback to the
boundary, as required. ∎

######  Remark 2.5.

More generally, the Hodge groups $H^{k,0}(\overline{\mathcal{M}}_{1,k})$
correspond to the space of cusp forms for $\operatorname{SL}_{2}(\mathbb{Z})$
of weight $k+1$. Essentially the same argument shows that the STE generated by
any subset of these cusp form classes is closed under all of the tautological
operations.

####  2.2.2. Genus 2

Here, we summarize what we need about the pure weight cohomology of
$\mathcal{M}_{2,n}$ in low degrees, from [30, 28].

######  Proposition 2.6.

Let $k\leq 10$. Then
$W_{2k}H^{2k}(\mathcal{M}_{2,n})=RH^{2k}(\mathcal{M}_{2,n})$.

###### Proof.

When $n<20$, we have
$H^{2k}(\overline{\mathcal{M}}_{2,n})=RH^{2k}(\overline{\mathcal{M}}_{2,n})$
by [28, Theorem 3.8]. Hence,
$W_{2k}H^{2k}(\mathcal{M}_{2,n})=RH^{2k}(\mathcal{M}_{2,n})$ by restriction.
For $n=20$ and $k\leq 10$, the same result holds (see [28, Remark 3.10]). For
$n>20$ and $k\leq 10$, all of $W_{2k}H^{2k}(\mathcal{M}_{2,n})$ is pulled back
from $W_{2k}H^{2k}(\mathcal{M}_{2,m})$ where $m\leq 20$ (see Lemma 3.1,
below). Because the tautological ring is closed under forgetful pullbacks, the
lemma follows. ∎

Applying Lemma 2.1 to the STE $RH^{*}$ immediately implies Conjecture 1.8 for
$g=2$.

######  Corollary 2.7.

If $k\leq 10$, then
$H^{2k}(\overline{\mathcal{M}}_{2,n})=RH^{2k}(\overline{\mathcal{M}}_{2,n})$
for all $n$.

In odd degrees, there are restrictions on the possible motivic structures
coming from the cohomology of local systems on the moduli space of principally
polarized abelian surfaces.

######  Proposition 2.8.

In the category of Galois representations, the pure weight cohomology of
$\mathcal{M}_{2,n}$ in degrees 13 and 15 is of the form

$W_{13}H^{13}(\mathcal{M}_{2,n})^{\mathrm{ss}}\cong\bigoplus\mathsf{L}\mathsf{S}_{12}\mbox{
\ \ and \ \
}W_{15}H^{15}(\mathcal{M}_{2,n})^{\mathrm{ss}}\cong\bigoplus\mathsf{L}^{2}\mathsf{S}_{12}.$

This follows by standard arguments with the Künneth formula, Leray spectral
sequence, and the long exact sequence in cohomology induced by the Torelli map
$\mathcal{M}_{2}\to\mathcal{A}_{2}$ to reduce to Petersen’s classification of
the cohomology of local systems on $\mathcal{A}_{2}$ [27].

######  Corollary 2.9.

On $\overline{\mathcal{M}}_{2,n}$, in the category of Galois representations,
we have

$H^{13}(\overline{\mathcal{M}}_{2,n})^{\mathrm{ss}}\cong\bigoplus\mathsf{L}\mathsf{S}_{12}\mbox{
\ \ and \ \
}H^{15}(\overline{\mathcal{M}}_{2,n})^{\mathrm{ss}}\cong\bigoplus\mathsf{L}^{2}\mathsf{S}_{12}.$

###### Proof.

For degree 13, consider the right exact sequence

$H^{11}(\widetilde{\partial\mathcal{M}_{2,n}})\rightarrow
H^{13}(\overline{\mathcal{M}}_{2,n})\rightarrow
W_{13}H^{13}(\mathcal{M}_{2,n})\rightarrow 0.$

By [8], the semisimplification of the left-hand side is a sum of terms
$\mathsf{S}_{12}$. Proposition 2.8 shows that the semisimplification of the
right-hand side consists only of Tate twists of $\mathsf{S}_{12}$. Thus, the
middle term does too. The argument for degree 15 is similar, using the degree
11 and 13 results, and the fact that $H^{2}(\overline{\mathcal{M}}_{g,n})$ is
pure Hodge–Tate. ∎

######  Remark 2.10.

See Lemma 8.1 for an analogous result in the category of Hodge structures.

##  3\. Finite generation

Let $S^{*}$ be an STE. By Lemma 2.1, $S^{k}(\overline{\mathcal{M}}_{g,n})$
contains $H^{k}(\overline{\mathcal{M}}_{g,n})$ for all $g$ and $n$ if and only
if it surjects onto
$W_{k}H^{k}(\mathcal{M}_{g,n})/\big{(}\Phi^{k}_{g,n}+RH^{k}(\mathcal{M}_{g,n})\big{)}$
for all $g$ and $n$. Below we give a sufficient criterion for the vanishing of
$W_{k}H^{k}(\mathcal{M}_{g,n})/\Phi^{k}_{g,n}.$ The argument is similar to the
proof of Proposition 2.2, using the Künneth decomposition of the Leray
spectral sequence for the $n$-fold fiber product
$\mathcal{C}^{n}\to\mathcal{M}_{g}$.

We are grateful to Dan Petersen for explaining how an earlier version of this
lemma could be strengthened to the version presented here. This stronger
version will also be useful for controlling some additional base cases in
Section 7. To state it, we define the subspace

$\Psi_{g,n}^{k}:=\psi_{1}\Phi^{k-2}_{g,n}+\ldots+\psi_{n}\Phi^{k-2}_{g,n}\subset
W_{k}H^{k}(\mathcal{M}_{g,n}).$

In other words, $\Psi^{k}_{g,n}$ is generated by pullbacks from moduli spaces
with fewer markings multiplied with $\psi$-classes. Given a $g$-tuple of
integers $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{g}\geq
0)$, let $\mathbb{V}_{\lambda}$ be the associated symplectic local system on
$\mathcal{M}_{g}$, as in [29]. Let $|\lambda|=\lambda_{1}+\ldots+\lambda_{g}$,
which is the weight of the local system $\mathbb{V}_{\lambda}$. When
$|\lambda|=n$, as in the previous section, we write $V_{\lambda}$ for the
irreducible $\mathbb{S}_{n}$ representation corresponding to $\lambda$.

######  Lemma 3.1.

(a) If $g\geq 2$ then

$W_{k}H^{k}(\mathcal{M}_{g,n})/(\Phi_{g,n}^{k}+\Psi_{g,n}^{k})\cong\bigoplus_{|\lambda|=n}W_{k}H^{k-n}(\mathcal{M}_{g},\mathbb{V}_{\lambda})\otimes
V_{\lambda^{T}}.$

(b) Moreover, if $n>0$ and $n\geq k$ then
$W_{k}H^{k}(\mathcal{M}_{g,n})=\Phi^{k}_{g,n}$.

###### Proof.

(a) Let $\pi\colon\mathcal{C}\to\mathcal{M}_{g}$ be the universal curve. There
is an open inclusion $\mathcal{M}_{g,n}\subset\mathcal{C}^{n}$, and hence
restriction gives a surjection from $W_{k}H^{k}(\mathcal{C}^{n})$ to
$W_{k}H^{k}(\mathcal{M}_{g,n})$. Given a subset $A\subset\\{1,\ldots,n\\}$,
let $\mathcal{C}^{n}\to\mathcal{C}^{A}$ be the projection onto the factors
indexed by $A$. We consider the following three subspaces of
$H^{k}(\mathcal{C}^{n})$:

  1. (i)

$\widetilde{\Phi}$, the span of the pullbacks of
$H^{k}(\mathcal{C}^{\\{i\\}^{c}})$ along projection
$\mathcal{C}^{n}\to\mathcal{C}^{\\{i\\}^{c}}$

  2. (ii)

$\widetilde{\Psi}$, the span of $\psi_{i}\cdot
H^{k-2}(\mathcal{C}^{\\{i\\}^{c}})$

  3. (iii)

$\widetilde{\Delta}$, the span of $\Delta_{ij}\cdot
H^{k-2}(\mathcal{C}^{\\{i,j\\}^{c}})$, where $\Delta_{ij}$ denotes the
pullback of the class of the diagonal in $H^{2}(\mathcal{C}^{\\{i,j\\}})$.

Note that $\widetilde{\Delta}$ lies in the kernel of
$H^{k}(\mathcal{C}^{n})\to H^{k}(\mathcal{M}_{g,n})$, and the images of the
weight $k$ parts of $\widetilde{\Phi}$ and $\widetilde{\Psi}$ under
$W_{k}H^{k}(\mathcal{C}^{n})\to W_{k}H^{k}(\mathcal{M}_{g,n})$ are the
subspaces $\Phi^{k}_{g,n}$ and $\Psi^{k}_{g,n}$ respectively. To prove part
(a), it thus suffices to show that

(3.1)
$H^{k}(\mathcal{C}^{n})/(\widetilde{\Phi}+\widetilde{\Psi}+\widetilde{\Delta})\cong\bigoplus_{|\lambda|=n}H^{k-n}(\mathcal{M}_{g},\mathbb{V}_{\lambda})\otimes
V_{\lambda^{T}}.$

We explain how (3.1) follows from [29].

Since $f\colon\mathcal{C}^{n}\to\mathcal{M}_{g}$ is smooth and proper, the
Leray spectral sequence degenerates at $E_{2}$, and by [12, Proposition 2.16],
we have a direct sum decomposition

$H^{k}(\mathcal{C}^{n})=\bigoplus_{p+q=k}H^{p}(\mathcal{M}_{g},R^{q}f_{*}\mathbb{Q}).$

Applying the Künneth formula, we obtain

(3.2) $H^{k}(\mathcal{C}^{n})=\bigoplus_{\begin{subarray}{c}p+q=k\\\
i_{1}+\ldots+i_{n}=q\end{subarray}}H^{p}\left(\mathcal{M}_{g},R^{i_{1}}\pi_{*}\mathbb{Q}\otimes\cdots\otimes
R^{i_{n}}\pi_{*}\mathbb{Q}\right).$

As observed in [29, Section 5.2.2], the subspace $\widetilde{\Phi}$
corresponds exactly to the span of the summands where some $i_{s}=0$.
Meanwhile, modulo $\widetilde{\Phi}$, the subspace $\widetilde{\Psi}$
corresponds to the span of summands where some $i_{s}=2$. This follows from
the formulas for the projector $\pi_{2}$ (which projects onto such summands)
in [29, Section 5.1]. Quotienting by $\widetilde{\Phi}+\widetilde{\Psi}$ thus
leaves the term with $i_{1}=\ldots=i_{n}=1$. The local system
$(R^{1}\pi_{*}\mathbb{Q})^{\otimes n}$ corresponds to the $n$th tensor power
of the standard representation of $\mathrm{Sp}_{2g}$. It thus decomposes into
a direct sum of irreducible local systems as in [29, Section 3.2]. As
explained there, we have a natural projection

(3.3) $(R^{1}\pi_{*}\mathbb{Q})^{\otimes
n}\to\bigoplus_{|\lambda|=n}\mathbb{V}_{\lambda}\otimes V_{\lambda^{T}}$

whose kernel is spanned by the image of the ${n\choose 2}$ insertion maps

$(R^{1}\pi_{*}\mathbb{Q})^{\otimes n-2}\to(R^{1}\pi_{*}\mathbb{Q})^{\otimes
n}$

given by inserting the class of the symplectic form. Modulo other Künneth
components, the class of the diagonal $\Delta_{i,j}$ is the class of the
symplectic form in $H^{1}(\mathcal{C})\otimes H^{1}(\mathcal{C})\subset
H^{2}(\mathcal{C}^{\\{i,j\\}})$. Summarizing, we have found

$H^{k}(\mathcal{C}^{n})/(\widetilde{\Phi}+\widetilde{\Psi})\cong
H^{k-n}(\mathcal{M}_{g},(R^{1}\pi_{*}\mathbb{Q})^{\otimes
n})\to\bigoplus_{|\lambda|=n}H^{k-n}(\mathbb{V}_{\lambda})\otimes
V_{\lambda^{T}},$

and the kernel is spanned by $\widetilde{\Delta}$. This establishes (3.1)

(b) Now assume $n\geq k$. If some $i_{s}=2$ and $i_{1}+\ldots+i_{n}=k$, we
must have some other $i_{s^{\prime}}=0$. It follows that
$\widetilde{\Psi}\subset\widetilde{\Phi}$ and hence
$\Psi^{k}_{g,n}\subset\Phi^{k}_{g,n}$. Thus it suffices to see that the right-
hand side of (a) vanishes when $n\geq k$ and $n>0$. This is immediate when
$n>k$. Meanwhile, for the case $n=k$, we have
$H^{0}(\mathcal{M}_{g},\mathbb{V}_{\lambda})=0$ since $\mathbb{V}_{\lambda}$
is non-trivial. ∎

We note a neat consequence of Lemma 3.1(a) in genus $2$, which was also
pointed out to us by Dan Petersen.

######  Lemma 3.2.

Suppose $n$ is odd. The STE generated by
$H^{k}(\overline{\mathcal{M}}_{2,n-1}),H^{k-2}(\overline{\mathcal{M}}_{2,n-1})$
and $H^{k-2}(\overline{\mathcal{M}}_{1,m})$ for $m\leq n+2$ contains
$H^{k}(\overline{\mathcal{M}}_{2,n})$.

