# The tautological ring of $\overline{\mathcal{M}}_{g,n}$ is rarely Gorenstein

Samir Canning

###### Abstract.

We prove that the tautological rings
$\mathsf{R}^{*}(\overline{\mathcal{M}}_{g,n})$ and
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ are not Gorenstein when $g\geq
2$ and $2g+n\geq 24$, extending results of Petersen and Tommasi in genus $2$.
The proof uses the intersection of tautological classes with non-tautological
bielliptic cycles. We conjecture the converse: the tautological rings should
be Gorenstein when $g=0,1$ or $g\geq 2$ and $2g+n<24$. The conjecture is known
for $g=0,1$ by work of Keel and Petersen, and we prove several new cases of
this conjecture for $\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ when
$g\geq 2$.

##  1\. Introduction

Let $\overline{\mathcal{M}}_{g,n}$ denote the moduli space of stable curves of
genus $g$ with $n$ markings. The tautological ring
$\mathsf{R}^{*}(\overline{\mathcal{M}}_{g,n})$ is the $\mathbb{Q}$-subalgebra
of the Chow ring of $\overline{\mathcal{M}}_{g,n}$ generated by the decorated
strata classes, and its image in cohomology under the cycle class map is
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$. Decorated strata classes are
Chow classes corresponding to stable graphs of genus $g$ with $n$ legs,
decorated by $\kappa$ classes on the vertices and $\psi$ classes on the legs
and half-edges [12, 21]. Finding a complete set of relations among these
generators is a fundamental open problem in the intersection theory of
$\overline{\mathcal{M}}_{g,n}$. Pixton [26] conjectured relations among the
tautological generators — the _3-spin relations_ 111The $3$-spin relations are
also known as the generalized Faber–Zagier relations or Pixton’s relations. —
that were shown to hold in cohomology by Pandharipande–Pixton–Zvonkine [22]
and in Chow by Janda [15]. No other relations have been found, despite
significant investigation [7, 16].

######  Conjecture 1 (Pixton).

The $3$-spin relations are complete for
$\mathsf{R}^{*}(\overline{\mathcal{M}}_{g,n})$ and
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$.

Before Pixton’s conjecture, it had been speculated that the tautological ring
is Gorenstein [9, 14, 20], extending Faber’s conjecture on
$\mathsf{R}^{*}(\mathcal{M}_{g})$ [11]. We say
$\mathsf{R}^{*}(\overline{\mathcal{M}}_{g,n})$ is Gorenstein if all the
intersection pairings

$\mathsf{R}^{i}(\overline{\mathcal{M}}_{g,n})\times\mathsf{R}^{3g-3+n-i}(\overline{\mathcal{M}}_{g,n})\rightarrow\mathsf{R}^{3g-3+n}(\overline{\mathcal{M}}_{g,n})\cong\mathbb{Q}$

are perfect, and similarly for
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$. If the Gorenstein property
holds, a tautological class is zero if and only if it intersects trivially
with every tautological class in complementary codimension, thereby providing
an algorithm to determine all of the relations in the tautological ring. The
Gorenstein property holds in genus $0$ and $1$ [17, 23], but Petersen and
Tommasi [25] showed it fails in genus $2$ when there are sufficiently many
marked points. Further investigation by Petersen [24] shows that
$\mathsf{R}^{*}(\overline{\mathcal{M}}_{2,n})$ and
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{2,n})$ are not Gorenstein when $n\geq
20$. Our main result is that the tautological rings are Gorenstein in only
finitely many cases when $g\geq 2$.

######  Theorem 2.

Neither $\mathsf{R}^{*}(\overline{\mathcal{M}}_{g,n})$ nor
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ is Gorenstein if $g\geq 2$ and
$2g+n\geq 24$.

In fact, it suffices to prove Theorem 2 for cohomology, as a direct
calculation [25, Corollary 2.5] shows that if
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ is not Gorenstein, then
neither is $\mathsf{R}^{*}(\overline{\mathcal{M}}_{g,n})$.

The central difficulty in Conjecture 1 is the study of classes in the
Gorenstein kernel, classes that intersect trivially with every tautological
class in complementary codimension. Conjecture 1 predicts the existence of
nonzero classes in the Gorenstein kernel, and proving that such classes are
nonzero thus requires the failure of the Gorenstein property. At the moment,
there are no known cases where Conjecture 1 holds, but the Gorenstein property
fails. The proof of Theorem 2 uses a new method for detecting nonzero classes
in the Gorenstein kernel: intersecting with non-tautological classes.
Recently, several ways of studying the cycle theory of
$\overline{\mathcal{M}}_{g,n}$ beyond the tautological ring have appeared [27,
18, 4]. Lian and Schmitt–van Zelm propose adding Hurwitz cycles parametrizing
covers of curves of positive genus to the calculus. Such cycles tend to be
non-tautological [12, 28, 19, 1], whereas the Hurwitz cycles parametrizing
covers of genus $0$ curves are tautological [10].

######  Remark 3.

Conjecture 1 and the Gorenstein property have natural analogues for moduli
spaces of curves of compact type [26, 9, 20]. In [5], it is shown that if
$g\geq 2$ and $2g+n\geq 12$, then neither
$\mathsf{R}^{*}(\mathcal{M}_{g,n}^{\operatorname{ct}})$ nor
$\mathsf{RH}^{*}(\mathcal{M}_{g,n}^{\operatorname{ct}})$ is Gorenstein.
Moreover, the restriction of the $3$-spin relations to
$\mathcal{M}_{6}^{\operatorname{ct}}$ are complete [5], providing the first
known case where Pixton’s conjecture holds but the tautological ring is not
Gorenstein.

The proof of Theorem 2 is carried out in Sections 2, 3, and 4. In Section 2,
using direct calculations in the tautological ring (Lemmas 11 and 12), we
reduce to the cases when $2g+n=24$. To prove these cases, we recall in Section
3 the result of Petersen–Tommasi [25] and Petersen [24] when $g=2$: there
exists a nonzero tautological class
$\alpha\in\mathsf{RH}^{24}(\overline{\mathcal{M}}_{2,20})$ such that
$\alpha\cdot\beta=0$ for all
$\beta\in\mathsf{RH}^{22}(\overline{\mathcal{M}}_{2,20})$. Let
$\xi:\overline{\mathcal{M}}_{2,20}\rightarrow\overline{\mathcal{M}}_{g,n}$ be
an iterated self-gluing map, which exists for $(g,n)=(2+k,20-2k)$, $0\leq
k\leq 10$, where $k$ is the number of gluings, i.e. $g\geq 2$ and $2g+n=24$.
We show that $\xi_{*}\alpha$ pairs trivially with every tautological class in
complementary codimension. It thus suffices to show that $\xi_{*}\alpha\neq
0$, which we do in Section 4 by pulling back to
$\overline{\mathcal{M}}_{2,20}$ and intersecting with a non-tautological
bielliptic cycle. The intersection computation is done using pullback formulas
for Hurwitz cycles under $\xi^{*}$ developed by Schmitt–van Zelm [27].

A key feature of the proof is that it does not require any explicit knowledge
of $\mathsf{H}^{*}(\overline{\mathcal{M}}_{g,n})$ when $(g,n)\neq(2,20)$.
Petersen and Tommasi’s proof [25] that
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{2,20})$ is not Gorenstein requires
essentially complete knowledge of
$\mathsf{H}^{*}(\overline{\mathcal{M}}_{2,20})$, which is obtained through an
argument with the Leray spectral sequence and the study of the cohomology of
local systems on $\mathcal{A}_{2}$, the moduli space of principally polarized
abelian surfaces. It seems unlikely that this proof strategy will generalize
to higher genus. In particular, when $g\geq 4$, $\mathcal{M}_{g}$ is not an
open substack of $\mathcal{A}_{g}$, so drawing conclusions about the
cohomology of $\mathcal{M}_{g}$ from the cohomology of $\mathcal{A}_{g}$ is
much more difficult. We hope that the idea of reducing to a few pairs $(g,n)$
where both $g$ and $n$ are relatively small may also be useful for studying
Conjecture 1.

######  Remark 4.

The proof of Theorem 2 provides a geometric reason for the failure of the
Gorenstein property: the existence of the non-tautological bielliptic cycle in
$\mathsf{H}^{22}(\overline{\mathcal{M}}_{2,20})$. The failure of the
Gorenstein property for $\mathcal{M}_{g,n}^{\operatorname{ct}}$ does not
require the existence of a non-tautological cycle [5]. Nevertheless, the
failure of the Gorenstein property on $\mathcal{M}_{g}^{\operatorname{ct}}$ is
closely related to the existence of a non-tautological class on
$\mathcal{A}_{g}$, the moduli space of $g$-dimensional principally polarized
abelian varieties [6].

We conjecture the converse of Theorem 2 should also hold.

######  Conjecture 5.

The tautological rings $\mathsf{R}^{*}(\overline{\mathcal{M}}_{g,n})$ and
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ are Gorenstein if and only if
$g=0,1$ or $g\geq 2$ and $2g+n<24$.

######  Remark 6.

If Conjecture 5 is true, the remaining open cases can in principle be checked
computationally, using the Sage package admcycles [8] to compute pairing
matrices. Because of the number of decorated strata classes, this computation
is not currently feasible.

In cohomology, some cases of Conjecture 5 follow immediately from results in
the literature. In particular, the $g=0,1,2$ cases are known [17, 23, 25, 24].
For the $g\geq 3$ cases, one can use that the cohomology rings are generated
by tautological classes for many cases when $g$ and $n$ are small [2, Theorem
1.4]. It follows by Poincaré duality that the cohomological tautological ring
is Gorenstein in these cases. Table 1 summarizes what is already known from
[17, 23, 25, 24, 2].

$g$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---
$c(g)$ | $\infty$ | $\infty$ | $20$ | $9$ | $7$ | $5$ | $3$ | $1$
$24-2g$ | $24$ | $22$ | $20$ | $18$ | $16$ | $14$ | $12$ | $10$
Table 1. $\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ is Gorenstein for
$n<c(g)$. When $g\geq 2$ and $n<24-2g$, Conjecture 5 predicts
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ is Gorenstein.

######  Remark 7.

When $g=0,1$,
$\mathsf{R}^{*}(\overline{\mathcal{M}}_{g,n})=\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$.
We do not know if this equality holds when $g=2$ and $n<c(g)$, but it is true
for $g=2$ and $n<10$ [2, Theorem 1.4]. For $3\leq g\leq 7$ and $n<c(g)$,
$\mathsf{R}^{*}(\overline{\mathcal{M}}_{g,n})=\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$
[2, Theorem 1.4]. Thus, $\mathsf{R}^{*}(\overline{\mathcal{M}}_{g,n})$ is
Gorenstein in these cases.

We can prove several more cases by studying just the even cohomology. Given
$0\leq g\leq 8$, we define $d(g)$ as in Table 2.

######  Theorem 8.

$g$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$
---|---|---|---|---|---|---|---|---|---
$d(g)$ | $\infty$ | $\infty$ | $20$ | $12$ | $10$ | $8$ | $6$ | $4$ | $1$
$24-2g$ | $24$ | $22$ | $20$ | $18$ | $16$ | $14$ | $12$ | $10$ | $8$
Table 2. The even cohomology of $\overline{\mathcal{M}}_{g,n}$ is tautological
for $n<d(g)$. Hence, $\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ is
Gorenstein for $n<d(g)$. When $g\geq 2$ and $n<24-2g$, Conjecture 5 predicts
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ is Gorenstein.

For $g\leq 8$ and $n<d(g)$, the even cohomology of
$\overline{\mathcal{M}}_{g,n}$ is tautological. Therefore,
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ is Gorenstein for $g\leq 8$
and $n<d(g)$.

We do not know the analogous result for
$\mathsf{R}^{*}(\overline{\mathcal{M}}_{g,n})$. In Section 5, we prove Theorem
8 using the stratification of $\overline{\mathcal{M}}_{g,n}$ by topological
type and previously known results about the cohomology (both even and odd) of
moduli spaces of curves of low genus [2, 3, 4]. When $g\geq 2$, Conjecture 5
predicts that $\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ is Gorenstein
for $110$ pairs $(g,n)$. Theorem 8 shows that
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ is Gorenstein for $61$ of
those pairs.

When $\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ is known to have the
Gorenstein property, the even cohomology algebra
$\mathsf{H}^{2*}(\overline{\mathcal{M}}_{g,n})$ is tautologically generated.
It seems likely that the Gorenstein property is simply an accident, occurring
only for small $g$ and $n$ because of the non-existence of non-tautological
classes. Together with Theorem 2, Conjecture 9 below implies Conjecture 5 in
cohomology by Poincaré duality.

######  Conjecture 9.

If $2g+n<24$, $\mathsf{H}^{2*}(\overline{\mathcal{M}}_{g,n})$ is generated by
tautological classes.

### Acknowledgments

I was inspired to work on this problem after a conversation with Dan Petersen.
I am grateful to Emily Clader, Carel Faber, Hannah Larson, Dragos Oprea, Rahul
Pandharipande, Dan Petersen, Aaron Pixton, and Johannes Schmitt for many
related discussions. In particular, Johannes Schmitt suggested the proof of
Lemma 12. Thanks to Carel Faber, Hannah Larson, Rahul Pandharipande, and
Johannes Schmitt for comments on an earlier draft. This research was supported
by a Hermann-Weyl-instructorship from the Forschungsinstitut für Mathematik at
ETH Zürich.

##  2\. Reduction to the case $2g+n=24$

We first reduce Theorem 2 to the cases when $g\geq 2$ and $2g+n=24$.

######  Lemma 10.

The self-gluing and forgetful pushforward maps in top degree

$\mathsf{RH}^{6g-8+2n}(\overline{\mathcal{M}}_{g-1,n+2})\rightarrow\mathsf{RH}^{6g-6+2n}(\overline{\mathcal{M}}_{g,n})\quad\text{and}\quad\mathsf{RH}^{6g-4+2n}(\overline{\mathcal{M}}_{g,n+1})\rightarrow\mathsf{RH}^{6g-6+2n}(\overline{\mathcal{M}}_{g,n})$

are isomorphisms.

###### Proof.

We have
$\mathsf{H}^{6g-6+2n}(\overline{\mathcal{M}}_{g,n})=\mathsf{RH}^{6g-6+2n}(\overline{\mathcal{M}}_{g,n})\cong\mathbb{Q}$
for any $g$ and $n$ [13]. The result follows by Poincaré duality and the fact
that zero cycles in homology push forward nontrivially for degree reasons. ∎

######  Lemma 11.

If $\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ is not Gorenstein, then
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n+1})$ is not Gorenstein.

###### Proof.

Let
$\pi:\overline{\mathcal{M}}_{g,n+1}\rightarrow\overline{\mathcal{M}}_{g,n}$ be
the map forgetting the last marked point. Then

$\pi^{*}:\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})\rightarrow\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n+1})$

is injective because for any
$\alpha\in\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$,

$\frac{1}{2g-2+n}\pi_{*}(\pi^{*}\alpha\cdot\psi_{n+1})=\alpha.$

Let $\beta\in\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ be a nonzero class
such that

$\beta\cdot\gamma=0$

for all $\gamma\in\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n})$ of
complementary degree. Then $\pi^{*}\beta\neq 0$, and for all
$\delta\in\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n+1})$ of complementary
degree

$\pi_{*}(\pi^{*}\beta\cdot\delta)=\beta\cdot\pi_{*}\delta=0.$

Because $\pi_{*}$ is injective in the top degree by Lemma 10,
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,n+1})$ is not Gorenstein.

∎

######  Lemma 12.

If $\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g})$ is not Gorenstein, then
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g+1})$ is not Gorenstein.

###### Proof.

Let $\beta\in\mathsf{RH}^{k}(\overline{\mathcal{M}}_{g})$ be a nonzero class
such that

$\beta\cdot\gamma=0$

for all $\gamma\in\mathsf{RH}^{6g-6-k}(\overline{\mathcal{M}}_{g})$. Set
$\alpha=\pi^{*}\beta$, where
$\pi:\overline{\mathcal{M}}_{g,1}\rightarrow\overline{\mathcal{M}}_{g}$ is the
forgetful map. Let

$\varphi:\overline{\mathcal{M}}_{g,1}\times\overline{\mathcal{M}}_{1,1}\rightarrow\overline{\mathcal{M}}_{g+1}$

be the map attaching an elliptic tail. For
$\zeta\in\mathsf{R}\mathsf{H}^{6g-2-k}(\overline{\mathcal{M}}_{g+1})$, we have

$\varphi_{*}(\alpha\otimes 1)\cdot\zeta=\varphi_{*}((\alpha\otimes
1)\cdot\varphi^{*}\zeta)\in\mathsf{RH}^{6g}(\overline{\mathcal{M}}_{g+1})\cong\mathbb{Q}.$

Consider the class $(\alpha\otimes 1)\cdot\varphi^{*}\zeta$. Because $\zeta$
is tautological,

$\varphi^{*}\zeta\in(\mathsf{RH}^{6g-2-k}(\overline{\mathcal{M}}_{g,1})\otimes\mathsf{RH}^{0}(\overline{\mathcal{M}}_{1,1}))\oplus(\mathsf{RH}^{6g-4-k}(\overline{\mathcal{M}}_{g,1})\otimes\mathsf{RH}^{2}(\overline{\mathcal{M}}_{1,1}))$

under the Künneth decomposition [12, Proposition 1]. The product of
$\alpha\otimes 1$ with any class in the first summand is trivial for dimension
reasons. The product of $\alpha\otimes 1$ with the second summand is zero by
the argument in the proof of Lemma 11.

It suffices to show that $\varphi_{*}(\alpha\otimes 1)\neq 0$. By the self-
intersection formula [12, Equation 11],

$\varphi^{*}\varphi_{*}(\alpha\otimes
1)=\alpha\cdot(\delta_{1,\emptyset}-\psi_{1})\otimes
1+\alpha\otimes(-\psi_{1}).$

Here, $\delta_{1,\emptyset}$ is the boundary divisor on
$\overline{\mathcal{M}}_{g,1}$ given by the gluing map
$\overline{\mathcal{M}}_{1,1}\times\overline{\mathcal{M}}_{g-1,2}\rightarrow\overline{\mathcal{M}}_{g,1}$.
We see that

$\mathrm{pr}_{1*}(\varphi^{*}\varphi_{*}(\alpha\otimes
1))=-\frac{1}{24}\alpha\neq 0,$

where
$\mathrm{pr}_{1}:\overline{\mathcal{M}}_{g,1}\times\overline{\mathcal{M}}_{1,1}\rightarrow\overline{\mathcal{M}}_{g,1}$
is projection onto the first factor. Thus, $\varphi_{*}(\alpha\otimes 1)\neq
0$. ∎

Combining Lemmas 11 and 12, we see it suffices to prove Theorem 2 in the cases
when $g\geq 2$ and $2g+n=24$.

##  3\. Admissible covers and cohomology in genus $2$

We collect background information for the proof of Theorem 2 on moduli spaces
of admissible double covers and the cohomology of
$\overline{\mathcal{M}}_{2,n}$.

###  3.1. Admissible covers and non-tautological cycles

We consider the stacks $\overline{\mathcal{H}}_{g,G,\xi}$ of admissible
$G$-covers of genus $g$ with monodromy data $\xi$, where $G$ is a finite
group, as in [27, Definitions 3.2 and 3.5]. See [27] for complete details. In
this paper, we always take $G=\mathbb{Z}/2\mathbb{Z}$. In this special case,
the monodromy data is simply a tuple $\xi=(a_{1},\dots,a_{b})$ of generators
for the stabilizer group of points in the fibers of the source curve over $b$
ordered points of the target curve, controlling the ramification behavior. The
group law will be written additively, and so $\xi$ has entries in $\\{0,1\\}$.
An entry $a_{i}=1$ corresponds to marking a ramification point of the source
curve over the $i$th marked point of the target curve. An entry $a_{j}=0$
corresponds to marking two points of the source pairwise switched by the
covering involution over the $j$th marked point of the target.

As a rule, we will mark every point of ramification when considering moduli of
admissible double covers. That is, if we are given the genus $h$ of the target
curve, the monodromy data $\xi$ will contain $2g-4h+2$ entries $1$, the
maximal number allowed by the Riemann–Hurwitz formula. Conversely, $\xi$ and
$g$ determine $h$.

The source map

$\phi:\overline{\mathcal{H}}_{g,G,\xi}\rightarrow\overline{\mathcal{M}}_{g,2g-4h+2+2n_{2}}$

sends an admissible cover to its source curve together with all of the marked
points determined by the monodromy data $\xi$ [27, Theorem 3.7]. When
$\xi=(1^{2g-4h+2},0^{n_{2}})$, we denote the image of the source map by
$\overline{\mathcal{B}}_{g\rightarrow h,2g-4h+2,2n_{2}}$. One can further
compose with the forgetful map
$\overline{\mathcal{M}}_{g,2g-4h+2+2n_{2}}\rightarrow\overline{\mathcal{M}}_{g,n_{1}+2n_{2}}$,
forgetting the first $2g-4h+2-n_{1}$ points. The image of
$\overline{\mathcal{B}}_{g\rightarrow h,2g-4h+2,2n_{2}}$ under this map is
denoted $\overline{\mathcal{B}}_{g\rightarrow h,n_{1},2n_{2}}$. When $h=1$,
these loci play a fundamental role because of the following result of
Graber–Pandharipande [12] (when $g=2$ and $n_{2}=10$) and van Zelm [28].

######  Theorem 13 (Graber–Pandharipande, van Zelm).

The class $[\overline{\mathcal{B}}_{g\rightarrow
1,0,2n_{2}}]\in\mathsf{H}^{22}(\overline{\mathcal{M}}_{g,2n_{2}})$ is non-
tautological if $g\geq 2$ and $g+n_{2}=12$.

######  Remark 14.

The proof of Theorem 2 only uses Theorem 13 when $g=2$. In fact, the $g>2$
cases of Theorem 13 will follow from the proof of Theorem 2, using equation
(1) below and the fact that tautological classes pull back to tautological
classes. The argument is not the same as van Zelm’s proof of the $g\geq 3$
part of Theorem 13, although it is quite similar.

On the other hand, classes of hyperelliptic loci are tautological [10,
Proposition 1].

######  Theorem 15 (Faber–Pandharipande).

The class $[\overline{\mathcal{B}}_{g\rightarrow 0,n_{1},2n_{2}}]$ is
tautological for any $g,n_{1},n_{2}$.

###  3.2. Cohomology in genus $2$

The cohomology of $\overline{\mathcal{M}}_{2,n}$ for $n\leq 20$ is completely
understood by Petersen–Tommasi [25, Theorems A and B] and Petersen [24,
Theorem 3.8].

######  Theorem 16 (Petersen–Tommasi, Petersen).

Let $k\in\mathbb{N}$ be even.

  1. (1)

For $n<20$,
$\mathsf{RH}^{k}(\overline{\mathcal{M}}_{2,n})=\mathsf{H}^{k}(\overline{\mathcal{M}}_{2,n})$.

  2. (2)

For $k\neq 22$,
$\mathsf{RH}^{k}(\overline{\mathcal{M}}_{2,20})=\mathsf{H}^{k}(\overline{\mathcal{M}}_{2,20})$.