###### Proof.

The STE generated by
$H^{k}(\overline{\mathcal{M}}_{2,n-1}),H^{k-2}(\overline{\mathcal{M}}_{2,n-1})$
and $H^{k-2}(\overline{\mathcal{M}}_{1,m})$ for $m\leq n+2$ contains all
classes in $H^{k}(\overline{\mathcal{M}}_{2,n})$ that are pushed forward from
the boundary, pulled back from less marked points, or products of $\psi$
classes with classes pulled back from less marked points. It therefore
suffices to show that
$W_{k}H^{k}(\mathcal{M}_{2,n})/(\Phi^{k}_{2,n}+\Psi^{k}_{2,n})=0$. By Lemma
3.1(a), this quotient is a sum of the pure cohomology of local systems of
weight $n$. But on $\mathcal{M}_{2}$, the hyperelliptic involution acts on
local systems of weight $n$ by $(-1)^{n}$, so local systems of odd weight have
no cohomology. ∎

Figure 1. The argument in the proof of Theorem 1.3 shows that
$W_{17}H^{17}(\mathcal{M}_{g,n})/(\Phi^{17}_{g,n}+RH^{17}(\mathcal{M}_{g,n}))$
vanishes for $(g,n)$ outside the gray shaded region. Note that this quotient
does _not_ vanish for $(g,n)$ equal to $(1,17)$ and $(2,14)$, which are
pictured by purple dots.

###### Proof of Theorem 1.3.

Let $S^{*}$ be the STE generated by the cohomology groups listed in the
statement of the theorem. By Lemma 2.1, it suffices to check that
$S^{*}(\overline{\mathcal{M}}_{g^{\prime},n^{\prime}})$ surjects onto
$W_{k^{\prime}}H^{k^{\prime}}(\overline{\mathcal{M}}_{g^{\prime},n^{\prime}})/\left(\Phi^{k}_{g^{\prime},n^{\prime}}+RH^{k^{\prime}}(\mathcal{M}_{g^{\prime},n^{\prime}})\right)$
for all $(g^{\prime},n^{\prime},k^{\prime})$ with $k^{\prime}\leq k$. By Lemma
3.1(b), the target vanishes when $n^{\prime}>k^{\prime}$. By the vcd of
$\overline{\mathcal{M}}_{g^{\prime},n^{\prime}}$, it also vanishes when
$k^{\prime}>4g^{\prime}-4+n^{\prime}$. Finally,
$H^{k^{\prime}}(\mathcal{M}_{g^{\prime},n^{\prime}})$ is tautological for
$k^{\prime}\leq\frac{2g^{\prime}-2}{3}$ [35]. ∎

##  4\. The Chow–Künneth generation Property

In this section we prove a key lemma about the cycle class map for spaces that
have the following property.

######  Definition 4.1.

Let $X$ be a smooth algebraic stack of finite type over a field, stratified by
quotient stacks. We say that $X$ has the _Chow–Künneth generation Property_
(CKgP) if for all algebraic stacks $Y$ (of finite type, stratified by quotient
stacks) the exterior product

$A_{*}(X)\otimes A_{*}(Y)\rightarrow A_{*}(X\times Y)$

is surjective.

For convenience, we record here several properties of the CKgP, all of which
are proven in [6, Section 3.1].

######  Proposition 4.2.

Let $X$ be a smooth algebraic stack of finite type over a field, stratified by
quotient stacks.

  1. (1)

if $U\subset X$ is open and $X$ has the CKgP, then $U$ has the CKgP;

  2. (2)

if $Y\rightarrow X$ is proper, surjective, representable by DM stacks, and $Y$
has the CKgP, then $X$ has the CKgP;

  3. (3)

if $X$ admits a finite stratification $X=\coprod_{S\in\mathcal{S}}S$ such that
each $S$ has the CKgP, then $X$ has the CKgP;

  4. (4)

if $V\rightarrow X$ is an affine bundle, then $V$ has the CKgP if and only if
$X$ has the CKgP;

  5. (5)

if $X$ has the CKgP, and $G\rightarrow X$ is a Grassmann bundle, then $G$ has
the CKgP;

  6. (6)

if $X=\operatorname{BGL}_{n},\operatorname{BSL}_{n}$, or
$\operatorname{BPGL}_{n}$, then $X$ has the CKgP.

If $X$ is smooth and proper and has the CKgP, then the cycle class map for $X$
is an isomorphism [6, Lemma 3.11].

When $X$ is smooth, but not necessarily proper (e.g., for
$X=\mathcal{M}_{g,n}$), the cycle class map is not necessarily an isomorphism.
Nevertheless, we have the following useful substitute (cf. [20, 33] for
slightly different statements with similar proofs).

######  Lemma 4.3.

Let $X$ be an open substack of a smooth proper Deligne–Mumford stack
$\overline{X}$ over the complex numbers. If $X$ has the CKgP, then the cycle
class map

$\operatorname{cl}\colon\bigoplus_{i}A_{i}(X)\rightarrow\bigoplus_{k}W_{k}H^{k}(X)$

is surjective. In particular, if $k$ is odd then $W_{k}H^{k}(X)=0$, and if $k$
is even then $W_{k}H^{k}(X)$ is pure Hodge–Tate.

###### Proof.

Set $d:=\dim\overline{X}$ and $D:=\overline{X}\smallsetminus X$. Let
$\Delta\subset\overline{X}\times\overline{X}$ denote the diagonal. Because $X$
has the CKgP, the exterior product map

$\bigoplus_{\ell=0}^{d}A^{\ell}(\overline{X})\otimes A^{d-\ell}(X)\rightarrow
A^{d}(\overline{X}\times X)$

is surjective. We have the excision exact sequence

$A^{d}(\overline{X}\times D)\rightarrow
A^{d}(\overline{X}\times\overline{X})\rightarrow A^{d}(\overline{X}\times
X)\rightarrow 0.$

It follows that we can write the class of the diagonal in
$A^{d}(\overline{X}\times\overline{X})$ as

(4.1) $\Delta=\Gamma+\Delta^{0}+\Delta^{1}+\cdots+\Delta^{d},$

where $\Gamma$ is supported on $\overline{X}\times D$ and each $\Delta^{\ell}$
is a linear combination of cycles of the form $V^{\ell}\times W^{d-\ell}$,
where $V^{\ell}$ and $W^{d-\ell}$ are subvarieties of $\overline{X}$ of
codimension $\ell$ and $d-\ell$, respectively.

Let $p_{1}$ and $p_{2}$ be the projections of $\overline{X}\times\overline{X}$
to the first and second factors respectively. Given a class $\Psi\in
H^{*}(\overline{X}\times\overline{X})$, we write $\Psi_{*}\colon
H^{*}(\overline{X})\to H^{*}(\overline{X})$ for the associated correspondence,
defined by $\Psi_{*}\alpha=p_{2*}(p_{1}^{*}\alpha\cdot\Psi)$.

Let $a\in W_{k}H^{k}(X)$. Let $\alpha$ be a lift of $a$ in
$W_{k}H^{k}(\overline{X})=H^{k}(\overline{X})$. Then

$\alpha=\Delta_{*}\alpha=(\Gamma_{*}+\Delta^{0}_{*}+\Delta^{1}_{*}+\cdots+\Delta^{d}_{*})\alpha.$

First, we study
$\Delta^{\ell}_{*}\alpha=p_{2*}(p_{1}^{*}\alpha\cdot\Delta^{\ell}).$ Note that
$p_{1}^{*}\alpha\cdot\Delta^{\ell}$ vanishes for dimension reasons if
$k+2\ell>2d$. Furthermore, since $p_{2}$ is of relative dimension $d$, the
pushforward by $p_{2}$ of any cycle will vanish if $k+2\ell<2d$. Thus, the
only non-zero terms occur when $k=2d-2\ell$. Moreover,
$\Delta^{\ell}_{*}\alpha$ is a linear combination of the form $\sum
a_{i}W^{d-\ell}$, so it lies in the image of the cycle class map.

Next, we study $\Gamma_{*}\alpha$. It suffices to treat the case that $\Gamma$
is the class of a subvariety of $\overline{X}\times\overline{X}$ contained in
$\overline{X}\times D$, as it is a linear combination of such subvarieties. In
this case, the map $p_{2}|_{\Gamma}\colon\Gamma\rightarrow\overline{X}$
factors through $D\rightarrow\overline{X}$. Thus, $\Gamma_{*}\alpha$ maps to
zero under the restriction to $W_{k}H^{k}(X)$ because the correspondence map
factors through the cohomology of the boundary $D$. Thus, $\alpha|_{X}=a$ is
in the image of the cycle class map. ∎

######  Remark 4.4.

Essentially the same argument (only with $\mathbb{Q}_{\ell}$-coefficients)
gives a similar statement for the cycle class map to $\ell$-adic étale
cohomology over an arbitrary field.

The first two authors have previously given many examples of of moduli spaces
$\mathcal{M}_{g,n}$ that have the CKgP and also satisfy
$A^{*}(\mathcal{M}_{g,n})=R^{*}(\mathcal{M}_{g,n})$ [6, Theorem 1.4]. For the
inductive arguments in this paper, we need more base cases in genus $7$. In
the next section, we prove that $\mathcal{M}_{7,n}$ has the CKgP and
$A^{*}(\mathcal{M}_{7,n})=R^{*}(\mathcal{M}_{7,n})$, for $n\leq 3$. The table
below records the previously known results from [6, Theorem 1.4] together with
Theorem 1.10.

$g$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---
$c(g)$ | $\infty$ | $10$ | $10$ | $11$ | $11$ | $7$ | $5$ | $3$
Table 1. $\mathcal{M}_{g,n}$ has the CKgP and
$A^{*}(\mathcal{M}_{g,n})=R^{*}(\mathcal{M}_{g,n})$, for $n\leq c(g)$, by [6,
Theorem 1.4] and Theorem 1.10.

The following is a consequence of Lemma 4.3.

######  Proposition 4.5.

For all $g\leq 7$ and $n\leq c(g)$ as specified in Table 1, we have

$W_{k}H^{k}(\mathcal{M}_{g,n})=RH^{k}(\mathcal{M}_{g,n}).$

In particular, if $k$ is odd and $n\leq c(g)$, then

$W_{k}H^{k}(\mathcal{M}_{g,n})=\mathrm{gr}^{W}_{k}H^{k}_{c}(\mathcal{M}_{g,n})=0.$

###### Proof of Theorem 1.9, assuming Theorem 1.1.

By the Hard Lefschetz theorem, it suffices to prove that
$H^{k}(\overline{\mathcal{M}}_{3,n})^{\mathrm{ss}}$ is a polynomial in
$\mathsf{L}$ and $\mathsf{S}_{12}$ for fixed $k\leq 17$ and $n\leq 11$.
Consider the right exact sequence

$H^{k-2}(\widetilde{\partial\mathcal{M}_{3,n}})\rightarrow
H^{k}(\overline{\mathcal{M}}_{3,n})\rightarrow
W_{k}H^{k}(\mathcal{M}_{3,n})\rightarrow 0.$

By Theorem 1.1 and [8, Theorems 1.1, 1.2 and Lemma 2.1], for $k^{\prime}\leq
15$,
$H^{k^{\prime}}(\overline{\mathcal{M}}_{g^{\prime},n^{\prime}})^{\mathrm{ss}}$
is a polynomial in $\mathsf{L}$ and $\mathsf{S}_{12}$ for all $g^{\prime}\neq
1$ and all $n^{\prime}$ as well as for $g^{\prime}=1$ and $n^{\prime}\leq 14$.
It follows that,
$H^{k-2}(\widetilde{\partial\mathcal{M}_{3,n}})^{\mathrm{ss}}$ is a polynomial
in $\mathsf{L}$ and $\mathsf{S}_{12}$. By [6, Theorem 1.4],
$\mathcal{M}_{3,n}$ has the CKgP and
$A^{*}(\mathcal{M}_{3,n})=R^{*}(\mathcal{M}_{3,n})$ for $n\leq 11$. Thus
$W_{k}H^{k}(\mathcal{M}_{3,n})=RH^{k}(\mathcal{M}_{3,n})$ by Lemma 4.3. Hence,
$W_{k}H^{k}(\mathcal{M}_{3,n})$ is pure Hodge–Tate, and the theorem follows. ∎

##  5\. The CKgP in genus 7 with at most $3$ marked points

In order to prove Theorems 1.1, 1.5(3) for $k=14$, and 1.7, we need more base
cases in genus $7$. The required base cases are given by Theorem 1.10, which
we now prove.

In order to prove Theorem 1.10, we filter $\mathcal{M}_{7,n}$ by gonality:

$\mathcal{M}^{2}_{7,n}\subset\mathcal{M}^{3}_{7,n}\subset\mathcal{M}^{4}_{7,n}\subset\mathcal{M}^{5}_{7,n}.$

Here, $\mathcal{M}^{k}_{7,n}$ is the locus parametrizing smooth curves $C$
with $n$ marked points that admit a map of degree at most $k$ to
$\mathbb{P}^{1}$. Standard results from Brill–Noether theory show that the
maximal gonality of a genus $7$ curve is $5$. By Proposition 4.2(3), to show
that $\mathcal{M}_{7,n}$ has the CKgP, it suffices to show that each gonality
stratum

$\mathcal{M}^{k}_{7,n}\smallsetminus\mathcal{M}^{k-1}_{7,n}.$

has the CKgP. Moreover, to show that
$A^{*}(\mathcal{M}_{7,n})=R^{*}(\mathcal{M}_{7,n})$, it suffices to show for
each $k$ that all classes supported on $\mathcal{M}^{k}_{7,n}$ are
tautological up to classes supported on $\mathcal{M}^{k-1}_{7,n}$. In other
words, we must show that every class in
$A^{*}(\mathcal{M}_{7,n}^{k}\smallsetminus\mathcal{M}_{7,n}^{k-1})$ pushes
forward to a class in
$A^{*}(\mathcal{M}_{7,n}\smallsetminus\mathcal{M}^{k-1}_{7,n})$ that is the
restriction of a tautological class on $\mathcal{M}_{7,n}$.