  3. (3)

The image of the pushforward map
$\mathsf{H}^{k-2}(\widetilde{\partial\mathcal{M}_{2,20}})\rightarrow\mathsf{H}^{k}(\overline{\mathcal{M}}_{2,20})$
lies in $\mathsf{RH}^{k}(\overline{\mathcal{M}}_{2,20})$, where
$\widetilde{\partial\mathcal{M}_{2,20}}$ is the normalization of the boundary.

Theorem 16(2) combined with Theorem 13 implies that
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{2,20})$ is not Gorenstein.

In contrast to the essentially complete result of Theorem 16, little is known
about the structure of the cohomology of $\overline{\mathcal{M}}_{g,2n_{2}}$
beyond Theorem 13 when $g\geq 3$ and $g+n_{2}=12$. The proof of Theorem 2
relies on pulling back to $\overline{\mathcal{M}}_{2,20}$ and leveraging the
complete description of its cohomology, as follows.

By Poincaré duality and Theorem 16(2), the intersection pairing

$\mathsf{H}^{22}(\overline{\mathcal{M}}_{2,20})\times\mathsf{RH}^{24}(\overline{\mathcal{M}}_{2,20})\rightarrow\mathbb{Q}$

is perfect. Hence, we can identify
$\mathsf{RH}^{24}(\overline{\mathcal{M}}_{2,20})$ with the dual of
$\mathsf{H}^{22}(\overline{\mathcal{M}}_{2,20})$. Pick
$\alpha\in\mathsf{RH}^{24}(\overline{\mathcal{M}}_{2,20})$ such that
$\alpha\cdot[\overline{\mathcal{B}}_{2\rightarrow 1,0,20}]=1$, but
$\alpha\cdot\beta=0$ for every tautological class
$\beta\in\mathsf{RH}^{22}(\overline{\mathcal{M}}_{2,20})$.222The class
$\alpha$ can be obtained by completing $[\overline{\mathcal{B}}_{2\rightarrow
1,0,20}]$ together with a basis for the tautological subspace
$\mathsf{RH}^{22}(\overline{\mathcal{M}}_{2,20})$ to a basis for
$\mathsf{H}^{22}(\overline{\mathcal{M}}_{2,20})$ and choosing $\alpha$ to be
the first basis vector in the dual basis.

Consider the gluing map

$\varphi_{g}:\overline{\mathcal{M}}_{2,20}\rightarrow\overline{\mathcal{M}}_{g,2n_{2}},$

given by identifying the last $g-2$ pairs of marked points on
$\overline{\mathcal{M}}_{2,20}$. Note that $g+n_{2}=12$. Let
$\gamma\in\mathsf{RH}^{24}(\overline{\mathcal{M}}_{g,2n_{2}})$. By the
projection formula,

$\varphi_{g*}\alpha\cdot\gamma=\varphi_{g*}(\alpha\cdot\varphi_{g}^{*}\gamma)=0$

because $\varphi_{g}^{*}\gamma$ is tautological. By iteratively applying Lemma
10, $\varphi_{g*}$ is an isomorphism in the top degree. Therefore, if
$\varphi_{g*}\alpha\neq 0$,
$\mathsf{RH}^{*}(\overline{\mathcal{M}}_{g,2n_{2}})$ is not Gorenstein. In
fact, it suffices to show that $\varphi_{12*}\alpha\neq 0$, as $\varphi_{g}$
factors $\varphi_{12}$. We will show that $\varphi_{12*}\alpha\neq 0$ by
computing the intersection of $\varphi_{12*}\alpha$ with the bielliptic locus
in genus $12$. By the projection formula,

$\varphi_{12*}\alpha\cdot[\overline{\mathcal{B}}_{12\rightarrow
1,0,0}]=\varphi_{12*}(\alpha\cdot\varphi_{12}^{*}[\overline{\mathcal{B}}_{12\rightarrow
1,0,0}]).$

By Lemma 10 applied iteratively, it suffices to show that
$\alpha\cdot\varphi_{12}^{*}[\overline{\mathcal{B}}_{12\rightarrow 1,0,0}]\neq
0$. We claim that

(1) $\varphi_{12}^{*}[\overline{\mathcal{B}}_{12\rightarrow
1,0,0}]=c[\overline{\mathcal{B}}_{2\rightarrow 1,0,20}]+T,$

where $c\neq 0$ and $T\in\mathsf{RH}^{22}(\overline{\mathcal{M}}_{2,20})$.
Assuming (1), we have

$\alpha\cdot\varphi_{12}^{*}[\overline{\mathcal{B}}_{12\rightarrow
1,0,0}]=\alpha\cdot c[\overline{\mathcal{B}}_{2\rightarrow
1,0,20}]+\alpha\cdot T=c\neq 0.$

To finish the proof of Theorem 2, we thus only need to show that equation (1)
holds.

######  Remark 17.

The non-tautological part of $\mathsf{H}^{22}(\overline{\mathcal{M}}_{2,20})$
spans a copy of the irreducible $\mathbb{S}_{20}$ representation associated to
the partition $(2^{10})$ [24, Remark 3.10], which makes equation (1)
surprisingly simple.

##  4\. Pulling back $\overline{\mathcal{B}}_{12\rightarrow 1,0,0}$: proof of
equation (1)

Schmitt and van Zelm [27] developed a method for pulling back admissible
covers cycles to boundary strata $\overline{\mathcal{M}}_{\Gamma}$ of
$\overline{\mathcal{M}}_{g,n}$ associated to a stable graph $\Gamma$. We apply
their method to prove equation (1).

Let $A$ be the stable graph with one vertex of genus $2$ and $10$ loop edges,
with the half-edges labeled by $\\{1,\dots,20\\}$ and edges given by
$(1,2),(3,4),\dots,(19,20)$. Then
$\overline{\mathcal{M}}_{A}=\overline{\mathcal{M}}_{2,20}$, and the gluing
morphism

$\varphi_{A}:\overline{\mathcal{M}}_{A}\rightarrow\overline{\mathcal{M}}_{12}$

is the map $\varphi_{12}$, defined in the previous section. Let $B$ denote the
graph obtained from $A$ by adding $22$ legs, corresponding to the ramification
points of a double cover of a genus $1$ curve by a genus $12$ curve. The
associated gluing map is denoted by

$\varphi_{B}:\overline{\mathcal{M}}_{B}\rightarrow\overline{\mathcal{M}}_{12,22}.$

Set $\xi=(1^{22})$. We have the composition

$\overline{\mathcal{H}}_{12,G,\xi}\xrightarrow{\phi}\overline{\mathcal{M}}_{12,22}\xrightarrow{\pi}\overline{\mathcal{M}}_{12},$

where the first map is the source curve map and the second map forgets the
markings. The image in $\overline{\mathcal{M}}_{12}$ is by definition
$\overline{\mathcal{B}}_{12\rightarrow 1,0,0}$. We want to compute the
pullback

$\varphi_{A}^{*}[\overline{\mathcal{B}}_{12\rightarrow
1,0,0}]=\varphi_{A}^{*}\pi_{*}\phi_{*}[\overline{\mathcal{H}}_{12,G,\xi}].$

###  4.1. The fiber product and admissible $G$-graphs

The first step is to describe the fiber product of $\varphi_{B}$ and $\phi$,
which Schmitt and van Zelm do combinatorially in terms of admissible
$G$-graphs [27, Proposition 4.3]. We summarize what is needed from their work
here. We continue to take $G=\mathbb{Z}/2\mathbb{Z}$, as it simplifies the
discussion of the monodromy data.

Given an admissible cover in $\overline{\mathcal{H}}_{g,G,\xi}$, we take the
dual graph $\Gamma$ of the source curve $C$. The graph $\Gamma$ inherits an
action of $G$. In addition to the graph $\Gamma$ with its $G$ action, we
record the stabilizer group of each half-edge and leg. This data allows us to
remember the ramification behavior at the nodes and markings. The data of
$\Gamma$ with the $G$ action and the stabilizer group of each half-edge and
leg is called an admissible $G$ graph and will be denoted by $(\Gamma,G)$. See
[27, Section 3.5] for a completely combinatorial definition.

To each admissible $G$ graph $(\Gamma,G)$, there is a space
$\overline{\mathcal{H}}_{(\Gamma,G)}$ and a morphism

$\varphi_{(\Gamma,G)}:\overline{\mathcal{H}}_{(\Gamma,G)}\rightarrow\overline{\mathcal{H}}_{g,G,\xi}$

whose image is the closure of the locus of admissible covers with admissible
$G$ graph $(\Gamma,G)$, defined as follows. Let $V$ be a set of vertices of
$\Gamma$ consisting of one vertex from each orbit of the $G$ action. Let
$G_{v}\subset G$ denote the stabilizer of $v\in V$, and let $\xi_{v}$ be the
vector of monodromy data associated to the half-edges and legs incident to $v$
(recall that an admissible $G$ graph includes the data of the stabilizer group
of each half-edge and leg, allowing for the assignment of monodromy data).
Then

$\overline{\mathcal{H}}_{(\Gamma,G)}=\prod_{v\in
V}\overline{\mathcal{H}}_{g(v),G_{v},\xi_{v}},$

and the morphism
$\varphi_{(\Gamma,G)}:\overline{\mathcal{H}}_{(\Gamma,G)}\rightarrow\overline{\mathcal{H}}_{g,G,\xi}$
is obtained by gluing according to the graph $\Gamma$. That
$\varphi_{(\Gamma,G)}$ is well-defined is established in [27, Page 30]. There
is also a natural morphism

$\phi_{(\Gamma,G)}:\overline{\mathcal{H}}_{(\Gamma,G)}\rightarrow\overline{\mathcal{M}}_{\Gamma}$

obtained by remembering the source curve and forgetting the $G$ action such
that the diagram

${{\overline{\mathcal{H}}_{(\Gamma,G)}}}$${{\overline{\mathcal{H}}_{g,G,\xi}}}$${\overline{\mathcal{M}}_{\Gamma}}$${{\overline{\mathcal{M}}_{g,n}}}$$\scriptstyle{\varphi_{(\Gamma,G)}}$$\scriptstyle{\phi_{(\Gamma,G)}}$$\scriptstyle{\phi}$$\scriptstyle{\varphi_{\Gamma}}$

commutes, where $\varphi_{\Gamma}$ is the gluing map associated to the stable
graph $\Gamma$.

Recall that a morphism of stable graphs $f:\Gamma\rightarrow\Gamma^{\prime}$
(also called a $\Gamma^{\prime}$ structure on $\Gamma$) induces a gluing
morphism

$\varphi_{f}:\overline{\mathcal{M}}_{\Gamma}\rightarrow\overline{\mathcal{M}}_{\Gamma^{\prime}}.$

If $(\Gamma,G)$ is an admissible $G$ graph, then a $\Gamma^{\prime}$ structure
on $(\Gamma,G)$ is simply a morphism of the underlying stable graphs
$f:\Gamma\rightarrow\Gamma^{\prime}$. Such a morphism includes the data of an
injection $\beta:E(\Gamma^{\prime})\hookrightarrow E(\Gamma)$ on the sets of
edges. The $\Gamma^{\prime}$ structure $f:\Gamma\rightarrow\Gamma^{\prime}$ is
called generic if $G\cdot\mathrm{Im}(\beta)=E(\Gamma)$. That is, every edge of
$\Gamma$ is in the $G$ orbit of an edge of $\Gamma^{\prime}$. Given a generic
$\Gamma^{\prime}$ structure $f:\Gamma\rightarrow\Gamma^{\prime}$, there is a
morphism

$\phi_{f}:\overline{\mathcal{H}}_{(\Gamma,G)}\xrightarrow{\phi_{(\Gamma,G)}}\overline{\mathcal{M}}_{\Gamma}\xrightarrow{\varphi_{f}}\overline{\mathcal{M}}_{\Gamma^{\prime}}.$

Let $\mathfrak{H}_{\Gamma^{\prime}}$ denote the set of isomorphism classes of
admissible $G$ graphs $\Gamma$ with generic $\Gamma^{\prime}$ structures
$f:\Gamma\rightarrow\Gamma^{\prime}$. Setting $\Gamma^{\prime}=B$, we have the
following specialization of [27, Proposition 4.3].

######  Proposition 18.

The diagram

(2)
${{\coprod_{(\Gamma,G,f)\in\mathfrak{H}_{B}}\overline{\mathcal{H}}_{(\Gamma,G)}}}$${{\overline{\mathcal{H}}_{12,G,\xi}}}$${\overline{\mathcal{M}}_{B}}$${{\overline{\mathcal{M}}_{12,22}}}$$\scriptstyle{\coprod\phi_{f}}$$\scriptstyle{\coprod\varphi_{(\Gamma,G)}}$$\scriptstyle{\phi}$$\scriptstyle{\varphi_{B}}$

is Cartesian.

###  4.2. Excess intersection and the pullback formula

The excess intersection class associated to the diagram (2) can be computed
separately on each component. The excess bundle $E_{f}$ is by definition the
quotient of the normal bundles
$E_{f}=\phi_{f}^{*}N_{\varphi_{B}}/N_{\varphi_{(\Gamma,G)}}$. The top Chern
class of $E_{f}$ is computed in [27, Proposition 4.7]. Let
$\beta:E(B)\rightarrow E(\Gamma)$ be the map on edges induced by the generic
$B$ structure $(\Gamma,G,f)$. Let $N$ be a set of representatives for the
orbit of the $G$ action on $E(\Gamma)$ such that
$N\subset\operatorname{Im}(\beta)$. Then

$c_{\mathrm{top}}(E_{f})=\prod_{(h,h^{\prime})\in\operatorname{Im}(\beta)\smallsetminus
N}(-\psi_{h}-\psi_{h}^{\prime}).$

By definition, these $\psi$ classes are pulled back from
$\overline{\mathcal{M}}_{B}$ under $\phi_{f}$, as noted in the discussion
following [27, Theorem 4.9]. In fact, since they are associated to edges of
$B$, they are pulled back from $\overline{\mathcal{M}}_{A}$ under the
composition

$\overline{\mathcal{H}}_{(\Gamma,G)}\xrightarrow{\phi_{f}}\overline{\mathcal{M}}_{B}\xrightarrow{\pi_{B}}\overline{\mathcal{M}}_{A},$

where the map
$\pi_{B}:\overline{\mathcal{M}}_{B}\rightarrow\overline{\mathcal{M}}_{A}$
simply forgets legs. Now consider the diagram

${{\coprod_{(\Gamma,G,f)\in\mathfrak{H}_{B}}\overline{\mathcal{H}}_{(\Gamma,G)}}}$${{\overline{\mathcal{H}}_{12,G,\xi}}}$${\overline{\mathcal{M}}_{B}}$${{\overline{\mathcal{M}}_{12,22}}}$${\overline{\mathcal{M}}_{A}}$${\overline{\mathcal{M}}_{12}\,.}$$\scriptstyle{\coprod\phi_{f}}$$\scriptstyle{\coprod\varphi_{(\Gamma,G)}}$$\scriptstyle{\phi}$$\scriptstyle{\pi_{B}}$$\scriptstyle{\varphi_{B}}$$\scriptstyle{\pi}$$\scriptstyle{\varphi_{A}}$

Specializing [27, Theorem 4.12] to the case at hand, we have

(3) $\varphi_{A}^{*}[\overline{\mathcal{B}}_{12\rightarrow
1,0,0}]=\varphi_{A}^{*}\pi_{*}\phi_{*}[\overline{\mathcal{H}}_{12,G,\xi}]=\sum_{(\Gamma,G,f)\in\mathfrak{H}_{B}}\pi_{B*}\left(c_{\mathrm{top}}(E_{f})\cdot\phi_{f*}[\overline{\mathcal{H}}_{(\Gamma,G)}]\right).$

###  4.3. Pullback computation

We will now establish equation (1) using equation (3). We examine each term in
the summation on the right hand side of equation (3):

$\sum_{(\Gamma,G,f)\in\mathfrak{H}_{B}}\pi_{B*}\left(c_{\mathrm{top}}(E_{f})\cdot\phi_{f*}[\overline{\mathcal{H}}_{(\Gamma,G)}]\right).$

We first study a large class of $(\Gamma,G,f)\in\mathfrak{H}_{B}$ that
contribute only tautological classes.

######  Lemma 19.

Suppose $(\Gamma,G,f)\in\mathfrak{H}_{B}$ satisfies any of the following
properties:

  1. (1)

$\overline{\mathcal{H}}_{(\Gamma,G)}$ is not of the expected codimension;

  2. (2)

$\Gamma$ has no vertices of genus $2$;

  3. (3)

$G$ acts nontrivially on the vertices;

  4. (4)

The quotient graph $\Gamma/G$ has first Betti number $1$;

  5. (5)

$\Gamma$ has loop edges;

  6. (6)

$G$ acts trivially on an edge;

  7. (7)

$\Gamma$ has vertices $v_{1},v_{2}$ connected by $n\geq 1$ edges and $n\neq
2$.

Then
$\pi_{B*}\left(c_{\mathrm{top}}(E_{f})\cdot\phi_{f*}[\overline{\mathcal{H}}_{(\Gamma,G)}]\right)$
is tautological.

###### Proof.

Suppose $(\Gamma,G,f)$ is such that (1) holds. Then the excess bundle $E_{f}$
is of rank at least $1$. Because $c_{\mathrm{top}}(E_{f})$ is pulled back from
$\overline{\mathcal{M}}_{A}$, the term

$\pi_{B*}\left(c_{\mathrm{top}}(E_{f})\cdot\phi_{f*}[\overline{\mathcal{H}}_{(\Gamma,G)}]\right)$

is a product of algebraic classes of lower codimension on
$\overline{\mathcal{M}}_{A}=\overline{\mathcal{M}}_{2,20}$ by the projection
formula. Hence, by Theorem 16(2),
$\pi_{B*}\left(c_{\mathrm{top}}(E_{f})\cdot\phi_{f*}[\overline{\mathcal{H}}_{(\Gamma,G)}]\right)$
is tautological.

From now on, we suppose $(\Gamma,G,f)$ does not satisfy property (1). Then
there is no excess contribution, and we study only the terms of the form
$\pi_{B*}\phi_{f*}[\overline{\mathcal{H}}_{(\Gamma,G)}]$. Now suppose
$(\Gamma,G,f)$ is such that (2) holds. First, assume that $\Gamma$ has a
vertex of genus $1$. Then
$\pi_{B*}\phi_{f*}[\overline{\mathcal{H}}_{(\Gamma,G)}]$ is supported on the
boundary of $\overline{\mathcal{M}}_{A}$, and hence is tautological by Theorem
16(3). We argue similarly if all vertices of $\Gamma$ are of genus $0$.

From now on, we assume $(\Gamma,G,f)$ does not satisfy property (2), and so
has a unique vertex of genus $2$. Suppose that $(\Gamma,G,f)$ is such that $G$
acts non-trivially on the vertices of $\Gamma$. We pick a pair $v_{1}$ and
$v_{2}$ of genus $0$ vertices interchanged by the $G$ action. The $G$ action
fixes all of the legs because the monodromy data is $\xi=(1^{22})$. Therefore,
there can be no legs attached to $v_{1}$ and $v_{2}$. Under the pushforward
$\pi_{B*}$ forgetting the legs, the vertices $v_{1}$ and $v_{2}$ are not
contracted, meaning $\pi_{B*}\phi_{f*}[\overline{\mathcal{H}}_{(\Gamma,G)}]$
is supported on the boundary of $\overline{\mathcal{M}}_{A}$, and is thus
tautological.

From now on, we assume $(\Gamma,G,f)$ does not satisfy property (3). Suppose
that $\Gamma/G$ has first Betti number $1$. It follows that every vertex of
the quotient graph is of genus $0$, as the target curve has genus $1$.
Therefore, in the product decomposition

$\overline{\mathcal{H}}_{(\Gamma,G)}=\prod_{v\in
V}\overline{\mathcal{H}}_{g(v),G_{v},\xi_{v}},$

the term corresponding to the genus $2$ vertex parametrizes admissible
hyperelliptic covers. Such spaces contribute only tautological classes by
Theorem 15.

From now on, we assume $(\Gamma,G,f)$ does not satisfy property (4). Suppose
that $\Gamma$ has a vertex with a loop edge. The quotient graph $\Gamma/G$
must have a loop edge because $G$ acts trivially on the vertices. This
contradicts the assumption that $(\Gamma,G,f)$ does not satisfy property (4).

From now on, we assume $(\Gamma,G,f)$ does not satisfy property (5). Consider
the case when $\Gamma$ has vertices $v_{1}$ and $v_{2}$ connected by an edge
$e$ that is fixed under the $G$ action. Because the $B$ structure is generic,
$e$ must be an edge of $B$. But $e$ is not a loop edge, whereas all the edges
of $B$ are. Therefore, there must exist another path in $\Gamma$ connecting
$v_{1}$ and $v_{2}$, in which all constituent edges are contracted by the $B$
structure $f:\Gamma\rightarrow B$. But then the quotient graph $\Gamma/G$ must
have nonzero first Betti number, as the images of $v_{1}$ and $v_{2}$ are
connected by at least two edges, contradicting that $(\Gamma,G,f)$ does not
satisfy property (4).

From now on, we assume $(\Gamma,G,f)$ does not satisfy property (6). Suppose
that $\Gamma$ has vertices $v_{1}$ and $v_{2}$ connected by $n\geq 1$ edges
$e_{1},\dots,e_{n}$. If $n$ is odd, then because $G$ acts trivially on the
vertices, it must also fix an edge, violating the assumption that it does not
satisfy property (6). Therefore, $n$ is even. If $n\geq 4$, then the quotient
graph $\Gamma/G$ has first Betti number at least $1$, violating the assumption
that it does not satisfy property (4).

∎

###### Proof of Equation (1).

We calculate using equation (3):

$\varphi_{A}^{*}[\overline{\mathcal{B}}_{12\rightarrow
1,0,0}]=\varphi_{A}^{*}\pi_{*}\phi_{*}[\overline{\mathcal{H}}_{12,G,\xi}]=\sum_{(\Gamma,G,f)\in\mathfrak{H}_{B}}\pi_{B*}\left(c_{\mathrm{top}}(E_{f})\cdot\phi_{f*}[\overline{\mathcal{H}}_{(\Gamma,G)}]\right).$

We need only to consider the terms on the right hand side satisfying none of
the properties listed in Lemma 19. By the negation of property (1), such terms
are of the form

$\pi_{B*}\phi_{f*}[\overline{\mathcal{H}}_{(\Gamma,G)}].$

By the negation of properties (2) and (5), there is one genus $2$ vertex, all
other vertices are of genus $0$, and there are no loops on any vertex. By the
negation of property (4), the first Betti number of the quotient graph
$\Gamma/G$ is $0$. Hence, the image of the unique vertex of genus $2$ is a
vertex of genus $1$. By the negation of property (3), the vertices of genus
$0$ map bijectively to vertices of genus $0$ of $\Gamma/G$. By
Riemann–Hurwitz, we see that every vertex of $\Gamma$ has two markings. By the
negation of properties (6) and (7), any two vertices $v_{1}$ and $v_{2}$ of
$\Gamma$ are either not connected by edges or are connected by exactly two
edges pairwise switched by the $G$ action.