###  5.1. Hyperelliptic and trigonal loci

By [6, Lemma 9.9], if $n\leq 14$, then $\mathcal{M}^{3}_{7,n}$ has the CKgP
and all classes in $A^{*}(\mathcal{M}_{7,n})$ supported on
$\mathcal{M}^{3}_{7,n}$ are tautological. Note that this includes the
hyperelliptic locus.

###  5.2. The tetragonal locus

To study the tetragonal locus
$\mathcal{M}^{4}_{7,n}\smallsetminus\mathcal{M}^{3}_{7,n}$, we will use the
Hurwitz stack $\mathcal{H}_{4,g,n}$ parametrizing degree $4$ covers $f\colon
C\rightarrow\mathbb{P}^{1}$, where $C$ is a smooth curve of genus $g$ with $n$
marked points. There is a forgetful morphism

$\beta_{n}\colon\mathcal{H}_{4,g,n}\rightarrow\mathcal{M}_{g,n}.$

Restricting to curves of gonality exactly $4$, we obtain a proper morphism

$\beta_{n}^{\prime}\colon\mathcal{H}_{4,g,n}^{\Diamond}:=\mathcal{H}_{4,g,n}\smallsetminus\beta_{n}^{-1}(\mathcal{M}^{3}_{g,n})\rightarrow\mathcal{M}_{g,n}\smallsetminus\mathcal{M}^{3}_{g,n}$

with image $\mathcal{M}_{g,n}^{4}\smallsetminus\mathcal{M}_{g,n}^{3}$. To show
that $\mathcal{M}^{4}_{7,n}\smallsetminus\mathcal{M}^{3}_{7,n}$ has the CKgP,
it suffices to show that $\mathcal{H}_{4,7,n}^{\Diamond}$ has the CKgP by
Proposition 4.2(2). We will do so by further stratifying
$\mathcal{H}_{4,7,n}^{\Diamond}$.

In [5, Section 4.4], the first two authors studied a stratification of
$\mathcal{H}_{4,7}:=\mathcal{H}_{4,7,0}$ with no markings. Here we carry out a
similar analysis with marked points. The Casnati–Ekedahl structure theorem [9]
associates to a point in $\mathcal{H}_{4,g}$ a rank $3$ vector bundle $E$ and
a rank $2$ vector bundle $F$ on $\mathbb{P}^{1}$, both of degree $g+3$,
equipped with a canonical isomorphism $\det E\cong\det F$ [7, Section 3]. Let
$\mathcal{B}$ be the moduli stack of pairs of vector bundles $(E,F)$ on
$\mathbb{P}^{1}$ of degree $g+3$, together with an isomorphism of their
determinants as in [7, Definition 5.2]. Let
$\pi\colon\mathcal{P}\to\mathcal{B}$ be the universal
$\mathbb{P}^{1}$-fibration, and let $\mathcal{E}$ and $\mathcal{F}$ be the
universal bundles on $\mathcal{P}$. There is a natural morphism
$\mathcal{H}_{4,g}\to\mathcal{B}$ that sends a degree $4$ cover to its
associated pair of vector bundles. Moreover, the Casnati–Ekedahl construction
gives an embedding of the universal curve $\mathcal{C}$ over
$\mathcal{H}_{4,g}$ into $\mathbb{P}\mathcal{E}^{\vee}$.

Consider the natural commutative diagram

(5.1)
${\mathcal{C}_{n}}$${\mathcal{C}}$${\mathbb{P}\mathcal{E}^{\vee}}$${\mathcal{H}_{4,g,n}}$${\mathcal{H}_{4,g}}$${\mathcal{B},}$$\scriptstyle{b}$$\scriptstyle{a}$$\scriptstyle{\sigma_{i}}$

where $\mathcal{C}_{n}$ is the universal curve over $\mathcal{H}_{4,g,n}$. For
each $i$, the map $a\circ b\circ\sigma_{i}$ sends a pointed curve to the image
of the $i$th marking under the Casnati–Ekedahl embedding. Taking the product
of these maps for $i=1,\ldots,n$, we obtain a commutative diagram

(5.2)
${\mathcal{H}_{4,g,n}^{\Diamond}}$${\mathcal{H}_{4,g,n}}$${(\mathbb{P}\mathcal{E}^{\vee})^{n}}$${\mathcal{H}_{4,g}^{\Diamond}}$${\mathcal{H}_{4,g}}$${\mathcal{B}.}$

######  Lemma 5.1 (Lemma 10.5 of [6]).

Suppose $x\in A^{*}(\mathcal{H}_{4,g,n}^{\Diamond})$ lies in the image of the
map $A^{*}((\mathbb{P}\mathcal{E}^{\vee})^{n})\to
A^{*}(\mathcal{H}_{4,g,n}^{\Diamond})$. Then $\beta^{\prime}_{n*}x$ is
tautological on $\mathcal{M}_{g,n}\smallsetminus\mathcal{M}_{g,n}^{3}$.

The splitting types of the Casnati–Ekedahl bundles $E$ and $F$ induce a
stratification on $\mathcal{H}_{4,7}$. We write $E=(e_{1},e_{2},e_{3})$ and
$F=(f_{1},f_{2})$ to indicate that the bundles have splitting types

$E=\mathcal{O}(e_{1})\oplus\mathcal{O}(e_{2})\oplus\mathcal{O}(e_{3})\qquad\text{and}\qquad
F=\mathcal{O}(f_{1})\oplus\mathcal{O}(f_{2}).$

We will consider a stratification into three pieces

$\mathcal{H}_{4,7,n}=X_{n}\sqcup Y_{n}\sqcup Z_{n}.$

The three strata correspond to unions of splitting types of $E$ and $F$. The
possible splitting types are recorded in [5, Section 4.4]. The locus $Z_{n}$
is the set of covers with maximally unbalanced splitting types and
parametrizes hyperelliptic curves [5, Equation 4.5]. Its image in
$\mathcal{M}_{7,n}$ is contained in $\mathcal{M}_{7,n}^{3}$, which has the
CKgP when $n\leq 14$, as noted above. We will show that $X_{n}$ and $Y_{n}$
have the CKgP and that $A^{*}((\mathbb{P}\mathcal{E}^{\vee})^{n})$ surjects
onto the Chow ring of their union, which is $\mathcal{H}_{4,7,n}^{\Diamond}$.
We start with $X_{n}$.

Let $X_{n}\subset\mathcal{H}_{4,7,n}$ denote the locus of covers with
splitting types $E=(3,3,4)$ and $F=(5,5)$, or $E=(3,3,4)$ and $F=(4,6)$.

######  Lemma 5.2.

If $n\leq 3$, then $X_{n}$ has the CKgP and
$A^{*}((\mathbb{P}\mathcal{E}^{\vee})^{n})\to A^{*}(X_{n})$ is surjective.

###### Proof.

Taking $f=4$ in [6, Definition 10.8], we have $X_{n}=\mathcal{H}_{4,7,n}^{4}$.
The result then follows from [6, Lemmas 10.11 and 10.12] with $g=7$ and
$f=f_{1}=4$. ∎

Now let $Y_{n}\subset\mathcal{H}_{4,7,n}$ be the union

$Y_{n}=\Sigma_{2,n}\sqcup\Sigma_{3,n},$

where $\Sigma_{2,n}$ parametrizes covers with splitting types $E=(2,4,4)$ and
$F=(4,6)$, and $\Sigma_{3,n}$ parametrizes covers with splitting types
$E=(2,3,5)$ and $F=(4,6)$.

Recall that $\pi\colon\mathcal{P}\rightarrow\mathcal{B}$ is the structure map
for the $\mathbb{P}^{1}$ bundle $\mathcal{P}$. Let
$\gamma\colon\mathbb{P}\mathcal{E}^{\vee}\to\mathcal{P}$ denote the structure
map and
$\eta_{i}\colon(\mathbb{P}\mathcal{E}^{\vee})^{n}\to\mathbb{P}\mathcal{E}^{\vee}$
denote the $i$th projection. Define

$z_{i}:=\eta_{i}^{*}\gamma^{*}c_{1}(\mathcal{O}_{\mathcal{P}}(1))\mbox{ \ \
and \ \
}\zeta_{i}:=\eta_{i}^{*}c_{1}(\mathcal{O}_{\mathbb{P}\mathcal{E}^{\vee}}(1)).$

Then $z_{i}$ and $\zeta_{i}$ generate
$A^{*}((\mathbb{P}\mathcal{E}^{\vee})^{n})$ as an algebra over
$A^{*}(\mathcal{B})$. Write $\Sigma_{\ell}:=\Sigma_{\ell,0}$.

######  Lemma 5.3.

For $\ell=2,3$ and $n\leq 3$, there is a surjection

$A^{*}(\Sigma_{\ell})[z_{1},\ldots,z_{n},\zeta_{1},\ldots,\zeta_{n}]\to
A^{*}(\Sigma_{\ell,n})$

induced by $\Sigma_{\ell,n}\rightarrow\Sigma_{\ell}$ and restriction from
$(\mathbb{P}\mathcal{E}^{\vee})^{n}$. Moreover, $\Sigma_{\ell,n}$ has the
CKgP.

###### Proof.

Let $\vec{e}$ and $\vec{f}$ be the splitting types associated to
$\Sigma_{\ell}$ and let $\mathcal{B}_{\vec{e},\vec{f}}\subset\mathcal{B}$ be
the locally closed substack that parametrizes pairs of bundles $(E,F)$ with
locally constant splitting types $\vec{e}$ and $\vec{f}$. We write
$\mathcal{O}(\vec{e}):=\mathcal{O}(e_{1})\oplus\cdots\oplus\mathcal{O}(e_{k})$.
By construction, $\Sigma_{\ell,n}$ is the preimage of
$\mathcal{B}_{\vec{e},\vec{f}}$ along $\mathcal{H}_{4,7,n}\to\mathcal{B}$. We
therefore study the base change of (5.2) along
$\mathcal{B}_{\vec{e},\vec{f}}\to\mathcal{B}$:

(5.3)
${\Sigma_{\ell,n}}$${(\mathbb{P}\mathcal{E}^{\vee}|_{\mathcal{B}_{\vec{e},\vec{f}}})^{n}}$${\Sigma_{\ell}}$${\mathcal{B}_{\vec{e},\vec{f}}.}$

We now recall the description of $\Sigma_{\ell}$ as an open substack of a
vector bundle on $\mathcal{B}_{\vec{e},\vec{f}}$, as in [5, Lemma 3.10]. Let

$U\subset
H^{0}(\mathbb{P}^{1},\mathcal{O}(\vec{f})^{\vee}\otimes\operatorname{Sym}^{2}\mathcal{O}(\vec{e}))=H^{0}(\mathbb{P}\mathcal{O}(\vec{e})^{\vee},\gamma^{*}\mathcal{O}(\vec{f})^{\vee}\otimes\mathcal{O}_{\mathbb{P}\mathcal{O}(\vec{e})^{\vee}}(2))$

be the open subset of equations that define a smooth curve, as in [5, Lemma
3.10]. Then

$\Sigma_{\ell}=[(U\times\mathbb{G}_{m})/\operatorname{SL}_{2}\ltimes(\operatorname{Aut}(\mathcal{O}(\vec{e}))\times\operatorname{Aut}(\mathcal{O}(\vec{f})))].$

The stack $\mathcal{B}_{\vec{e},\vec{f}}$ is the part obtained by forgetting
$U$:

$\mathcal{B}_{\vec{e},\vec{f}}=[\mathbb{G}_{m}/\operatorname{SL}_{2}\ltimes(\operatorname{Aut}(\mathcal{O}(\vec{e}))\times\operatorname{Aut}(\mathcal{O}(\vec{f})))].$

As explained in [5, Equation 3.1], there is a product of stacks
$\operatorname{BGL}_{n}$ which is an affine bundle over
$\mathrm{B}\\!\operatorname{Aut}(\mathcal{O}(\vec{e}))$. As such,
$\mathcal{B}_{\vec{e},\vec{f}}$ has CKgP by Proposition 4.2(4) and (6). It
follows that $\Sigma_{\ell}$ also has the CKgP by Proposition 4.2(1) and (4).

Let us define the rank $2$ vector bundle
$\mathcal{W}:=\gamma^{*}\mathcal{F}\otimes\mathcal{O}_{\mathbb{P}\mathcal{E}^{\vee}}(2)$
on $\mathbb{P}\mathcal{E}^{\vee}$. Write $\mathcal{W}_{\vec{e},\vec{f}}$ for
the restriction of $\mathcal{W}$ to
$\mathbb{P}\mathcal{E}^{\vee}|_{\mathcal{B}_{\vec{e},\vec{f}}}$. The
discussion above says that $\Sigma_{\ell}$ is an open substack of the vector
bundle $(\pi\circ\gamma)_{*}\mathcal{W}_{\vec{e},\vec{f}}$ on
$\mathcal{B}_{\vec{e},\vec{f}}$.