We consider the case where there is a genus $0$ vertex $v_{1}$ not connected
to the genus $2$ vertex. If $v_{1}$ is connected to at least two other
vertices, then the component corresponding to $v_{1}$ is not contracted under
the map forgetting the marked points. It follows that
$\pi_{B*}\phi_{f*}[\overline{\mathcal{H}}_{(\Gamma,G)}]$ is supported on the
boundary, and is thus tautological. If $v_{1}$ is connected to only one other
vertex, then under the map forgetting the marked points, $v_{1}$ is
contracted. The resulting graph has a genus $0$ vertex with a loop, and thus
$\pi_{B*}\phi_{f*}[\overline{\mathcal{H}}_{(\Gamma,G)}]$ is supported on the
boundary.

Finally, we consider the case when no two genus $0$ vertices are connected by
edges. Upon forgetting the marked points, all of the genus $0$ components are
contracted, and the cycle
$\pi_{B*}\phi_{f*}[\overline{\mathcal{H}}_{(\Gamma,G)}]$ is a positive
multiple of $[\overline{\mathcal{B}}_{2\rightarrow 1,0,20}]$.

∎

##  5\. Proof of Theorem 8

We prove Theorem 8 by induction on $g$ and $n$. The proof idea is analogous to
[2, Lemma 4.1], but extra care is needed for possible contributions from odd
cohomology in the boundary. The statement holds for $g=0$ [17], $g=1$ [23],
and $g=2$ [24]. For any $g,n$ and $k$, there is an exact sequence

(4)
$\mathsf{H}^{2k-2}(\widetilde{\partial\mathcal{M}_{g,n}})\rightarrow\mathsf{H}^{2k}(\overline{\mathcal{M}}_{g,n})\rightarrow\mathsf{W}_{2k}\mathsf{H}^{2k}(\mathcal{M}_{g,n})\rightarrow
0,$

where $\widetilde{\partial\mathcal{M}_{g,n}}$ is the normalization of the
boundary. For $3\leq g\leq 8$, $n<d(g)$, and $k$ arbitrary, we have that

$\mathsf{W}_{2k}\mathsf{H}^{2k}(\mathcal{M}_{g,n})=\mathsf{RH}^{2k}(\mathcal{M}_{g,n})$

by [2, Theorem 1.4] and [4, Theorem 1.10 and Lemma 4.3]. Therefore, using the
exact sequence (4), it suffices to study the boundary contribution.

$g$ | $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$
---|---|---|---|---|---|---|---|---
$e(g)$ | $\infty$ | $11$ | $10$ | $9$ | $7$ | $5$ | $3$ | $1$
Table 3. The odd cohomology of $\overline{\mathcal{M}}_{g,n}$ vanishes for
$n<e(g)$.

Up to quotients by finite groups, the normalization of the boundary
$\widetilde{\partial\mathcal{M}_{g,n}}$ is the disjoint union of spaces of the
form $\overline{\mathcal{M}}_{g-1,n+2}$ and
$\overline{\mathcal{M}}_{g_{1},n_{1}+1}\times\overline{\mathcal{M}}_{g_{2},n_{2}+1}$,
where $g_{1}+g_{2}=g$ and $n_{1}+n_{2}=n$. By induction on $g$ and $n$, the
even cohomology of $\overline{\mathcal{M}}_{g-1,n+2}$ and
$\overline{\mathcal{M}}_{g_{1},n_{1}+1}$ and
$\overline{\mathcal{M}}_{g_{2},n_{2}+1}$ are tautological. We thus only need
to consider the summands in the Künneth decomposition

$\mathsf{H}^{2k-2}(\overline{\mathcal{M}}_{g_{1},n_{1}+1}\times\overline{\mathcal{M}}_{g_{2},n_{2}+1})\cong\bigoplus_{p+q=2k-2}\mathsf{H}^{p}(\overline{\mathcal{M}}_{g_{1},n_{1}+1})\otimes\mathsf{H}^{q}(\overline{\mathcal{M}}_{g_{2},n_{2}+1})$

when both $p$ and $q$ are odd. It follows from [2, Theorem 1.4] that for
$n_{1}+1<e(g_{1})$ (respectively, $n_{2}+1<e(g_{2})$), as defined in Table 3,
all of the odd cohomology of $\overline{\mathcal{M}}_{g_{1},n_{1}+1}$
(respectively, $\overline{\mathcal{M}}_{g_{2},n_{2}+1}$) vanishes. Therefore,
for $3\leq g\leq 8$ and $n<d(g)$, the terms

$\mathsf{H}^{p}(\overline{\mathcal{M}}_{g_{1},n_{1}+1})\otimes\mathsf{H}^{q}(\overline{\mathcal{M}}_{g_{2},n_{2}+1})$

with $p$ and $q$ odd vanish identically, as either
$\mathsf{H}^{p}(\overline{\mathcal{M}}_{g_{1},n_{1}+1})$ or
$\mathsf{H}^{q}(\overline{\mathcal{M}}_{g_{2},n_{2}+1})$ is zero. ∎

## References

  * [1] Veronica Arena, Samir Canning, Emily Clader, Richard Haburcak, Amy Q Li, Siao Chi Mok, and Carolina Tamborini, _Holomorphic forms and non-tautological cycles on moduli spaces of curves_ , arXiv preprint arXiv:2402.03874 (2024).
  * [2] Samir Canning and Hannah Larson, _On the Chow and cohomology rings of moduli spaces of stable curves_ , arXiv preprint arXiv:2208.02357 (2022).
  * [3] Samir Canning, Hannah Larson, and Sam Payne, _The eleventh cohomology group of $\overline{\mathcal{M}}_{g,n}$_, Forum Math. Sigma 11 (2023), Paper No. e62, 18. MR 4616435
  * [4] by same author, _Extensions of tautological rings and motivic structures in the cohomology of $\overline{\mathcal{M}}_{g,n}$_, arXiv preprint arXiv:2307.08830 (2023).
  * [5] Samir Canning, Hannah Larson, and Johannes Schmitt, _The Gorenstein property and Pixton’s conjecture for compact type moduli_ , in preparation (2024).
  * [6] Samir Canning, Dragos Oprea, and Rahul Pandharipande, _Cycles on the moduli space of principally polarized abelian varieties_ , in preparation (2024).
  * [7] Emily Clader and Felix Janda, _Pixton’s double ramification cycle relations_ , Geom. Topol. 22 (2018), no. 2, 1069–1108. MR 3748684
  * [8] Vincent Delecroix, Johannes Schmitt, and Jason van Zelm, _admcycles—a Sage package for calculations in the tautological ring of the moduli space of stable curves_ , J. Softw. Algebra Geom. 11 (2021), no. 1, 89–112. MR 4387186
  * [9] C. Faber and R. Pandharipande, _Logarithmic series and Hodge integrals in the tautological ring_ , vol. 48, 2000, With an appendix by Don Zagier, Dedicated to William Fulton on the occasion of his 60th birthday, pp. 215–252. MR 1786488
  * [10] by same author, _Relative maps and tautological classes_ , J. Eur. Math. Soc. (JEMS) 7 (2005), no. 1, 13–49. MR 2120989
  * [11] Carel Faber, _A conjectural description of the tautological ring of the moduli space of curves_ , Moduli of curves and abelian varieties, Aspects Math., vol. E33, Friedr. Vieweg, Braunschweig, 1999, pp. 109–129. MR 1722541
  * [12] T. Graber and R. Pandharipande, _Constructions of nontautological classes on moduli spaces of curves_ , Michigan Math. J. 51 (2003), no. 1, 93–109. MR 1960923
  * [13] Tom Graber and Ravi Vakil, _On the tautological ring of $\overline{\mathscr{M}}_{g,n}$_, Turkish J. Math. 25 (2001), no. 1, 237–243. MR 1829089
  * [14] Richard Hain and Eduard Looijenga, _Mapping class groups and moduli spaces of curves_ , Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Part 2, Amer. Math. Soc., Providence, RI, 1997, pp. 97–142. MR 1492535
  * [15] Felix Janda, _Relations on $\overline{M}_{g,n}$ via equivariant Gromov-Witten theory of $\mathbb{P}^{1}$_, Algebr. Geom. 4 (2017), no. 3, 311–336. MR 3652083
  * [16] by same author, _Frobenius manifolds near the discriminant and relations in the tautological ring_ , Lett. Math. Phys. 108 (2018), no. 7, 1649–1675. MR 3802725
  * [17] Sean Keel, _Intersection theory of moduli space of stable $n$-pointed curves of genus zero_, Trans. Amer. Math. Soc. 330 (1992), no. 2, 545–574. MR 1034665
  * [18] Carl Lian, _The $\mathcal{H}$-tautological ring_, Selecta Math. (N.S.) 27 (2021), no. 5, Paper No. 96, 74. MR 4320845
  * [19] by same author, _Non-tautological Hurwitz cycles_ , Math. Z. 301 (2022), no. 1, 173–198. MR 4405647
  * [20] R. Pandharipande, _Three questions in Gromov-Witten theory_ , Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 503–512. MR 1957060
  * [21] Rahul Pandharipande, _A calculus for the moduli space of curves_ , Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., vol. 97.1, Amer. Math. Soc., Providence, RI, 2018, pp. 459–487. MR 3821159
  * [22] Rahul Pandharipande, Aaron Pixton, and Dimitri Zvonkine, _Relations on $\overline{\mathscr{M}}_{g,n}$ via $3$-spin structures_, J. Amer. Math. Soc. 28 (2015), no. 1, 279–309. MR 3264769
  * [23] Dan Petersen, _The structure of the tautological ring in genus one_ , Duke Math. J. 163 (2014), no. 4, 777–793. MR 3178432
  * [24] by same author, _Tautological rings of spaces of pointed genus two curves of compact type_ , Compos. Math. 152 (2016), no. 7, 1398–1420. MR 3530445
  * [25] Dan Petersen and Orsola Tommasi, _The Gorenstein conjecture fails for the tautological ring of $\overline{\mathcal{M}}_{2,n}$_, Invent. Math. 196 (2014), no. 1, 139–161. MR 3179574
  * [26] Aaron Pixton, _The tautological ring of the moduli space of curves_ , ProQuest LLC, Ann Arbor, MI, 2013, Thesis (Ph.D.)–Princeton University. MR 3153424
  * [27] Johannes Schmitt and Jason van Zelm, _Intersections of loci of admissible covers with tautological classes_ , Selecta Math. (N.S.) 26 (2020), no. 5, Paper No. 79, 69. MR 4177576
  * [28] Jason van Zelm, _Nontautological bielliptic cycles_ , Pacific J. Math. 294 (2018), no. 2, 495–504. MR 3770123


# A Novel Low Complexity High Resolution Spectrum Hole Detection Technique for
Cognitive Radio

Sushmitha Sajeevua and Sakthivel Vellaisamyb CONTACT Sushmitha Sajeeevu Email:
<EMAIL_ADDRESS>aNational Institute of Technology, Calicut,
Kerala, India; b National Institute of Technology, Calicut, Kerala, India;

###### Abstract

Cognitive radio is a potential solution to meet the upcoming spectrum crunch
issue. In a cognitive radio, spectrum holes can be identified using spectrum
sensing techniques. A high resolution spectrum hole detection can ensure even
the smallest inactive portion in the spectrum is efficiently utilized. In this
paper, a spectrum hole detection technique is proposed in which coarse sensing
is done initially so as to detect occupied channels simultaneously. Spectrum
holes in the occupied band can be efficiently detected using a fine sensing
method. A two stage Frequency Response Masking (FRM) filter sandwiched between
two Pascal structure based sampling rate converters results in arbitrary
variation of bandwidth. This arbitrary variation of bandwidth can be utilized
for fine sensing the spectrum such that the spectrum holes can be detected
with high resolution. In the proposed method, high resolution in spectrum hole
detection can be achieved without increasing the hardware complexity of the
design. The hardware complexity of the proposed method is compared with the
state of the art and is found to be significantly less.

###### keywords:

Cognitive radio, Spectrum holes, Frequency Response Masking (FRM), Pascal
structure.

††articletype: RESEARCH ARTICLE

##  1 Introduction

With the advances in wireless technology, use of wireless devices has elevated
significantly. A remarkable growth of wireless connected devices is expected
in the near future, with the extensive adoption of Internet of things (IOT).
Enormous amount of radio spectrum is required to support these wireless
devices. But the available spectrum is scarce. Cognitive radio is a concept
introduced to resolve the upcoming spectrum scarcity issue. With Cognitive
radio, an unlicensed user, also known as secondary user (SU) is able to
opportunistically or concurrently access spectrum bands owned by the licensed
users also known as primary users (PU).
In a country, spectrum is allocated by Government agencies for defense,
satellite operators, aviation, railways, service providers etc. Federal
Communications Commission (FCC) in United States, Telecom Regularity Authority
in India, the Infocomm Development Authority (IDA) in Singapore, the Office of
Communications (Ofcom) in the United Kingdom (UK) etc are the authorities
allocating spectrum in respective countries [1]. Conventionally, different
parts of the radio spectrum with certain bandwidth is allocated to different
services. With such an allocation only the licensed user can utilize the
spectrum assigned to that particular service irrespective of whether the
licensed user is active or not. Even though this particular allocation reduces
the interference among different services, efficient utilization of spectrum
cannot be achieved through this conventional allocation. Studies conducted
worldwide such as Singapore, US, New Zealand, Germany and China have revealed
that a huge amount of the allocated spectrum is underutilized [2]. It can be
inferred that inefficient utilization of the spectrum contributes to the
spectrum scarcity issue even more than the physical shortage of the spectrum.
Cognitive radio can offer an efficient communication with proper integration
of the radio spectrum. In a cognitive radio, spectrum sharing schemes can be
classified into different categories. Spectral sharing in a cognitive radio
can be classified as centralized spectrum sharing and distributed spectrum
sharing [3]. In centralized spectrum sharing, there will be a centralized
entity which controls the spectrum allocation procedure. In distributed
spectrum sharing, there will be no centralized operation, each and every node
is responsible for spectral allocation. Based on the access behaviour,
spectrum sharing can be classified into cooperative spectrum sharing and non-
cooperative spectrum sharing. In cooperative spectrum sharing, interference
information is shared between nodes and the spectrum is accessed in a
cooperative fashion. Unlike the cooperative spectrum sharing, in non-
cooperative spectrum sharing, nodes will not share information with other
nodes and hence non-cooperative spectrum sharing will not result in efficient
spectrum utilization. Based on the spectrum access technique, spectrum sharing
can be classified into mainly three types; underlay, overlay and interweave.
In underlay spectrum sharing technique [4, 5], secondary users access the
channel concurrently with the primary user as long as the interference is
below an acceptable level. The transmit power of the secondary user is limited
by the interference constraint. In this technique, the secondary user need not
wait for the transmission until the primary user is inactive and channel is
idle. However this paradigm has increased complexity which is the main
disadvantage of this spectrum access technique. In overlay spectrum sharing
technique [6], secondary user can concurrently transmit with the primary user
however the secondary user must have complete knowledge of the primary user.
In interweave spectrum sharing technique, transmission will be interference
free. The basic principle of interweave spectrum sharing technique is listen
and talk. Here the secondary users utilizes spectrum holes and end their
transmissions when the sensing algorithms indicate that primary users are
resuming. The importance of spectrum sensing comes in this scenario.
Spectrum sensing is the process through which the status of the spectrum is
checked. If only one frequency band is taken into consideration it is called
as narrowband spectrum sensing [7] whereas if multiple frequency bands are
taken into consideration, it is called as wideband spectrum sensing. Matched
filter detection [8], cyclo-stationary feature detection [9], covariance based
detection [10], machine learning based spectrum sensing [11], energy detection
[12] etc. are some of the narrowband spectrum sensing techniques. Optimal
sensing can be achieved through matched filter detection even at low SNR
region. Prior knowledge of the user signal is required for performing this
detection but it is not always available. Hence matched filter detection is
not always a practical solution. Cyclo-stationary feature detection can
distinguish between signal and noise and it has got decreased probability of
false alarm at low SNR environment. Meanwhile large sensing time is required
to achieve good performance in this method. On the other hand, prior knowledge
of the primary user signal and noise is not required in covariance based
detection. But the computational complexity of this method is very high. If
trained correctly, machine learning based spectrum sensing is a good approach.
But this method requires complex techniques and large data set. Energy
detection based spectrum sensing is the simplest spectrum sensing scheme in
the literature. It has low complexity when compared to other spectrum sensing
schemes. Prior knowledge of the licensed user characteristics is not required
in this method which resulted in the wide acceptance of this method. The
energy consumption of a cognitive radio depends on the complexity of the
spectrum sensing scheme adopted. In battery powered cognitive radios, low
energy consumption spectrum sensing schemes are preferred. Hence energy
detection based cognitive radio is an ideal choice in power efficient
cognitive radio systems [13]. Wideband spectrum sensing (WSS) can be achieved
by splitting the entire bandwidth into different frequency bands and sensing
each frequency band using narrowband spectrum sensing techniques.
Using a spectrum sensing scheme, we can sense a band is occupied or not.
Numerous filter bank based techniques [14],[15] were introduced to efficiently
sense different bands present in a bandwidth. But the problem with filter bank
based methods is that complexity of the design increases considerably when the
number of bands increases. Hence proper sensing of narrow unoccupied bands
fails using filter bank based methods. Later in [16,17], introduced a design
in which the unoccupied portions of an occupied band can be sensed. Here in
this design, without increasing the number of bands fine sensing can be
achieved. A variable bandwidth filter was used for the sake of fine sensing in
this method. The resolution to which fine sensing is done is directly
proportional to the complexity of the design in [16,17].
This paper proposes a novel spectrum sensing technique using Pascal structure
based continuous variable bandwidth filter and energy detection technique. In
this design, high resolution in spectrum sensing can be achieved without
increasing the complexity of the design.
The paper is organized as given below. Section 2 briefly explains the Pascal
structure based continuously variable bandwidth filter. Section 3 explains the
proposed method for spectrum hole detection. Section 4 includes the design
examples and result analysis. Section 5 and 6 describes hardware complexity
calculation and evaluation respectively of the proposed scheme. Section 7
concludes the paper.

##  2 Pascal Structure based continuously variable bandwidth filter

Arbitrary variation of bandwidth can be achieved using Pascal structure based
continuously variable bandwidth filter [18]. This can be used to get tunable
frequency responses with any arbitrary bandwidth.

###  2.1 Basic Pascal structure

Figure 1: Pascal Structure [19]

Pascal structure is a variable fractional delay filter introduced in [19]. The
transfer function of the fractional delay filter can be represented by the
equation

$H(z,f)=\sum_{k=0}^{N}P(f,k)(1-z^{-1})^{k}$ (1)

where, $H(z,f)$ is the transfer function of Pascal structure, $f$ is the
fractional delay parameter, $P(f,k)$ is the Pascal polynomial of k-th degree.

$P(f,k)$ can be written as

$P(f,k)=(1-\frac{f+1}{1})(1-\frac{f+1}{2})(1-\frac{f+1}{3})...(1-\frac{f+1}{k})$
(2)

The implementation of equation 1 is shown in Figure 1. Low hardware complexity
when compared with the state of the art [20],[21] and high modularity are the
special features of the Pascal structure.

###  2.2 Continuously Variable Bandwidth digital filter

Figure 2: Block diagram of the Pascal structure based continuously variable
bandwidth FIR filter

Figure 2 shows the basic block diagram of the Pascal structure based
continuously variable bandwidth FIR filter. It is through the Pascal structure
based sampling rate converter, the continuous variation of bandwidth is
achieved. There is a fixed filter which is designed using two stage FRM fixed
filter in which Parks McClellan (PM) algorithm is employed. The design of
Pascal structure based sampling rate converter is explained in section 2.3.

$RF=\frac{BW_{Orig}}{BW_{Des}}$ (3)

An arbitrary bandwidth ($BW_{Des}$) can be obtained from the original fixed
filter bandwidth ($BW_{Orig}$) using the above equation. The RF (Reduction
factor) value has to be adjusted so as to get desirable bandwidth from
original bandwidth. In Figure 2, the input is an impulse function and the
output is the filter with bandwidth ($BW_{Des}$).

###  2.3 Design of Sampling rate converter using Pascal structure

Figure 3: Proposed Sampling rate converter (SRC)

Sampling rate conversion is a technique for converting the initial discrete
time samples for the same continuous time signal to a different set of time
samples.In order to delay the input signal by any desired amount of time,
fractional delay structures are used. As a result, it is possible to determine
the samples at non-integer multiples of the sampling period. Pascal structure
is a fractional delay structure that has high modularity and low complexity.
Figure 3 shows the proposed design of a sampling rate converter using Pascal
structure. The Sampling Rate Converter’s operation is described in the
following steps.

  1. 1.

The index values for each sampling rate conversion are the values in the
accumulator. The accumulator (ACC) is first loaded with zero.

  2. 2.

The output pointer (dotted lines) points to the first location (n=0) of the
output register (REG).

  3. 3.

The fractional part (d) of the accumulator is separated from accumulator
value.

$d=ACC-\lfloor{ACC}\rfloor$ (4)

  4. 4.

The fractional delay (f) of the Pascal structure is calculated by

$f=1-d\iff d\neq 0$ (5) $f=d\iff d=0$ (6)

The ceil and the multiplier block having (1-d) is meant to perform the
equations 5 and 6.

  5. 5.

The OUTPUT of the Pascal structure corresponding to the fractional delay (f)
is calculated.

  6. 6.

The ceil of the value in the accumulator, gives an index(i) value.

$i=\lceil{ACC}\rceil$ (7)

  7. 7.

The value at the $i^{th}$ index of the Pascal structure OUTPUT is saved to the
pointed location of the register (REG).

  8. 8.

The output pointer (dotted lines) points to the next location of the register
(REG).

  9. 9.

Accumulator is accumulated with sampling rate conversion factor( SRCF) value.

$ACC=ACC+SRCF$ (8)

  10. 10.

Repeat steps 3-9 until the value in the accumulator is not greater than
$(L_{I}-1)$, where $L_{I}$ is the length of the input signal (The signal to be
sampling rate converted).

The Pascal structure based sampling rate converter can be used for
continuously varying bandwidth digital filters. This continuously varying
bandwidth digital filter can be efficiently used in the context of spectrum
hole detection.

##  3 Proposed Method for spectrum hole detection

Figure 4: Block diagram of the Proposed Method

Figure 5: 64 bandwidths that can be obtained within the band.

Figure 6: 64 bandwidths that can be obtained within the band from the other
side of the band.

The basic block diagram of the proposed spectrum hole detection is Figure 4.
In the proposed method, there are two levels of sensing. Initial coarse
sensing can be done in multiple bands in parallel. The coarse sensing is done
to identify whether a particular band is occupied or not. Later fine sensing
can be done in each occupied band to detect spectrum holes in the occupied
bands. The bandwidth of the coarse sensing filter is determined based on two
factors:
a) The bandwidth of the coarse sensing filter is obtained by the frequency
modulation of the continuously variable bandwidth filter design.
b) The modulated filters should be able to cover a range of frequency.
In the fine sensing stage, using the continuously variable bandwidth design,
sensing is done from two sides of each band. Any arbitrary bandwidth within
the band can be obtained using the continuously variable bandwidth design.
Figure 5 shows 64 bandwidths that can be obtained within the band. Figure 6
shows 64 bandwidths within the band from the other side of the band. It is
through low pass to high pass frequency translation, sensing is done from the
other side of the band. In Figure 5 and Figure 6, a frequency range from 5 to
15 MHz is sensed. The coarse sensing filters in these Figures with bandwidth
3.33 MHz are derived from the DFT modulation of a low pass filter with
bandwidth 1.665 MHz which can be readily obtained using the continuously
variable bandwidth design.