Next, we give a similar description with marked points. Consider the
evaluation map

$(\pi\circ\gamma)^{*}(\pi\circ\gamma)_{*}\mathcal{W}_{\vec{e},\vec{f}}\rightarrow\mathcal{W}_{\vec{e},\vec{f}}.$

Pulling back to the fiber product
$(\mathbb{P}\mathcal{E}^{\vee}|_{\mathcal{B}_{\vec{e},\vec{f}}})^{n}$, we
obtain

(5.4)
$\eta_{j}^{*}(\pi\circ\gamma)^{*}(\pi\circ\gamma)_{*}\mathcal{W}_{\vec{e},\vec{f}}\rightarrow\bigoplus_{i=1}^{n}\eta_{i}^{*}\mathcal{W}_{\vec{e},\vec{f}}.$

Note that
$\eta_{j}^{*}(\pi\circ\gamma)^{*}(\pi\circ\gamma)_{*}\mathcal{W}_{\vec{e},\vec{f}}$
is independent of $j$. The kernel $\mathcal{Y}$ of (5.4) parametrizes tuples
of $n$ points on $\mathbb{P}\mathcal{O}(\vec{e})^{\vee}$ together with a
section of

$H^{0}(\mathbb{P}\mathcal{O}(\vec{e})^{\vee},\gamma^{*}\mathcal{O}(\vec{f})^{\vee}\otimes\mathcal{O}_{\mathbb{P}\mathcal{O}(\vec{e})^{\vee}}(2))$

that vanishes on the points. There is therefore a natural map
$\Sigma_{\ell,n}\hookrightarrow\mathcal{Y}$ defined by sending a pointed curve
to the images of the points on $\mathbb{P}\mathcal{O}(\vec{e})^{\vee}$ and the
equation of the curve. The kernel $\mathcal{Y}$ is not locally free.
Nevertheless, its restriction to the open substack
$U_{n}\subset(\mathbb{P}\mathcal{E}^{\vee}|_{\mathcal{B}_{\vec{e},\vec{f}}})^{n}$
where (5.4) is surjective is locally free.

We claim that the map $\Sigma_{\ell,n}\to\mathcal{Y}$ factors through
$\mathcal{Y}|_{U_{n}}$. To see this, suppose
$C\subset\mathbb{P}\mathcal{O}(\vec{e})^{\vee}$ is the vanishing of a section
of
$\gamma^{*}\mathcal{O}(\vec{f})\otimes\mathcal{O}_{\mathbb{P}\mathcal{O}(\vec{e})^{\vee}}(2)$
and $C$ is smooth and irreducible. By [6, Lemma 10.6], if $n\leq 3$, the
evaluation map (5.4) is surjective at any tuple of $n$ distinct points on $C$.
It follows that the image of the composition
$\Sigma_{\ell,n}\rightarrow\mathcal{Y}\rightarrow(\mathbb{P}\mathcal{E}^{\vee}|_{\mathcal{B}_{\vec{e},\vec{f}}})^{n}$
is contained in $U_{n}$. Hence,
$\Sigma_{\ell,n}\hookrightarrow\mathcal{Y}|_{U_{n}}$.

In summary, we have a sequence of maps

$\Sigma_{\ell,n}\hookrightarrow\mathcal{Y}|_{U_{n}}\rightarrow
U_{n}\hookrightarrow(\mathbb{P}\mathcal{E}^{\vee}|_{\mathcal{B}_{\vec{e},\vec{f}}})^{n}\rightarrow(\mathcal{P}|_{\mathcal{B}_{\vec{e},\vec{f}}})^{n}\rightarrow\mathcal{B}_{\vec{e},\vec{f}}.$

each of which is an open inclusion, vector bundle, or product of projective
bundles. Since $\mathcal{B}_{\vec{e},\vec{f}}$ has the CKgP, it follows that
$\Sigma_{\ell,n}$ also has the CKgP by Proposition 4.2(1),(4), and (5).
Moreover, we see that

$A^{*}(\mathcal{B}_{\vec{e},\vec{f}})[z_{1},\ldots,z_{n},\zeta_{1},\ldots,\zeta_{n}]\to
A^{*}(\Sigma_{\ell,n})$

is surjective. Finally note that
$\Sigma_{\ell,n}\to\mathcal{B}_{\vec{e},\vec{f}}$ factors through
$\Sigma_{\ell}$, so $A^{*}(\mathcal{B}_{\vec{e},\vec{f}})\to
A^{*}(\Sigma_{\ell,n})$ factors through $A^{*}(\Sigma_{\ell})$. This proves
the claim. ∎

######  Corollary 5.4.

For $n\leq 3$, $\mathcal{H}_{4,7,n}^{\Diamond}$ has the CKgP. Hence,
$\mathcal{M}^{4}_{7,n}$ has the CKgP.

###### Proof.

We have $\mathcal{H}_{4,7,n}^{\Diamond}=X_{n}\sqcup Y_{n}$. By Lemmas 5.2 and
5.3, each of these pieces has the CKgP. Note that
$\mathcal{H}^{\Diamond}_{4,7,n}$ maps properly onto
$\mathcal{M}^{4}_{7,n}\smallsetminus\mathcal{M}^{3}_{7,n}$ and
$\mathcal{M}_{7,n}^{3}$ has the CKgP. Thus the result follows by Proposition
4.2(2)-(3). ∎

The next step is to show that all classes in $A^{*}(\Sigma_{\ell,n})$ are
restrictions from $(\mathbb{P}\mathcal{E}^{\vee})^{n}$.

######  Lemma 5.5.

For $\ell=2,3$ and $n\leq 3$, the restriction
$A^{*}((\mathbb{P}\mathcal{E}^{\vee})^{n})\to A^{*}(\Sigma_{\ell,n})$ is
surjective.

###### Proof.

Consider the following diagram:

(5.5)
${A^{*}((\mathbb{P}\mathcal{E}^{\vee})^{n})}$${A^{*}(\Sigma_{\ell,n})}$${A^{*}(\mathcal{B})[z_{1},\ldots,z_{n},\zeta_{1},\ldots,\zeta_{n}]}$${A^{*}(\Sigma_{\ell})[z_{1},\ldots,z_{n},\zeta_{1},\ldots,\zeta_{n}].}$

Note that the map $A^{*}(\mathcal{B})\to A^{*}(\Sigma_{\ell})$ is surjective
by [5, Lemma 4.2] for $\ell=2$, and by [5, Lemma 4.3(1)] for $\ell=3$.
Therefore, the bottom horizontal arrow is surjective. By Lemma 5.3, the right
vertical arrow is also surjective. Hence, the top horizontal arrow is
surjective. ∎

######  Lemma 5.6.

For $n\leq 3$, the pullback map $A^{*}((\mathbb{P}\mathcal{E}^{\vee})^{n})\to
A^{*}(\mathcal{H}_{4,7,n}^{\Diamond})$ is surjective.

###### Proof.

First, we fix some notation. Let

$\Sigma_{3,n}\xhookrightarrow{\jmath}Y_{n}\xhookrightarrow{\iota}\mathcal{H}_{4,7,n}^{\Diamond}$

denote the natural closed inclusion maps. Let
$\phi\colon\mathcal{H}_{4,7,n}^{\Diamond}\to(\mathbb{P}\mathcal{E}^{\vee})^{n}$
be the map from the top left to top right in diagram (5.1). We let
$\phi^{\prime}:X_{n}\rightarrow(\mathbb{P}\mathcal{E}^{\vee})^{n}$ be
composite of the open inclusion
$X_{n}\hookrightarrow\mathcal{H}_{4,7,n}^{\Diamond}$ and $\phi$. Let
$\psi:=\phi\circ\iota$, and let $\psi^{\prime}$ be the composite of the open
inclusion $\Sigma_{2,n}\hookrightarrow Y_{n}$ and $\psi$.

Consider the following commutative diagram, where the bottom row is exact.

(5.6)
${A^{*}((\mathbb{P}\mathcal{E}^{\vee})^{n})}$${A^{*-2}(Y_{n})}$${A^{*}(\mathcal{H}_{4,7,n}^{\Diamond})}$${A^{*}(X_{n})}$${0.}$$\scriptstyle{\phi^{*}}$$\scriptstyle{\phi^{\prime*}}$$\scriptstyle{\iota_{*}}$

By Lemma 5.2, $\phi^{\prime*}$ is surjective. It thus suffices to show that
the image of $\iota_{*}$ is contained in the image of $\phi^{*}$. To do so, we
consider another commutative diagram where the bottom row is exact.

(5.7)
${A^{*}((\mathbb{P}\mathcal{E}^{\vee})^{n})}$${A^{*-1}(\Sigma_{3,n})}$${A^{*}(Y_{n})}$${A^{*}(\Sigma_{2,n})}$${0.}$$\scriptstyle{\psi^{*}}$$\scriptstyle{\psi^{\prime*}}$$\scriptstyle{\jmath_{*}}$

By Lemma 5.5, $\psi^{\prime*}$ is surjective. Moreover, by the projection
formula and Lemma 5.5, the image of $\jmath$ is generated as an
$A^{*}((\mathbb{P}\mathcal{E}^{\vee})^{n})$ module by the fundamental class
$[\Sigma_{3,n}]\in A^{*}(Y_{n})$. Therefore, any class $\alpha\in
A^{*}(Y_{n})$ can be written as

$\alpha=\psi^{*}\alpha_{0}+[\Sigma_{3,n}]\psi^{*}\alpha_{1}=\iota^{*}\phi^{*}\alpha_{0}+[\Sigma_{3,n}]\iota^{*}\phi^{*}\alpha_{1},$

where $\alpha_{i}\in A^{*}((\mathbb{P}\mathcal{E}^{\vee})^{n})$. By the
projection formula,

$\iota_{*}\alpha=[Y_{n}]\phi^{*}\alpha_{0}+[\Sigma_{3,n}]\phi^{*}\alpha_{1},$

where now the fundamental class $[\Sigma_{3,n}]$ is a class on
$\mathcal{H}^{\Diamond}_{4,7,n}$. It thus suffices to show that the classes
$[Y_{n}]$ and $[\Sigma_{3,n}]$ are in the image of $\phi^{*}$.

By [5, Lemma 4.8], $[\overline{\Sigma_{\ell}}]$ is in the image of
$A^{*}(\mathcal{B})\to A^{*}(\mathcal{H}_{4,7}^{\Diamond})$. Because
$[\Sigma_{3,n}]$ is the pullback of $[\Sigma_{3}]$ along
$A^{*}(\mathcal{H}_{4,7}^{\Diamond})\to A^{*}(\mathcal{H}_{4,7,n}^{\Diamond})$
and $[Y_{n}]$ is the pullback of $[\overline{\Sigma_{2}}]$, both
$[\Sigma_{3,n}]$ and $[Y_{n}]$ are in the image of $A^{*}(\mathcal{B})\to
A^{*}(\mathcal{H}_{4,7,n}^{\Diamond})$. Hence, they are in the image of
$\phi^{*}$. ∎

Recall that proper, surjective maps induce surjective maps on rational Chow
groups. Since the map
$\beta_{n}^{\prime}\colon\mathcal{H}_{4,7,n}^{\Diamond}\to\mathcal{M}_{7,n}\smallsetminus\mathcal{M}_{7,n}^{3}$
is proper with image
$\mathcal{M}_{7,n}^{4}\smallsetminus\mathcal{M}_{7,n}^{3}$, every class
supported on the tetragonal locus is the pushforward of a class from
$\mathcal{H}_{4,7,n}^{\Diamond}$. Combining Lemmas 5.1 and 5.6 therefore
proves the following.

######  Lemma 5.7.

If $n\leq 3$, then all classes supported on
$\mathcal{M}_{7,n}^{4}\smallsetminus\mathcal{M}_{7,n}^{3}$ are tautological.

###  5.3. The pentagonal locus

It remains to study the locus
$\mathcal{M}_{7,n}^{\circ}=\mathcal{M}_{7,n}\smallsetminus\mathcal{M}_{7,n}^{4}$
of curves of gonality exactly $5$. Mukai showed that every curve in
$\mathcal{M}_{7}^{\circ}$ is realized as a linear section of the orthogonal
Grassmannian in its spinor embedding
$\operatorname{OG}(5,10)\hookrightarrow\mathbb{P}^{15}$ [21]. To take
advantage of this construction, we first develop a few lemmas about the
orthogonal Grassmannian.

####  5.3.1. The orthogonal Grassmannian

Let $\mathcal{V}$ be the universal rank $10$ bundle on
$\operatorname{BSO}_{10}$. The universal orthogonal Grassmannian is the
quotient stack $[\operatorname{OG}(5,10)/\mathrm{SO}_{10}]$, which we think of
as the orthogonal Grassmann bundle with structure map
$\pi\colon\operatorname{OG}(5,\mathcal{V})\to\operatorname{BSO}_{10}$. By
construction, the pullback of $\mathcal{V}$ along $\pi$ satisfies
$\pi^{*}\mathcal{V}=\mathcal{U}\oplus\mathcal{U}^{\vee}$ where $\mathcal{U}$
is the universal rank $5$ subbundle on $\operatorname{OG}(5,\mathcal{V})$.

######  Lemma 5.8.

The stack

$\operatorname{OG}(5,\mathcal{V})\cong[\operatorname{OG}(5,10)/\mathrm{SO}_{10}]$

has the CKgP. Moreover, its Chow ring is freely generated by the Chern classes
of $\mathcal{U}$.