Figure 7: Flowchart of the fine sensing stage

The user occupancy within a bandwidth can be checked using the energy
detection method described in subsection 3.1. If the bandwidth is not
occupied, it can be considered as a spectrum hole. Since any arbitrary
bandwidth can be obtained using the continuously variable bandwidth design,
spectrum holes can be detected with high resolution.
An efficient fine sensing can be done using the steps in the flow chart shown
in Figure 7. Fine sensing is done such a way that initially the full bandwidth
within the band is considered. If spectral occupancy is detected, take the
half bandwidth. Check the spectral occupancy in the half bandwidth. If the
spectral occupancy is detected in the half bandwidth, take again the half
bandwidth and the process continues. If the spectral occupancy is not detected
in the half bandwidth, it is evident that the other half of the full bandwidth
will be occupied. Hence choose a bandwidth between half bandwidth and full
bandwidth and check the occupancy. If spectrum is detected in this bandwidth,
take a bandwidth between the half bandwidth and the new bandwidth and if
spectrum is not detected, take a bandwidth between new bandwidth and the full
band width. This process is repeated until we get a proper resolution at which
the spectrum begins within the band. The same process is repeated from the
other side of the band. Hence with a proper resolution the spectrum beginning
and ending portion within the band can be detected. So spectrum holes form two
sides of the band can be detected with high resolution. Figure 7 shows a
flowchart of this.

###  3.1 Energy Detection

Energy detection based spectrum sensing is the most widely used spectrum
sensing method. In this method, prior knowledge of the primary user (PU)
characteristics is not required. The binary hypothesis $H_{0}$ and $H_{1}$ can
be defined as follows:

$\begin{split}H_{0}:&y_{k}(n)=w(n)\\\
H_{1}:&y_{k}(n)=h_{k}(n).x_{k}(n)+w(n)\end{split}$ (9)

where $y_{k}(n)$ is the $k^{th}$ signal received by the CVBW filter,
$x_{k}(n)$ is the PU signal, $h_{k}(n)$ is the channel gain and $w(n)$ is
Additive White Gaussian Noise (AWGN) signal with zero mean and variance
$\sigma_{\omega}^{2}$. $H_{0}$ denotes the absence of the primary user signal
and $H_{1}$ denotes the presence of the primary user signal.
Energy of the signal is given by

$E=\frac{1}{N_{s}}\sum_{n=1}^{N_{s}}|y_{k}|^{2}$ (10)

where $N_{s}$ is the number of samples of the signal in the available time.
The performance of this scheme can be evaluated using probability of detection
and probability of false alarm.

$\begin{split}P_{d}&=P(E>\lambda|H_{1})\\\
P_{f}&=P(E>\lambda|H_{0})\end{split}$ (11)

where $\lambda$ is the threshold value. $P_{d}$ and $P_{f}$ can be expressed
as:

$P_{d}=\int_{\lambda}^{\infty}p_{1}(x)dx=Q\big{(}\frac{\lambda-\mu_{1}}{\sigma_{1}}\big{)}$
(12)
$P_{f}=\int_{\lambda}^{\infty}p_{0}(x)dx=Q\big{(}\frac{\lambda-\mu_{0}}{\sigma_{0}}\big{)}$
(13)

Using the Central Limit Theorem, the probability density function under
hypothesis $H_{1}$ can be approximated by a normal distribution with mean
$\mu_{1}=(\sigma_{x}^{2}+\sigma_{w}^{2})$ and variance
$\sigma_{1}^{2}=\frac{2}{N_{s}}(\sigma_{x}^{2}+\sigma_{w}^{2})^{2}$. Similarly
$H_{0}$ can be approximated by a normal distribution with mean
$\mu_{0}=\sigma_{w}^{2}$ and $\sigma_{0}^{2}=\frac{2}{N_{s}}\sigma_{w}^{4}$.
Hence $P_{d}$ and $P_{f}$ can be expressed as:

$P_{d}=Q\Big{(}\frac{\lambda-(\sigma_{x}^{2}+\sigma_{w}^{2})}{(\sigma_{x}^{2}+\sigma_{w}^{2})/{\sqrt{N_{s}/2}}}\Big{)}$
(14)
$P_{f}=Q\Big{(}\frac{\lambda-\sigma_{w}^{2}}{\sigma_{w}^{2}/{\sqrt{N_{s}/2}}}\Big{)}$
(15)

From equations 14 and 15 it can be analyzed that the decision threshold can be
derived for either a target $P_{d}$ or $P_{f}$. The threshold for a constant
detection rate ($\lambda_{P_{d}}$) and the threshold for a constant false
alarm rate ($\lambda_{P_{f}}$) can be derived from equations 14 and 15 and are
shown below.

$\lambda_{P_{d}}=(\sigma_{x}^{2}+\sigma_{w}^{2})\Big{(}1+\frac{Q^{-1}(P_{d})}{\sqrt{N_{s}/2}}\Big{)}$
(16)

$\lambda_{P_{f}}=\sigma_{w}^{2}\Big{(}1+\frac{Q^{-1}(P_{f})}{\sqrt{N_{s}/2}}\Big{)}$
(17)

If $P_{d}$ is high, it means that primary users are given importance, since
primary users are better protected then. If $P_{f}$ is low, more chances a
channel can be utilized by secondary users. In fixed threshold method, we are
fixing the threshold either favour of $P_{d}$ or $P_{f}$. So a high $P_{d}$
and low $P_{f}$ cannot be achieved together. An adaptive threshold method is
proposed in [22] where the threshold level in each sub-band is calculated with
respect to the signal in that sub-band. This is used for spectrum hole
detection in the proposed method. This results in a better spectrum efficiency
for both primary user and secondary user. Hence spectrum holes can be detected
more accurately.

Figure 8: The probability of error versus spectrum utilization for different
SNR Values

Adaptive threshold is found by solving [22]

$min(P_{e})=min((1-\alpha)P_{f}+\alpha(1-P_{d}))$ (18)

where $P_{e}$ represents the minimum probability of the error decision,
$(1-P_{d})$ is the probability of the missed detection, $\alpha$ is the
spectrum utilization ratio by PU, where $0<\alpha<1$. A threshold is found in
such a way that $P_{f}$ is kept minimum when the spectrum is not utilized by
the primary user and $(1-P_{d})$ is kept minimum when the spectrum is utilized
by the primary user.
A threshold is found such a way that the probability of error is minimum which
is given by

$\lambda_{adap}=\frac{1+\sqrt{1+\frac{4}{N_{s}}\big{(}1+\frac{2}{\gamma}\big{)}\ln\big{(}\frac{(1-\alpha)(1+\gamma)}{\alpha}\big{)}}}{\sigma_{\omega}^{-2}\big{(}\frac{2+\gamma}{1+\gamma}\big{)}}$
(19)

where $\sigma_{\omega}^{2}$ is the noise variance, $\gamma$ is the SNR and
$N_{s}$ is the sample points. This adaptive threshold is used for spectrum
hole detection in this proposed method. The probability of error versus
spectrum utilization for different SNR values is shown in Figure 8.

##  4 Results and Discussion

In this section, the working of the proposed spectrum hole detection technique
is demonstrated. To test the proposed scheme for spectrum hole detection, a
continuously variable bandwidth filter using Pascal structure and two stage
FRM is first designed using PM algorithm. The following specifications are
used to design the fixed filter in the continuously variable bandwidth filter
design:
_Passband edge frequency_ : $0.14\pi$
_Stopband edge frequency_ : $0.141\pi$
_Maximum passband ripple_ : 0.03 dB
_Minimum stopband attenuation_ : 50 dB

A Quadrature Phase Shift Keying (QPSK) modulated signal with different center
frequencies is used as the primary user signal for evaluating the spectrum
sensing model. AWGN channel is considered here with a noise power known a
priori. For calculating the adaptive threshold, $\alpha$ value is chosen as
0.5. Three cases are discussed in the section. In the first case, spectrum
holes in a frequency range where a single user is occupied is discussed.
Spectrum hole detection while multiple users occupy the spectrum is discussed
in the second case. In the third case, spectrum hole detection in a frequency
range where the multiple users occupy the spectrum in close proximity is
discussed. A channel of bandwidth 10 MHz is considered for evaluating the
spectrum hole detection model.

###  4.1 Case 1

A Quadrature Phase Shift Keying (QPSK) modulated signal with center frequency
at 6.875 MHz and sampled at 10 MHz is used as the primary user signal in this
case. Initially the coarse sensing is done by modulating the continuously
variable bandwidth filter into 3 bands. The bandwidth of the coarse sensing
filter is derived using the continuously variable bandwidth design through
which a low pass filter is obtained with 1.665 MHz. This is modulated to
obtain 3.33 MHz. Three such band pass filters will cover the entire 10 MHz.

Figure 9: Input Signal and the coarse sensing stage

Figure 10:  Fine sensing of the occupied band

For fine sensing the algorithm described in section 3 is used. Spectrum is
detected in the frequency range 6.77 MHz to 6.9789 MHz in the first band. The
unoccupied portions in the occupied band can be considered as a spectrum hole.
Figure 9 shows the input signal and the coarse sensing stage. Figure 10 shows
the corresponding fine sensing stage of the occupied band. Table 1 shows the
energy detection associated with the coarse sensing of the spectrum holes.
Table 2 shows the energy detection associated with the fine sensing of the
spectrum holes. In Table 1, $B_{1}$ is the low frequency band and $B_{3}$ is
the high frequency band. In Table 2, $B_{L1}$, $B_{L2}$, $B_{L3}$ etc are the
bands in which the energy is detected so as to find the starting of the
spectrum. Here $B_{L1}$ is the lowest frequency band and $B_{L10}$ is the full
bandwidth. The threshold, energy and occupancy corresponding to each band is
shown in Table 2. $B_{R1}$, $B_{R2}$, $B_{R3}$ etc are the bands in which the
energy is detected so as to find the ending of the spectrum. Here $B_{R1}$ is
the highest frequency band and $B_{R9}$ is the full bandwidth. Corresponding
threshold, energy and occupancy is shown in Table 2.

Table 1: Energy detection associated with coarse sensing  Band | Threshold | Energy | occupancy
---|---|---|---
$B_{1}$ | 0.2008 | 31.6343 | Yes
$B_{2}$ | 0.1098 | 0.0212 | No
$B_{3}$ | 0.1073 | 0.0152 | No
Table 2: Energy detection associated with fine sensing Band | Threshold | Energy | occupancy | Band | Threshold | Energy | occupancy
---|---|---|---|---|---|---|---
$B_{L1}$ | 0.1089 | 0.0189 | No | $B_{R1}$ | 0.1047 | 0.0093 | No
$B_{L2}$ | 0.1318 | 0.0919 | No | $B_{R2}$ | 0.1169 | 0.0397 | No
$B_{L3}$ | 0.1424 | 0.1447 | Yes | $B_{R3}$ | 0.1493 | 0.1362 | No
$B_{L4}$ | 0.1566 | 0.2560 | Yes | $B_{R4}$ | 0.1527 | 0.1492 | No
$B_{L5}$ | 0.1742 | 0.5582 | Yes | $B_{R5}$ | 0.1527 | 0.2192 | Yes
$B_{L6}$ | 0.1976 | 5.6810 | Yes | $B_{R6}$ | 0.1994 | 0.2189 | Yes
$B_{L7}$ | 0.2004 | 20.4982 | Yes | $B_{R7}$ | 0.2007 | 10.9126 | Yes
$B_{L8}$ | 0.2008 | 31.4512 | Yes | $B_{R8}$ | 0.2007 | 28.3152 | Yes
$B_{L9}$ | 0.2008 | 31.2684 | Yes | $B_{R9}$ | 0.2007 | 28.9599 | Yes
$B_{L10}$ | 0.2008 | 31.6343 | Yes | - | - | - | -

###  4.2 Case 2

A Quadrature Phase Shift Keying (QPSK) modulated signal with center
frequencies at 6.875 MHz and 12.75 MHz and sampled at 10 MHz is used as the
primary user signal in this case. By coarse sensing, spectrum is detected in
the first band and third band. Then fine sensing is done in each of these
bands separately. Fine sensing in first band is the same as in case 1. The
coarse sensing stage and the fine sensing in the third band is shown in Figure
11.