###### Proof.

Let $V=\mathrm{span}\\{e_{1},\ldots,e_{10}\\}$ be a fixed $10$-dimensional
vector space with quadratic form

$Q=\left(\begin{matrix}0&I_{5}\\\ I_{5}&0\end{matrix}\right).$

Let $U=\mathrm{span}\\{e_{1},\ldots,e_{5}\\}$, which is an isotropic subspace.
The stabilizer of $\mathrm{SO}_{10}$ acting on $\operatorname{OG}(5,10)$ at
$U$ is

$\mathrm{Stab}_{U}=\left\\{M=\left(\begin{matrix}A&B\\\
0&D\end{matrix}\right):M^{T}QM=Q\right\\}\subset\mathrm{SO}_{10}.$

Expanding, we have

$M^{T}QM=\left(\begin{matrix}A^{T}&0\\\
B^{T}&D^{T}\end{matrix}\right)\left(\begin{matrix}0&I_{5}\\\
I_{5}&0\end{matrix}\right)\left(\begin{matrix}A&B\\\
0&D\end{matrix}\right)=\left(\begin{matrix}0&A^{T}D\\\
D^{T}A&B^{T}D+D^{T}B\end{matrix}\right).$

Thus, $\mathrm{Stab}_{U}$ is defined by the conditions $D=(A^{T})^{-1}$ and
$B^{T}D+D^{T}B=0$.

Note that $\mathrm{Stab}_{U}$ is a maximal parabolic subgroup and
$\operatorname{OG}(5,10)=\mathrm{SO}_{10}/\mathrm{Stab}_{U}$. As such, the
quotient $[\operatorname{OG}(5,10)/\mathrm{SO}_{10}]$ is equivalent to the
classifying stack $\mathrm{BStab}_{U}$. To gain a better understanding of the
latter, consider the group homomorphism

$\operatorname{GL}_{5}\hookrightarrow\mathrm{Stab}_{U},\qquad
A\mapsto\left(\begin{matrix}A&0\\\ 0&(A^{T})^{-1}\end{matrix}\right).$

For fixed $D$, the condition $B^{T}D+D^{T}B=0$ is linear in $B$. Specifically,
it says that, $B$ lies in the $(D^{T})^{-1}$ translation of the
$\mathbb{A}^{10}$ of skew symmetric $5\times 5$ matrices. In particular, the
cosets of the subgroup $\operatorname{GL}_{5}\hookrightarrow\mathrm{Stab}_{U}$
are isomorphic to affine spaces $\mathbb{A}^{10}$. In other words, the induced
map on classifying spaces
$\operatorname{BGL}_{5}\to\mathrm{BStab}_{U}\cong\operatorname{OG}(5,\mathcal{V})$
is an affine bundle. It follows that $\operatorname{OG}(5,\mathcal{V})$ has
the CKgP by Proposition 4.2(4) and (6).

Furthermore, by construction, the tautological subbundle $\mathcal{U}$ on
$\operatorname{OG}(5,\mathcal{V})$ pulls back to the tautological rank $5$
bundle on $\operatorname{BGL}_{5}$. It follows that
$A^{*}(\mathrm{BStab}_{U})\cong A^{*}(\mathrm{BGL}_{5})$, and is freely
generated by the Chern classes of the tautological bundle. ∎

######  Remark 5.9.

There is also a natural map $\mathrm{Stab}_{U}\to\operatorname{GL}_{5}$ that
sends $\left(\begin{matrix}A&B\\\ 0&D\end{matrix}\right)$ to $A$. The kernel
of $\mathrm{Stab}_{U}\to\operatorname{GL}_{5}$ is the subgroup
$G\cong(\mathbb{G}_{a})^{10}$ where $A=I_{5},D=I_{5}$ and $B+B^{T}=0$. This
shows that $\mathrm{Stab}_{U}$ is actually a semi-direct product
$G\rtimes\operatorname{GL}_{5}$. The map
$\mathrm{BStab}_{U}\to\operatorname{BGL}_{5}$ is a $\mathrm{B}G$-banded gerbe.

######  Lemma 5.10.

Let $V$ be a rank $2\nu$ vector bundle with quadratic form on $X$.

  1. (1)

The rational Chow ring of $\operatorname{OG}(\nu,V)$ is generated over the
Chow ring of $X$ by the Chern classes of the tautological subbundle.

  2. (2)

If $X$ has the CKgP, then $\operatorname{OG}(\nu,V)$ has the CKgP.

###### Proof.

The argument is very similar to that for Grassmannians in type A.

First consider the case when $X$ is a point. The orthogonal Grassmannian
$\operatorname{OG}(\nu,2\nu)$ is stratified by Schubert cells, each of which
is isomorphic to an affine space. By Proposition 4.2(3), it follows that
$\operatorname{OG}(\nu,2\nu)$ has the CKgP. Moreover, the fundamental classes
of these cells are expressed in terms of the Chern classes of the tautological
quotient or subbundle via a Giambelli formula [19, p. 1–2]. Note that this
formula involves dividing by $2$, so it is important that we work with
rational coefficients for this claim. In conclusion, the Chern classes of the
tautological subbundle generate the Chow ring of
$\operatorname{OG}(\nu,2\nu)$.

More generally for a fiber bundle $f\colon\operatorname{OG}(\nu,V)\to X$, to
prove (1), we stratify $X$ into locally closed subsets $X_{i}$ over which $V$
is trivial, such that $\overline{X}_{i}\supset X_{j}$ for $i\geq j$ and
$X_{i}$ has codimension at least $i$. To check that the desired classes
generate $A^{k}(\operatorname{OG}(\nu,V))$ for a given $k$, it suffices to
show that they generate $A^{k}(f^{-1}(X_{0}\cup X_{1}\cup\cdots\cup X_{k}))$.

Over each piece of the stratification,
$f^{-1}(X_{i})=X_{i}\times\operatorname{OG}(\nu,2\nu)$. Since
$\operatorname{OG}(\nu,2\nu)$ has the CKgP, the Chow ring of
$f^{-1}(X_{i})=X_{i}\times\operatorname{OG}(\nu,2\nu)$ is generated by
$A^{*}(X_{i})$ and restrictions of the Chern classes from the tautological
subbundle on $\operatorname{OG}(\nu,V)$. By excision and the push-pull
formula, the Chow ring of any finite union $f^{-1}(X_{0})\cup
f^{-1}(X_{1})\cup\cdots\cup f^{-1}(X_{k})$ is generated by the desired
classes.

Finally (2) follows from (1) exactly as in [6, Lemma 3.7]. ∎

######  Corollary 5.11.

For any $n\geq 1$, the $n$-fold fiber product

$\operatorname{OG}(5,\mathcal{V})^{n}:=\operatorname{OG}(5,\mathcal{V})\times_{\operatorname{BSO}_{10}}\cdots\times_{\operatorname{BSO}_{10}}\operatorname{OG}(5,\mathcal{V})$

has the CKgP and its Chow ring is generated by the Chern classes of the
tautological subbundles $\mathcal{U}_{1},\ldots,\mathcal{U}_{n}$ pulled back
from each factor.

###### Proof.

The case $n=1$ follows from Lemma 5.8. For $n>1$, the $n$-fold fiber product
is an orthogonal Grassmann bundle over the $(n-1)$-fold fiber product, so the
claim follows from Lemma 5.10. ∎

######  Remark 5.12.

By Proposition 4.2(2), the fact that $\operatorname{OG}(5,\mathcal{V})$ has
the CKgP implies that $\operatorname{BSO}_{10}$ also has the CKgP. It should
be possible to show that $\operatorname{BSO}_{10}$ has the CKgP (with integral
coefficients as well) using the calculation of its Chow ring by Field [14]. If
the calculation there holds over any field, it would show that
$\operatorname{BSO}_{10}$ has Totaro’s “weak Chow–Künneth Property,” which is
equivalent to the CKgP by the proof of [34, Theorem 4.1].

####  5.3.2. Review of the Mukai construction

We first review Mukai’s construction and then explain how to modify it for
pointed curves. The canonical model of a pentagonal genus $7$ curve
$C\subset\mathbb{P}^{6}$ lies on a $10$-dimensional space of quadrics. The
vector space $W$ of these quadrics is defined by the exact sequence

(5.8) $0\rightarrow W\rightarrow\operatorname{Sym}^{2}H^{0}(C,\omega_{C})\to
H^{0}(C,\omega_{C}^{\otimes 2})\rightarrow 0.$

For each $p\in C$, the subspace $W_{p}\subset W$ of quadrics that are singular
at $p$ is $5$-dimensional. It lies in the exact sequence (see [21, Section 3])

(5.9) $0\rightarrow
W_{p}\rightarrow\operatorname{Sym}^{2}H^{0}(C,\omega_{C}(-p))\to
H^{0}(C,\omega_{C}(-p)^{\otimes 2})\rightarrow 0.$

Mukai shows that $W^{\vee}$ has a canonical quadratic form and $W_{p}^{\perp}$
is an isotropic subspace, so one obtains a map
$C\to\operatorname{OG}(5,W^{\vee})$ via
$p\mapsto[W_{p}^{\perp}]\in\operatorname{OG}(5,W^{\vee})$ [21, Theorem 0.4].

Let $\operatorname{OG}(5,10)\hookrightarrow\mathbb{P}S^{+}$ be the spinor
embedding, as in [21, Section 1]. The composition

$C\to\operatorname{OG}(5,W^{\vee})\cong\operatorname{OG}(5,10)\hookrightarrow\mathbb{P}S^{+}$

realizes $C$ as a linear section
$C=\operatorname{OG}(5,10)\cap\mathbb{P}^{6}\subset\mathbb{P}S^{+}$, and
$C\subset\mathbb{P}^{6}$ is canonically embedded [21, Theorem 0.4].
Conversely, if a linear section $\mathbb{P}^{6}\cap\operatorname{OG}(5,10)$ is
a smooth curve, then it is a canonically embedded pentagonal genus $7$ curve
[21, Proposition 2.2]. This construction works in families, and Mukai proves
that there is an equivalence of stacks

(5.10)
$\mathcal{M}_{7}^{\circ}\cong[\operatorname{Gr}(7,S^{+})\smallsetminus\Delta/\mathrm{SO}_{10}],$

where $\Delta\subset\operatorname{Gr}(7,S^{+})$ is the closed locus of linear
subspaces $\mathbb{P}^{6}\subset\mathbb{P}S^{+}$ whose intersection with
$\operatorname{OG}(5,10)$ is not a smooth curve [21, Section 5].

The spinor representation of $\mathrm{SO}_{10}$ corresponds to a rank $16$
vector bundle $\mathcal{S}^{+}$ on the classifying stack
$\operatorname{BSO}_{10}$. Then, the equivalence (5.10) shows that
$\mathcal{M}_{7}^{\circ}$ is an open substack of the Grassmann bundle
$\operatorname{Gr}(7,\mathcal{S}^{+})$ over $\operatorname{BSO}_{10}$:

(5.11)
${\mathcal{M}_{7}^{\circ}}$${\operatorname{Gr}(7,\mathcal{S}^{+})}$${\operatorname{BSO}_{10}.}$$\scriptstyle{\alpha_{0}}$

We note that the tautological subbundle $\mathcal{E}$ on
$\operatorname{Gr}(7,\mathcal{S}^{+})$ restricts to the dual of the Hodge
bundle on $\mathcal{M}_{7}^{\circ}$. In other words, if
$f\colon\mathcal{C}\to\mathcal{M}_{7}^{\circ}$ is the universal curve, then
$\alpha_{0}^{*}\mathcal{E}=(f_{*}\omega_{f})^{\vee}$. The universal version of
the Mukai construction furnishes a map
$\mathcal{C}\to\operatorname{OG}(5,\mathcal{V})$ over
$\operatorname{BSO}_{10}$.

####  5.3.3. The Mukai construction with markings

For $n\leq 4$, we describe $\mathcal{M}_{7,n}^{\circ}$ in a similar fashion to
(5.11), but this time as an open substack of a Grassmann bundle over
$\operatorname{OG}(5,\mathcal{V})^{n}$. To do so, let
$\iota\colon\operatorname{OG}(5,\mathcal{V})\hookrightarrow\mathbb{P}\mathcal{S}^{+}$
be the universal spinor embedding and let
$\mathcal{L}:=\iota^{*}\mathcal{O}_{\mathbb{P}\mathcal{S}^{+}}(-1)$. The
embedding $\iota$ is determined by an inclusion of vector bundles
$\mathcal{L}\hookrightarrow\pi^{*}\mathcal{S}^{+}$ on
$\operatorname{OG}(5,\mathcal{V})$. On the $n$-fold fiber product
$\operatorname{OG}(5,\mathcal{V})^{n}$, let $\mathcal{L}_{i}$ and
$\mathcal{U}_{i}$ denote the pullbacks of $\mathcal{L}$ and $\mathcal{U}$
respectively from the $i$th factor. Let
$\pi_{n}\colon\operatorname{OG}(5,\mathcal{V})^{n}\to\operatorname{BSO}_{10}$
be the structure map. Consider the sum of the inclusions

(5.12)
$\bigoplus_{i=1}^{n}\mathcal{L}_{i}\xrightarrow{\phi_{n}}\pi_{n}^{*}\mathcal{S}^{+}.$

Let $Z_{n}\subset\operatorname{OG}(5,\mathcal{V})^{n}$ be the open substack
where $\phi_{n}$ has rank $n$. In other words, $Z_{n}$ is the locus where the
$n$ points on $\operatorname{OG}(5,\mathcal{V})$ have independent image under
the spinor embedding.