Figure 11: Spectrum hole detection with multiple primary users

###  4.3 Case 3

When multiple primary users are there in close proximity, chances are there
that the spectrum hole in between two primary users will be missed. In order
to avoid such situation, when a band in coarse sensing stage is having high
energy, chances are there that multiple bands are there within a band. In such
cases, number of bands of the coarse sensing stage can be increased. Here in
this case 6 bands are taken instead of 3 bands. This is shown in Figure 12.
Since using continuously variable bandwidth filter can give multiple bandwidth
filter as the prototype filter, increasing the number of bands in a coarse
sensing stage does not demand the design of a new prototype filter. Only the
complexity for the modulation of the filter will be increased. In this case a
Quadrature Phase Shift Keying (QPSK) modulated signal with center frequencies
at 9 MHz and 11 MHz and sampled at 10 MHz is used as the primary user signal.
Fine sensing is done similar to the cases 1 and 2.

Figure 12: Spectrum hole detection with multiple primary users in close proximity Table 3: Hardware Complexity Comparison Design Method | Transition | Multipliers in | Total
---|---|---|---
| Width | Prototype filter | Multipliers
CMFB based [14] | $0.0025\pi$ | 1396 | 2226
Farrow Structure [16] | $0.0025\pi$ | 700 | 888
Modified FRM based [15] | $0.0025\pi$ | 145 | 465
Proposed Method | $0.001\pi$ | 228 | 266

##  5 Hardware Complexity Calculation

Table 3 shows the hardware complexity comparison of the proposed scheme with
the state of the art. Hardware complexity is calculated by considering both
the prototype filter and multipliers for modulation. The complexity of
determining the threshold level to perform test statistics for the spectrum
hole detection is excluded in the computation as it’s not much significant
when compared to the complexity of the prototype filter and the modulation.
The hardware complexity is given by the equation:

$\mu_{Total}=\mu_{Prototype}+\mu_{modulation}+\mu_{HPF}$ (20)

$\mu_{Total}$ is the total number of multipliers, $\mu_{Prototype}$ is the
multipliers of the prototype filter, $\mu_{modulation}$ is the multipliers for
modulation and $\mu_{HPF}$ is the number of multipliers required for
generating the high pass filter from low pass filter. Since DFT modulation is
used, number of multipliers for modulation is

$\mu_{modulation}=N_{c}^{2}$ (21)

$N_{c}$ is the number of coarse sensing bands. Since in the proposed design’s
case, we are not restricting $N_{c}$ to be a power of two, FFT (Fast Fourier
Transform) is not guaranteed for DFT computation. Hence equation 21 is used
for calculating number of multipliers for modulation. Since the fixed filter
is a low pass filter, to sense from either side, high pass filter is used. Low
pass filter to high pass filter conversion can be considered as a case with
number of channels, M=2. Hence the number of multipliers for this conversion
is

$\mu_{HPF}=Mlog_{2}M$ (22)

This gives $\mu_{HPF}$ as 2.
The case where $N_{c}$ is taken as 6 and the prototype filter is designed as
per section 4 is taken for finding the number of multipliers in the proposed
scheme. This is shown in Table 3.

$228+36+2=266$ (23)

##  6 Hardware Implementation and Complexity Evaluation

The complexity of implementation, power consumption and area of the proposed
spectral hole detection scheme is evaluated by synthesizing and implementing
the prototype continuously variable bandwidth filter in both ASIC (Application
Specific Integrated Circuit) and FPGA (Field Programmable Gate Array)
platforms. 14-bit quantized prototype filter coefficients synthesized and
implemented in Xilinx Artix 7 AC 701 (XC7A200T-2FBG676C) FPGA using Vivado
design suit and the resource consumptions are evaluated at 20 MHz frequency.
The results are shown in Table 4.

Table 4: FPGA Synthesis results | Slice LUTS | FFs
---|---|---
Available resources | 1,34,600 | 2,69,200
Prototype filter in [23] | 50078 | 56636
Prototype filter in the proposed method | 29621 | 33254
Table 5: ASIC Synthesis results | No. of cells | Area | Power
---|---|---|---
|  | ($\mu m^{2}$) | (mW)
Prototype filter in [23] | 1670693 | 14681196 | 175.186
Prototype filter | 1214170 | 10554162 | 123.447
in the proposed method |  |  |

To evaluate the performance of the proposed spectrum hole detection scheme in
the ASIC platform, 14-bit quantized prototype filter coefficients synthesized
in 90 nm technology ‘fast’ library using Cadence Genus synthesis tool. The
results are shown in Table 5.

##  7 Conclusion

A high resolution spectrum hole detection method is proposed in this paper. A
continuously varying bandwidth filter designed using Pascal structure and two
stage FRM is deployed for this purpose. In the proposed method, the coarse
sensing is done using frequency modulation of the continuously variable
bandwidth filter for detecting the occupied bands. Fine sensing is done by
varying the bandwidth within an occupied band for detecting the spectrum
holes. Adaptive threshold based energy detection is used in the proposed
method for spectrum sensing. This results in a better spectrum efficiency for
both primary user and secondary user. The hardware complexity of the proposed
spectrum hole detection method outperforms the prior techniques in the
literature.

## References

  * S. Bhattarai et al. (2016) [1]. S. Bhattarai, J. -M. J. Park, B. Gao, K. Bian and W. Lehr, ”An Overview of Dynamic Spectrum Sharing: Ongoing Initiatives, Challenges, and a Roadmap for Future Research,” in IEEE Transactions on Cognitive Communications and Networking, vol. 2, no. 2, pp. 110-128, June 2016.
  * El et al. (2020) [2]. El Rharras, A., Saber, M., Chehri, A., Saadane, R., Hakem, N. and Jeon, G., 2020. Optimization of spectrum utilization parameters in cognitive radio using genetic algorithm. Procedia Computer Science, 176, pp.2466-2475.
  * Chik et al. (2016) [3]. Chikhale, S., Hate, S. and Chutke, S., 2016. An overview to cognitive radio spectrum sharing. Int. J. Latest Res. Eng. Technol.(IJLRET), 2(2), pp.20-15.
  * Baby et al. (2016) [4]. Baby, S.M. and James, M., 2016. A comparative study on various spectrum sharing techniques. Procedia Technology, 25, pp.613-620.
  * Agarwal et al. (2022) [5]. Agrawal, S.K., Samant, A. and Yadav, S.K., 2022. Spectrum sensing in cognitive radio networks and metacognition for dynamic spectrum sharing between radar and communication system: A review. Physical Communication, p.101673.
  * kusaladharma et al. (1999) [6]. Kusaladharma, S. and Tellambura, C., 1999. An overview of cognitive radio networks. Wiley encyclopedia of electrical and electronics engineering, pp.1-17.
  * Jain et al. (2019) [7]. Jain, P.P., Pawar, P.R., Patil, P. and Pradhan, D., 2019. Narrowband spectrum sensing in cognitive radio: Detection methodologies. International Journal of Computer Sciences and Engineering, 7(11), pp.105-113.
  * Soofi et al. (2017) [8]. Soofi, S., Potnis, A. and Dwivedy, P., 2017. Heterogeneous spectrum sensing in cognitive radio network using traditional energy detection and matched filter.
  * Yawada et al. (2016) [9]. Yawada, P.S. and Wei, A.J., 2016, June. Cyclostationary Detection Based on Non-cooperative spectrum sensing in cognitive radio network. In 2016 IEEE international conference on cyber technology in automation, control, and intelligent systems (CYBER) (pp. 184-187). IEEE.
  * Zeng et al. (2008) [10]. Zeng, Y. and Liang, Y.C., 2008. Spectrum-sensing algorithms for cognitive radio based on statistical covariances. IEEE transactions on Vehicular Technology, 58(4), pp.1804-1815.
  * Abbas et al. (2015) [11]. Abbas, N., Nasser, Y. and Ahmad, K.E., 2015. Recent advances on artificial intelligence and learning techniques in cognitive radio networks. EURASIP Journal on Wireless Communications and Networking, 2015(1), pp.1-20.
  * Dey et al. (2009) [12]. Dey, A.K. and Banerjee, A., 2009. On primary user detection using energy detection technique for cognitive radio. NCC 2009, pp.16-18.
  * Liu et al. (2013) [13]. Liu, X., Evans, B.G. and Moessner, K., 2013. Comparison of reliability, delay and complexity for standalone cognitive radio spectrum sensing schemes. IET Communications, 7(9), pp.799-807.
  * Raghu et al. (2016) [14]. Raghu, I., Chowdary, S.S. and Elias, E., 2016, November. Efficient spectrum sensing for cognitive radio using cosine modulated filter banks. In 2016 IEEE region 10 conference (TENCON) (pp. 2086-2089). IEEE.
  * Parvathi et al. (2017) [15]. Parvathi, A.K. and Vellaisamy, S., 2021. Low Complexity Adaptive Spectrum Sensing using Modified FRM Filter Bank. International Journal of Electronics, pp.1-20.
  * Raghu et al. (2019) [16]. Raghu, I. and Elias, E., 2019. Low complexity spectrum sensing technique for cognitive radio using Farrow Structure Digital Filters. Engineering Science and Technology, an International Journal, 22(1), pp.131-142.
  * Indrakanti et al. (2019) [17]. Indrakanti, R. and Elias, E., 2019. Design of low-complexity Farrow structure-based reconfigurable filters for parallel spectrum hole detection. Signal, Image and Video Processing, 13(4), pp.787-794.
  * Sajeevu et al. (2020) [18]. Sajeevu, S. and Sakthivel, V., 2021. A low complexity and high modularity design for continuously variable bandwidth digital filters. arXiv preprint arXiv:2109.11771.
  * Soontornwong et al. (2017) [19]. Soontornwong, P., Deng, T.B. and Chivapreecha, S., 2017. Low-complexity and high-modularity structure for implementing transient-free Pascal-delay filter. IEEE Transactions on Signal Processing, 65(23), pp.6233-6243.
  * Zeineddine et al. (2018) [20]. Zeineddine, A., Paquelet, S., Kanj, M., Moy, C., Nafkha, A. and Jezequel, P.Y., 2018, November. Reconfigurable Newton structure for sample rate conversion. In 2018 IEEE Global Conference on Signal and Information Processing (GlobalSIP) (pp. 271-275). IEEE.
  * Farrow et al. (1988) [21]. Farrow, C.W., 1988, June. A continuously variable digital delay element. In 1988., IEEE International Symposium on Circuits and Systems (pp. 2641-2645). IEEE.
  * Wang et al. (2013) [22].Wang, N., Gao, Y. and Zhang, X., 2013. Adaptive spectrum sensing algorithm under different primary user utilizations. IEEE communications letters, 17(9), pp.1838-1841.
  * Haridas et al. (2017) [23]. Haridas, N. and Elias, E., 2017. Low‐complexity technique to get arbitrary variation in the bandwidth of a digital FIR filter. IET Signal Processing, 11(4), pp.372-377.