Let $\mathcal{Q}_{n}$ be the cokernel of $\phi_{n}|_{Z_{n}}$, which is a rank
$16-n$ vector bundle on $Z_{n}$. The fiber of
$\operatorname{Gr}(7-n,\mathcal{Q}_{n})$ over
$(p_{1},\ldots,p_{n})\in\operatorname{OG}(5,\mathcal{V})^{n}$ parametrizes
linear spaces $\mathbb{P}^{6}\subset\mathbb{P}S^{+}$ that contain the $n$
points $p_{i}$. In other words, we can identify
$\operatorname{Gr}(7-n,\mathcal{Q}_{n})$ with the locally closed substack

(5.13) $\left\\{(p_{1},\ldots,p_{n},\Lambda):p_{i}\in\mathbb{P}\Lambda\text{
and $p_{i}$
independent}\right\\}\subset\operatorname{OG}(5,\mathcal{V})^{n}\times_{\operatorname{BSO}_{10}}\operatorname{Gr}(7,\mathcal{S}^{+}).$

######  Lemma 5.13.

For $n\leq 4$, there is an open embedding $\alpha_{n}$ of
$\mathcal{M}_{7,n}^{\circ}$ in the Grassmann bundle

${\mathcal{M}_{7,n}^{\circ}}$${\operatorname{Gr}(7-n,\mathcal{Q}_{n})}$${Z_{n}}$${\operatorname{OG}(5,\mathcal{V})^{n}.}$$\scriptstyle{\alpha_{n}}$

Hence, $\mathcal{M}_{7,n}^{\circ}$ has the CKgP.

###### Proof.

Let $f\colon\mathcal{C}\to\mathcal{M}_{7,n}^{\circ}$ be the universal curve
and let $\sigma_{i}\colon\mathcal{M}_{7,n}^{\circ}\to\mathcal{C}$ be the $i$th
section. The universal version of the Mukai construction furnishes a morphism
$\mathcal{C}\to\operatorname{OG}(5,\mathcal{V})$. Precomposing with each of
the sections $\sigma_{i}$ defines a map
$\mathcal{M}_{7,n}^{\circ}\to\operatorname{OG}(5,\mathcal{V})^{n}$ over
$\operatorname{BSO}_{10}$. We claim that, for $n\leq 4$, the images of
$\sigma_{1},\ldots,\sigma_{n}$ must be independent. Indeed, suppose the images
of the sections in a fiber, $p_{1},\ldots,p_{n}\in C$, are dependent under the
canonical embedding. Then $p_{1}+\ldots+p_{n}$ would give rise to a
$g^{1}_{n}$ on $C$, but curves in $\mathcal{M}_{7,n}^{\circ}$ have no
$g^{1}_{n}$ for $n\leq 4$ by definition.

We also have the map
$\mathcal{M}_{7,n}^{\circ}\to\mathcal{M}_{7}^{\circ}\to\operatorname{Gr}(7,\mathcal{S}^{+})$
that sends a curve to its span under the spinor embedding. Taking the product
of these maps over $\operatorname{BSO}_{10}$ yields a map

$\mathcal{M}_{7,n}^{\circ}\to\operatorname{OG}(5,\mathcal{V})^{n}\times_{\operatorname{BSO}_{10}}\operatorname{Gr}(7,\mathcal{S}^{+}).$

This map sends a family of pointed curves $(C,p_{1},\ldots,p_{n})$ over a
scheme $T$ to the data of sections $p_{i}\colon T\to
C\to\operatorname{OG}(5,W^{\vee})$ and the subbundle of the spinor
representation of $W^{\vee}$ determined by the span of the fibers of
$C\subset\operatorname{OG}(5,W^{\vee})\subset\mathbb{P}S^{+}$ over $T$. The
map evidently factors through the locally closed locus in (5.13), which we
identified with $\operatorname{Gr}(7-n,\mathcal{Q}_{n})$. In fact, the image
is precisely $\operatorname{Gr}(7-n,\mathcal{Q}_{n})\smallsetminus\Delta$,
where $\Delta$ is the closed locus such that
$\mathbb{P}\Lambda\cap\operatorname{OG}(5,W^{\vee})$ is not a family of smooth
curves. Indeed, on the complement of $\Delta$, an inverse map
$\operatorname{Gr}(7-n,\mathcal{Q}_{n})\smallsetminus\Delta\to\mathcal{M}_{7,n}^{\circ}$
is defined by sending $(p_{1},\ldots,p_{n},\Lambda)$ to the curve
$C=\mathbb{P}\Lambda\cap\operatorname{OG}(5,W^{\vee})$ together with the
sections $p_{i}\in\mathbb{P}\Lambda\cap\operatorname{OG}(5,W^{\vee})=C$.

By Corollary 5.11, we know $\operatorname{OG}(5,\mathcal{V})^{n}$ has the
CKgP. To complete the proof, apply Proposition 4.2(1) and (5). ∎

We now identify the restrictions of the universal bundles on
$\operatorname{OG}(5,\mathcal{V})^{n}$ and
$\operatorname{Gr}(7-n,\mathcal{Q}_{n})$ to $\mathcal{M}_{7,n}^{\circ}$ along
$\alpha_{n}$.

######  Lemma 5.14.

Let $f\colon\mathcal{C}\to\mathcal{M}_{7,n}^{\circ}$ be the universal curve
and let $\sigma_{i}$ denote the image of the $i$th section. The vector bundle
$\alpha_{n}^{*}\,\mathcal{U}_{i}$ sits in an exact sequence

(5.14)
$0\rightarrow\alpha_{n}^{*}\,\mathcal{U}_{i}\rightarrow\operatorname{Sym}^{2}f_{*}(\omega_{f}(-\sigma_{i}))\rightarrow
f_{*}((\omega_{f}(-\sigma_{i}))^{\otimes 2})\rightarrow 0.$

In particular, all classes pulled back from
$\operatorname{OG}(5,\mathcal{V})^{n}$ to $\mathcal{M}_{7,n}^{\circ}$ are
tautological.

###### Proof.

The composition of $\alpha_{n}$ with projection onto the $i$th factor of
$Z_{n}$ is the map that sends a pointed curve $(C,p_{1},\ldots,p_{n})$ to the
image of $p_{i}$ under the canonical map from $C$ to
$\operatorname{OG}(5,W^{\vee})\cong\operatorname{OG}(5,10)$. By construction,
the fiber at $p_{i}$ of the universal rank $5$ bundle on
$\operatorname{OG}(5,10)$ is $W_{p}^{\perp}$. Equation (5.14) is the relative
version of (5.9). By Grothendieck–Riemann–Roch, the middle and right terms in
(5.14) have tautological Chern classes. It follows that the Chern classes of
$\alpha_{n}^{*}\mathcal{U}_{i}$ are also tautological. The last claim now
follows from Corollary 5.11. ∎

######  Lemma 5.15.

We have $c_{1}(\alpha_{n}^{*}\mathcal{L}_{i})=-\psi_{i}$ in
$A^{1}(\mathcal{M}_{7,n}^{\circ})$.

###### Proof.

Let $f\colon\mathcal{C}\to\mathcal{M}_{7,n}^{\circ}$ be the universal curve.
The line bundle $\alpha^{*}\mathcal{L}_{i}$ is the pullback of
$\mathcal{O}_{\mathbb{P}\mathcal{S}^{+}}(-1)$ along the composition

$\mathcal{M}_{7,n}^{\circ}\xrightarrow{\sigma_{i}}\mathcal{C}\rightarrow\operatorname{OG}(5,\mathcal{V})\xrightarrow{\iota}\mathbb{P}\mathcal{S}^{+}.$

But the above composition also factors as

$\mathcal{M}_{7,n}^{\circ}\xrightarrow{\sigma_{i}}\mathcal{C}\xrightarrow{|\omega_{f}|}\mathbb{P}(f_{*}\omega_{f})^{\vee}\to\mathbb{P}\mathcal{S}^{+},$

and $\mathcal{O}_{\mathbb{P}\mathcal{S}^{+}}(-1)$ restricts to
$\mathcal{O}_{\mathbb{P}(f_{*}\omega_{f})^{\vee}}(-1)$. Hence,
$\alpha_{n}^{*}\mathcal{L}_{i}$ is
$\sigma_{i}^{*}\mathcal{O}_{\mathbb{P}(f_{*}\omega_{f})^{\vee}}(-1)=(\sigma_{i}^{*}\omega_{f})^{\vee}$.
∎

######  Lemma 5.16.

Let $\mathcal{E}_{n}$ be the tautological rank $7-n$ subbundle on
$\operatorname{Gr}(7-n,\mathcal{Q}_{n})$. The pullback
$\alpha_{n}^{*}\mathcal{E}_{n}$ sits in an exact sequence

(5.15)
$0\rightarrow\bigoplus_{i=1}^{n}\alpha_{n}^{*}\mathcal{L}_{i}\rightarrow(f_{*}\omega_{f})^{\vee}\rightarrow\alpha_{n}^{*}\mathcal{E}_{n}\rightarrow
0.$

In particular, the Chern classes of $\alpha_{n}^{*}\mathcal{E}_{n}$ are
tautological.

###### Proof.

Recall that we defined $\mathcal{Q}_{n}$ as the cokernel of (5.12). The map
onto the second factor in (5.13),
$p\colon\operatorname{Gr}(7-n,\mathcal{Q}_{n})\to\operatorname{Gr}(7,\mathcal{S}^{+})$,
is defined by sending a $7-n$ dimensional subspace to its preimage under the
quotient map $\mathcal{S}^{+}\to\mathcal{Q}_{n}$. Hence, there is an exact
sequence on $\operatorname{Gr}(7-n,\mathcal{Q}_{n})$

$0\rightarrow\bigoplus_{i=1}^{n}\mathcal{L}_{i}\to
p^{*}\mathcal{E}\to\mathcal{E}_{n}\rightarrow 0.$

Then note that $\alpha_{n}^{*}p^{*}\mathcal{E}$ is the same as the pullback of
$\mathcal{E}$ along
$\mathcal{M}_{7,n}^{\circ}\to\mathcal{M}_{7}^{\circ}\to\operatorname{Gr}(7,\mathcal{S}^{+})$,
which is the dual of the Hodge bundle. The last claim follows by combining the
exact sequence (5.15) with Lemma 5.15. ∎

######  Lemma 5.17.

The Chow ring of $\mathcal{M}_{7,n}^{\circ}$ is generated by tautological
classes.

###### Proof.

By Lemma 5.13, we know that $\mathcal{M}_{7,n}^{\circ}$ is an open substack of
$\operatorname{Gr}(7-n,\mathcal{Q}_{n})$. The Chow ring of the latter is
generated by pullbacks of classes from $\operatorname{OG}(5,\mathcal{V})^{n}$
and by the Chern classes of the tautological subbundle $\mathcal{E}_{n}$. By
Lemmas 5.14 and 5.16 respectively, both of these classes restrict to
tautological classes on $\mathcal{M}_{7,n}^{\circ}$. ∎

##  6\. Applications to even cohomology

Here we use the results from Sections 3, 4, and 5 to prove Theorem 1.5.

###  6.1. The degree $4$ cohomology of $\overline{\mathcal{M}}_{g,n}$ is
tautological

###### Proof of Theorem 1.5(1).

By [1], we know that $H^{k}(\overline{\mathcal{M}}_{g,n})$ is tautological for
$k\leq 3$. Therefore, using Theorem 1.3(1), we deduce that
$H^{4}(\overline{\mathcal{M}}_{g,n})$ is contained the STE generated by
$H^{4}(\overline{\mathcal{M}}_{g^{\prime},n^{\prime}})$ for $g^{\prime}<7$ and
$n^{\prime}\leq 4$. By Proposition 4.5,
$W_{4}H^{4}(\mathcal{M}_{g^{\prime},n^{\prime}})$ is tautological for
$g^{\prime}$ and $n^{\prime}$ in this range, and it follows that
$H^{4}(\overline{\mathcal{M}}_{g,n})$ is tautological. ∎

###### Proof of Theorem 1.5(2).

By Theorem 1.5(1) and [1], all classes in
$H^{6}(\overline{\mathcal{M}}_{g,n})$ that are pushed forward from the
boundary are tautological. Also, $H^{6}(\mathcal{M}_{g,n})$ is stable and
hence tautological for $g\geq 10$ [35]. It follows that
$H^{6}(\overline{\mathcal{M}}_{g,n})$ is tautological for $g\geq 10$. ∎

######  Remark 6.1.

By Lemma 3.1, to show that $H^{6}(\overline{\mathcal{M}}_{g,n})$ is
tautological for all $g$ and $n$, it would suffice to show this for $g\leq 9$
and $n\leq 6$. In principle, this can be checked computationally as follows.
By Theorem 1.5(3), we know that $H_{6}(\overline{\mathcal{M}}_{g,n})$ is
tautological for all $g$ and $n$. Thus, if the intersection pairing

$RH^{6}(\overline{\mathcal{M}}_{g,n})\times
RH_{6}(\overline{\mathcal{M}}_{g,n})\rightarrow\mathbb{Q}$

is perfect, then $H^{6}(\overline{\mathcal{M}}_{g,n})$ is tautological. In
principle, one could compute this pairing using the Sage package admcycles
[11]. In practice, however, this computation is too memory intensive to carry
out.

###  6.2. The low degree homology of $\overline{\mathcal{M}}_{g,n}$ is
tautological

Let $H_{k}$ denote Borel–Moore homology with coefficients in $\mathbb{Q}$ or
$\mathbb{Q}_{\ell}$, together with its mixed Hodge structure or Galois action.
By Poincaré duality, we have the identification of vector spaces

$H^{2d_{g,n}-k}(\mathcal{M}_{g,n})\cong H_{k}(\mathcal{M}_{g,n}).$

From the long exact sequence in Borel–Moore homology, we have a right exact
sequence

(6.1) $H_{k}(\widetilde{\partial\mathcal{M}_{g,n}})\rightarrow
H_{k}(\overline{\mathcal{M}}_{g,n})\rightarrow
W_{-k}H_{k}(\mathcal{M}_{g,n})\rightarrow 0$

for all $k$.

######  Remark 6.2.

One can prove analogues of Theorem 1.3 and Lemma 2.1 for homology. In
particular, for each $k$, there is a finitely generated STE that contains
$H_{k}(\overline{\mathcal{M}}_{g,n})$ for all $g$ and $n$, with explicit
bounds on the generators.

###### Proof of Theorem 1.5(3).

When $g=0$, all cohomology is tautological [18]. By [4, Proposition 2.1] and
the universal coefficient theorem, for $g\geq 1$ we have:

$H_{k}(\mathcal{M}_{g,n})=0\text{ for }\begin{cases}k<2g\text{ and }n=0,1;\\\
k<2g-2+n\text{ and }n\geq 2.\end{cases}$

Combining this with the exact sequence (6.1), we reduce inductively to the
finitely many cases $k\geq 2g$ and $n=0,1$ or $k\geq 2g-2+n$ and $n\geq 2$.
When $g\geq 3$, in all of these cases we know that $\mathcal{M}_{g,n}$ has the
CKgP and $A^{*}(\mathcal{M}_{g,n})=R^{*}(\mathcal{M}_{g,n})$ (see Table 1). By
Proposition 4.5, it follows that $W_{-k}H_{k}(\mathcal{M}_{g,n})$ is
tautological, so again by induction and the exact sequence (6.1) we reduce to
the cases of $g=1,2$. When $g=1$, all even cohomology is tautological [26].
When $g=2$, we apply Corollary 2.7. ∎

######  Corollary 6.3.

As Hodge structures or Galois representations, we have

$H^{2d_{g,n}-14}(\overline{\mathcal{M}}_{g,n})^{\mathrm{ss}}\cong\bigoplus\mathsf{L}^{d_{g,n}-7}$

###### Proof.

We proved $H_{14}(\overline{\mathcal{M}}_{g,n})$ is tautological and hence
algebraic. Applying Poincaré duality yields the corollary. ∎

##  7\. The thirteenth and fifteenth homology of
$\overline{\mathcal{M}}_{g,n}$

In this section, we prove Theorems 1.6 and 1.7. As a corollary, we obtain
restrictions on the Hodge structure and Galois representations appearing in
$H_{13}(\overline{\mathcal{M}}_{g,n})$ and
$H_{15}(\overline{\mathcal{M}}_{g,n})$.

###  7.1. Thirteenth homology

As usual, we argue by induction on $g$ and $n$. First, we must treat one extra
base case.

######  Lemma 7.1.

The group $H^{13}(\overline{\mathcal{M}}_{2,11})$ is in the STE generated by
$H^{11}(\overline{\mathcal{M}}_{1,11})$.

###### Proof.

We showed in [8] that $H^{11}(\overline{\mathcal{M}}_{g,n})$ is in the STE
generated by $H^{11}(\overline{\mathcal{M}}_{1,11})$ for all $g$ and $n$.
Next, we note that $\mathcal{M}_{2,10}$ has the CKgP (see Table 1), so
$H^{13}(\overline{\mathcal{M}}_{2,10})$ is pushed forward from the boundary by
Lemma 4.5. In particular, $H^{13}(\overline{\mathcal{M}}_{2,10})$ also lies in
this STE. The claim then follows from Lemma 3.2. ∎

###### Proof of Theorem 1.6.

The proof is similar to that of Theorem 1.5(3). When $g=0$, all odd cohomology
vanishes [18], so we can assume $g\geq 1$. By [4, Proposition 2.1] and the
universal coefficient theorem, we have that for $g\geq 1$

$H_{13}(\mathcal{M}_{g,n})=0\text{ for }\begin{cases}13<2g\text{ and }n=0,1\\\
13<2g-2+n\text{ and }n\geq 2.\end{cases}$

Combining this with the exact sequence (6.1), we reduce inductively to the
finitely many cases where $13\geq 2g$ and $n=0,1$ or $13\geq 2g-2+n$ and
$n\geq 2$. All such cases except for $(g,n)$ in
$Z=\\{(2,11),(1,11),(1,12),(1,13)\\}$ have the CKgP (see Table 1). By
Proposition 4.5, it follows that $W_{-13}H_{13}(\mathcal{M}_{g,n})=0$ for
$(g,n)\not\in Z$. By induction and the exact sequence (6.1), we reduce to the
exceptional cases of $(g,n)\in Z$. The case $(g,n)=(2,11)$ follows from Lemma
7.1 and Hard Lefschetz. The cases where $g=1$ follow from Proposition 2.2 and
induction on $n$. ∎

Following the proof strategy above, we obtain the following corollary.

######  Corollary 7.2.

As Hodge structures or Galois representations, we have

$H^{2d_{g,n}-13}(\overline{\mathcal{M}}_{g,n})^{\mathrm{ss}}\cong\bigoplus\mathsf{L}^{d_{g,n}-12}\mathsf{S}_{12}$

for all $g$ and $n$.

###### Proof.

From the proof above, we see that $W_{-13}H_{13}(\mathcal{M}_{g,n})=0$ for
$(g,n)\notin Z$, and so

$H_{13}(\widetilde{\partial\mathcal{M}_{g,n}})\rightarrow
H_{13}(\overline{\mathcal{M}}_{g,n})$

is surjective. Noting that pushforward along a gluing map just induces a Tate
twist, by induction on $g$ and $n$, we reduce to the cases $(g,n)\in Z$. The
case $(g,n)=(2,11)$ follows from Lemma 7.1 and Hard Lefschetz. The cases where
$g=1$ follow from Proposition 2.2 and induction on $n$. ∎

###  7.2. Fifteenth homology

Again, we will argue by induction on $g$ and $n$. We first need to treat two
additional base cases in genus $2$. The difficult one is with $12$ markings.

######  Lemma 7.3.

$H^{15}(\overline{\mathcal{M}}_{2,12})$ is generated by classes pushed forward
from the boundary. Hence, $H^{15}(\overline{\mathcal{M}}_{2,12})$ is contained
in the STE generated by $H^{11}(\overline{\mathcal{M}}_{1,11})$.

The proof of Lemma 7.3 involves calculating intersection pairings with
decorated graphs. In order to simplify pictures for and language surrounding
decorated graphs, we use the following conventions:

  1. (1)

A white circle $\circ$ is a vertex of genus $1$

  2. (2)

A black circle $\bullet$ is a vertex of genus $0$

  3. (3)

All markings that are not drawn elsewhere are understood to be on the leftmost
genus $1$ vertex

  4. (4)

The decoration $\omega_{i}$ on a genus $1$ vertex is short-hand for
$f_{i}^{*}\omega$ where
$f_{i}\colon\overline{\mathcal{M}}_{1,12}\to\overline{\mathcal{M}}_{1,11}$
forgets the $i$th marking and $\omega\in
H^{11}(\overline{\mathcal{M}}_{1,11})$. Similarly, $\omega_{ij}$ is short-hand
for $f_{ij}^{*}\omega$ where
$f_{ij}\colon\overline{\mathcal{M}}_{1,13}\to\overline{\mathcal{M}}_{1,11}$
forgets the $i$th and $j$th markings.

  5. (5)

As we vary the decoration $\omega$ over
$H^{11}(\overline{\mathcal{M}}_{1,11})\cong\mathsf{S}_{12}$, pushforward along
a map that glues $e$ pairs of marked points determines a morphism
$\mathsf{L}^{e}\mathsf{S}_{12}\to H^{11+2e}(\overline{\mathcal{M}}_{g,n})$. We
say that a collection of $\omega$-decorated graphs “is a basis for
$H^{k}(\overline{\mathcal{M}}_{g,n})$” if the corresponding morphisms
$\mathsf{L}^{e}\mathsf{S}_{12}\to H^{k}(\overline{\mathcal{M}}_{g,n})$ form a
basis for
$\operatorname{Hom}(\mathsf{L}^{e}\mathsf{S}_{12},H^{k}(\overline{\mathcal{M}}_{g,n}))$
and we have
$H^{k}(\overline{\mathcal{M}}_{g,n})\cong\operatorname{Hom}(\mathsf{L}^{e}\mathsf{S}_{12},H^{k}(\overline{\mathcal{M}}_{g,n}))\otimes\mathsf{L}^{e}\mathsf{S}_{12}$.

Bergström and Faber’s implementation [2] of the Getzler–Kapranov formula in
genus $2$ shows that, viewed as Galois representations,
$H^{15}(\overline{\mathcal{M}}_{2,12})=\operatorname{Hom}(\mathsf{L}^{2}\mathsf{S}_{12},H^{15}(\overline{\mathcal{M}}_{2,12}))\otimes\mathsf{L}^{2}\mathsf{S}_{12}$
and

(7.1)
$\dim\operatorname{Hom}(\mathsf{L}^{2}\mathsf{S}_{12},H^{15}(\overline{\mathcal{M}}_{2,12}))=836.$

Thus, our goal is to prove that there are $836$ independent copies of
$\mathsf{L}^{2}\mathsf{S}_{12}$ in $H^{15}(\overline{\mathcal{M}}_{2,12})$
that are pushed forward from the boundary. As discussed above, an
$\omega$-decorated graph with $e$ edges represents a morphism
$\mathsf{L}^{e}\mathsf{S}_{12}\to H^{11+2e}(\overline{\mathcal{M}}_{g,n})$. In
Figure 2 we list $891$ such decorated graphs for
$H^{15}(\overline{\mathcal{M}}_{2,12})$ and record particular Künneth
components of their pullbacks to boundary divisors. We then argue that there
are $55$ relations among these graphs, which allows us to conclude that there
are $836$ independent graphs among them.

Before proving Lemma 7.3, we require some explicit bases for
$H^{13}(\overline{\mathcal{M}}_{g,n})$ for small $g,n$. Recall that Getzler
[16] showed that

$H^{13}(\overline{\mathcal{M}}_{1,n})=\operatorname{Hom}(\mathsf{L}\mathsf{S}_{12},H^{13}(\overline{\mathcal{M}}_{1,n}))\otimes\mathsf{L}\mathsf{S}_{12}.$

for all $n$.

######  Lemma 7.4.

The following $11$ $\omega$-decorated graphs form a basis for
$H^{13}(\overline{\mathcal{M}}_{1,12})$.

###### Proof.

Consider the pairing against the basis $\omega_{2},\ldots,\omega_{12}\in
H^{11}(\overline{\mathcal{M}}_{1,12})$. ∎

######  Lemma 7.5.

The following $429$ $\omega$-decorated graphs form a basis for
$H^{13}(\overline{\mathcal{M}}_{1,13})$.

Above, $x,y$ are any two markings. In the last group of classes, $i$ or $j$
can by equal to $y$.

###### Proof.

Getzler [16] proved that
$\operatorname{Hom}(\mathsf{L}\mathsf{S}_{12},H^{13}(\overline{\mathcal{M}}_{1,13}))$
has dimension $429$, so it suffices to show that these classes span. We know
that $H^{13}(\overline{\mathcal{M}}_{1,13})$ is pushed forward from the
boundary by Proposition 2.2. Modulo classes of the first kind, the second two
kinds are pulled back from $H^{13}(\overline{\mathcal{M}}_{1,12})$. Pulling
back the relations from $H^{13}(\overline{\mathcal{M}}_{1,12})$, we can ensure
that a fixed marking such as $y$ in the first case or $x$ in the next case is
on the genus $0$ vertex. ∎

######  Lemma 7.6.

Let $\\{x,y,1,\ldots,12\\}$ be a set of $14$ markings. The following $6006$
$\omega$-decorated graphs form a basis for
$H^{13}(\overline{\mathcal{M}}_{1,14})$.

In the last grouping above, $i,j,$ or $k$ can equal $y$.

###### Proof.

Getzler [16] proved that
$\operatorname{Hom}(\mathsf{L}\mathsf{S}_{12},H^{13}(\overline{\mathcal{M}}_{1,14}))$
has dimension $6006$, so it suffices to show that these classes span. Modulo
classes of the first and second kinds above, the last three kinds are pulled
back from an $\overline{\mathcal{M}}_{1,12}$. Thus, using the relations on
$H^{13}(\overline{\mathcal{M}}_{1,12})$ we can ensure a fixed marking among
those $12$ is on the genus $0$ vertex. ∎

######  Lemma 7.7.

The following $264$ $\omega$-decorated graphs form a basis for
$H^{13}(\overline{\mathcal{M}}_{2,12})$.

###### Proof.

Bergström and Faber’s implementation [2] of the Getzler–Kapranov formula in
genus $2$ shows that, as Galois representations,
$H^{13}(\overline{\mathcal{M}}_{2,12})=\operatorname{Hom}(\mathsf{L}\mathsf{S}_{12},H^{13}(\overline{\mathcal{M}}_{2,12}))\otimes\mathsf{L}\mathsf{S}_{12}$
and

$\dim\operatorname{Hom}(\mathsf{L}\mathsf{S}_{12},H^{15}(\overline{\mathcal{M}}_{2,12}))=264.$

Thus, it suffices to show that the above $264$ classes are independent. We
verify this by computing their pullbacks to
$H^{13}(\overline{\mathcal{M}}_{1,14})$ and using Lemma 7.6 to see that those
classes are independent. Indeed, the pullback is given by the graph that
replaces the rightmost genus $1$ vertex with a genus $0$ vertex and adds two
markings labeled $x,y$ to that vertex. ∎

######  Remark 7.8.

Theorem 1.6 says that $H_{13}(\overline{\mathcal{M}}_{g,n})$ is contained in
the STE generated by $H^{11}(\overline{\mathcal{M}}_{1,11})$. Roughly
speaking, this means that it has a graphical presentations, in which the
generators are graphs of the sort appearing in Lemmas 7.4–7.7. In forthcoming
work, we will show that $H^{13}(\overline{\mathcal{M}}_{g,n})$ is also
contained in this STE. There, we take a more systematic approach and provide a
complete list of relations among the corresponding decorated graph generators.

###### Proof of Lemma 7.3.

Figure 2 represents a matrix with $891$ rows. We first show that the matrix
has rank $825$ and describe the $66$ relations among the rows.

First notice that the rows in A1 are the only rows with nonzero entries in the
first column. Next, rows in A2, A3, and A4 have entries in column 2 that are
independent from the other entries that appear in that column for later rows
(by Lemma 7.5). This shows that the rows in A1 through A4 are independent and
independent from the remaining 5 rows. Meanwhile, the bottom 4 rows are block
upper triangular. This shows that the rank of the matrix is at least $825$.

Figure 2. Selected Künneth components of pullbacks of the $891$ classes to
boundary divisors. In the chart, each row represents several rows of a block
matrix. The number of rows in each group is listed to its left. The two half-
edges glued to make a boundary divisor (pictured across the top) are labeled
$q$ on the left vertex and $p$ on the right vertex.

Meanwhile, it is not hard to check that the following $66$ relations exist
among the rows:

$R_{1}(ij)+\frac{1}{24}R_{2}(ij)+R_{3}[i,j]+R_{3}[j,i]-R_{4}(ij)+\frac{1}{12}R_{5}(ij)=0.$

To verify these relations one needs to make use of known relations among
tautological classes in $H^{4}(\overline{\mathcal{M}}_{1,3})$ and
$H^{2}(\overline{\mathcal{M}}_{1,2})$ and
$H^{2}(\overline{\mathcal{M}}_{1,1})$.

Now, let

We claim $M_{ij}\neq 0$. To see this, consider the intersection of $M_{ij}$
with $R_{5}(ik)$ for $k\neq j$. The intersection of $R_{5}(ik)$ with the first
$5$ terms in the above equation vanish since there is no $H^{11}$ on a genus
$1$ vertex with less than $11$ markings. Meanwhile, the intersection
$R_{5}(ij)\cdot R_{5}(ik)$ is nonzero using the usual rules for intersecting
decorated graphs as in [17]. To see this, one should take the $\omega$
decoration on the $\overline{\mathcal{M}}_{1,11}$ vertex in $R_{5}(ij)$ to be
dual to the decoration used on the $\overline{\mathcal{M}}_{1,11}$ vertex in
$R_{5}(ik)$.

Finally, the $\mathbb{S}_{12}$ action on the symbols $M_{ij}$ is
$\mathrm{Ind}_{\mathbb{S}_{10}}^{\mathbb{S}_{12}}(V_{1^{10}})=V_{2,1^{10}}\oplus
V_{3,1^{9}}$, which is a sum of irreducible representations of dimension $11$
and $55$. It follows that there are either $11$ or $55$ relations among the
$M_{ij}$. There cannot be only $11$ relations among the $M_{ij}$ because then
there would be more than $836$ independent copies of
$\mathsf{L}^{2}\mathsf{S}_{12}$ in $H^{15}(\overline{\mathcal{M}}_{2,12})$,
violating the dimension count (7.1). Hence, there must be exactly $55$
relations among the $M_{ij}$. This shows that the $891$ copies of
$\mathsf{L}^{2}\mathsf{S}_{12}$ listed on the left of the table generate an
$836$ dimensional space, so they span all of
$H^{15}(\overline{\mathcal{M}}_{2,12})$. ∎

######  Remark 7.9.

One could instead prove that the $891$ classes span a space of rank $836$ by
computing the pairing of this space against itself. By the argument above, the
pairing must have rank $836$.

Finally, applying Lemma 3.2(a) together with Lemmas 7.3 and 7.7, and Corollary
2.3, we obtain one more base case.

######  Lemma 7.10.

$H^{15}(\overline{\mathcal{M}}_{2,13})$ lies in the STE generated by
$H^{11}(\overline{\mathcal{M}}_{1,11})$.

###### Proof of Theorem 1.7.

The proof is similar to that of Theorem 1.6. When $g=0$, all odd cohomology
vanishes [18], so we can assume $g\geq 1$. By [4, Proposition 2.1] and the
universal coefficient theorem, we have that for $g\geq 1$

$H_{15}(\mathcal{M}_{g,n})=0\text{ for }\begin{cases}15<2g\text{ and }n=0,1\\\
15<2g-2+n\text{ and }n\geq 2.\end{cases}$

Combining this with the exact sequence (6.1), we reduce inductively to the
finitely many cases $15\geq 2g$ and $n=0,1$ or $15\geq 2g-2+n$ and $n\geq 2$.
In such cases when $g\geq 3$, $\mathcal{M}_{g,n}$ has the CKgP, and so by
induction and the exact sequence (6.1), we reduce to the cases where $g=1$ and
$11\leq n\leq 14$, or $g=2$ and $11\leq n\leq 13$. When $g=2$ and $n=11$, the
result follows from Hard Lefschetz and Lemma 7.1. When $g=2$ and $12\leq n\leq
13$, the result follows from Lemmas 7.3 and 7.10. For the genus $1$ cases, the
claim follows from Proposition 2.2. ∎

######  Corollary 7.11.

As Hodge structures or Galois representations, we have

$H^{2d_{g,n}-15}(\overline{\mathcal{M}}_{g,n})^{\mathrm{ss}}\cong\bigoplus\mathsf{L}^{d_{g,n}-13}\mathsf{S}_{12}\oplus\bigoplus\mathsf{L}^{d_{g,n}-15}\mathsf{S}_{16}$

for all $g$ and $n$.

###### Proof.

The proof is similar to that of Corollary 7.2, using that
$H^{15}(\overline{\mathcal{M}}_{1,15})\cong\mathsf{S}_{16}$. ∎

We expect that Theorem 1.7 can be improved as follows.

######  Conjecture 7.12.

The STE generated by $H^{11}(\overline{\mathcal{M}}_{1,11})$ contains
$H^{15}(\overline{\mathcal{M}}_{g,n})$ for $g\geq 2$.

The proof of Theorem 1.1 in the next section shows that for $g\geq 2$,
$H^{15}(\overline{\mathcal{M}}_{g,n})$ contains no copies of
$\mathsf{S}_{16}$. Conjecture 7.12 is therefore equivalent to the assertion
that the STE generated by $H^{11}(\overline{\mathcal{M}}_{1,11})$ and
$H^{15}(\overline{\mathcal{M}}_{1,15})$ contains
$H^{15}(\overline{\mathcal{M}}_{g,n})$ for all $g$ and $n$.

##  8\. Proof of Theorem 1.1

The first two statements of Theorem 1.1 follow immediately from Corollaries
6.3 and 7.2 by inverting copies of $\mathsf{L}$. In Corollary 7.11, we can
invert $\mathsf{L}$ to see that
$H^{15}(\overline{\mathcal{M}}_{g,n})^{\mathrm{ss}}$ is a direct sum of copies
of $\mathsf{L}^{2}\mathsf{S}_{12}$ and $\mathsf{S}_{16}$. To finish, it
suffices to show that when $g\geq 2$, no copies of $\mathsf{S}_{16}$ appear.

######  Lemma 8.1.

The STE generated by $H^{11}(\overline{\mathcal{M}}_{1,11})$ contains
$H_{15}(\overline{\mathcal{M}}_{2,n})$ for all $n$. Hence,
$H^{15}(\overline{\mathcal{M}}_{2,n})^{\mathrm{ss}}\cong\bigoplus\mathsf{L}^{2}\mathsf{S}_{12}$,
as Galois representations or Hodge structures.

###### Proof.

When $g=2$, as Galois representations, there are no copies of
$\mathsf{S}_{16}$ in $H^{15}(\overline{\mathcal{M}}_{2,n})^{\mathrm{ss}}$ by
Corollary 2.9. This shows that
$H_{15}(\overline{\mathcal{M}}_{2,n})^{\mathrm{ss}}$ is a direct sum of copies
of $\mathsf{L}^{d_{2,n}-13}\mathsf{S}_{12}$ in the category of Galois
representations. But we know that $H_{15}(\overline{\mathcal{M}}_{g,n})$ lies
in the STE generated by $H^{11}(\overline{\mathcal{M}}_{1,11})$ and
$H^{15}(\overline{\mathcal{M}}_{1,15})$ by Theorem 1.7. By considering the
Galois representations involved, it follows that
$H_{15}(\overline{\mathcal{M}}_{2,n})$ lies in the STE generated by
$H^{11}(\overline{\mathcal{M}}_{1,11})$. In particular,
$H_{15}(\overline{\mathcal{M}}_{2,n})^{\mathrm{ss}}$ is a direct sum of copies
of $\mathsf{L}^{d_{2,n}-13}\mathsf{S}_{12}$, either as Galois representations
or Hodge structures. Inverting copies $\mathsf{L}$ implies the second claim in
the statement. ∎

To show there are no copies of $\mathsf{S}_{16}$ when $g\geq 3$, we will use a
similar strategy to that of [8]. We study the first two maps in the weight
$15$ complex:

(8.1)
$H^{15}(\overline{\mathcal{M}}_{g,n})\xrightarrow{\alpha}\bigoplus_{|E(\Gamma)|=1}H^{15}(\overline{\mathcal{M}}_{\Gamma})^{\operatorname{Aut}(\Gamma)}\rightarrow\bigoplus_{|E(\Gamma)|=2}(H^{15}\big{(}\overline{\mathcal{M}}_{\Gamma})\otimes\det
E(\Gamma)\big{)}^{\operatorname{Aut}(\Gamma)}.$

Here, $\Gamma$ is a stable graph of genus $g$ with $n$ legs. The first map is
the pullback to the normalization of the boundary
$\widetilde{\partial\mathcal{M}_{g,n}}$. The second map is explicitly
described in [8, Section 4.1].

######  Lemma 8.2.

If $g\geq 3$, then the pullback map $\alpha$ in (8.1) is injective.

###### Proof.

The map $\alpha$ is injective when
$\mathrm{gr}^{W}_{15}H^{15}_{c}(\mathcal{M}_{g,n})=0$. By [4, Proposition
2.1], this holds whenever $2g-2+n>15$. When $2g-2+n\leq 15$,
$\mathcal{M}_{g,n}$ has the CKgP (see Table 1). The result follows from
Proposition 4.5. ∎

We now assume $g\geq 3$, and we study the second map in (8.1) as a morphism of
Hodge structures. In particular, we consider the $H^{15,0}$ part:

(8.2)
$\bigoplus_{|E(\Gamma)|=1}H^{15,0}(\overline{\mathcal{M}}_{\Gamma})^{\operatorname{Aut}(\Gamma)}\xrightarrow{\
\beta\
}\bigoplus_{|E(\Gamma)|=2}\big{(}H^{15,0}(\overline{\mathcal{M}}_{\Gamma})\otimes\det
E(\Gamma)\big{)}^{\operatorname{Aut}(\Gamma)}.$

By Lemma 8.2, to prove the theorem in the category of Hodge structures, it
suffices to show that $\beta$ is injective. The domain is a direct sum over
graphs with one edge. If $\Gamma$ has one vertex and a loop, then
$H^{15,0}(\overline{\mathcal{M}}_{\Gamma})=H^{15,0}(\overline{\mathcal{M}}_{g-1,n+2})=0$
by induction on $g$. If $\Gamma$ has two vertices with an edge connecting
them, then by the Künneth formula, we have

$H^{15,0}(\overline{\mathcal{M}}_{\Gamma})=H^{15,0}(\overline{\mathcal{M}}_{a,A\cup
p})\oplus H^{15,0}(\overline{\mathcal{M}}_{b,A^{c}\cup q}).$

Assume $a\geq b$. By induction, the above is non-vanishing only if $b=1$ and
$|A^{c}|\geq 14$. Because $g\geq 3$, it follows that
$H^{15,0}(\overline{\mathcal{M}}_{\Gamma})=H^{15,0}(\overline{\mathcal{M}}_{1,A^{c}\cup
q})$ in this case. Let $\Gamma^{\prime}$ be the graph obtained from $\Gamma$
by adding a loop on the genus $g-1$ vertex and decreasing the genus
accordingly, as in Figure 3. Then
$H^{15,0}(\overline{\mathcal{M}}_{\Gamma^{\prime}})$ has a summand
$H^{15,0}(\overline{\mathcal{M}}_{1,A^{c}\cup q})$, and the map
$H^{15,0}(\overline{\mathcal{M}}_{\Gamma})\rightarrow
H^{15,0}(\overline{\mathcal{M}}_{\Gamma^{\prime}})$ injects into that summand.

Figure 3. The graph $\Gamma$ on the left and $\Gamma^{\prime}$ on the right.

The proof in the category of Galois representations is similar to that for
Hodge structures. One applies $\operatorname{Hom}(\mathsf{S}_{16},-)$ to the
second map in (8.1) and argues by induction on $g$ and $n$ to see that this
map is injective. ∎

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