# Cohen-Macaulay monomial ideals of codimension 2

Muhammad Naeem  Muhammad Naeem, Abdus Salam School of Mathematical Sciences
(ASSMS), GC University, Lahore, Pakistan<EMAIL_ADDRESS>
###### Abstract.

We give a structure theorem for Cohen-Macaulay monomial ideals of codimension
2, and describe all possible relation matrices of such ideals. In case that
the ideal has a linear resolution, the relation matrices can be identified
with the spanning trees of a connected chordal graph with the property that
each distinct pair of maximal cliques of the graph has at most one vertex in
common.

Key words  : Monomial Ideals, Taylor Complexes, Linear Resolutions, Chordal
Graphs.

###### 1991 Mathematics Subject Classification:

13C14, 13D02, 13D25, 13P10

## Introduction

The purpose of the paper is to work out in detail a remark on the structure of
Cohen-Macaulay monomial ideals of codimension 2 which was made in the paper
[1]. There it was observed that the ‘generic’ ideals of this type, generated
by $n$ elements, are in bijective correspondence to the trees with $n$
vertices. In Proposition 1.2 we give an explicit description of the generators
of a generic Cohen-Macaulay monomial ideal of codimension 2 in terms of the
associated tree and describe the minimal prime ideals of such ideals in
Proposition 1.4. As a consequence of these two results we obtain as the main
result of Section 1 a full description of all Cohen-Macaulay monomial ideals
of codimension 2, see Theorem 1.5.

In Section 2 we study the possible relation trees of a Cohen-Macaulay monomial
ideals of codimension 2. This set of relation trees is always the set of bases
of a matriod (Proposition 2.4), which in case of a generic ideal consists of
only one tree as shown in Proposition 2.1. We call the graph $G$ whose set of
edges is the union of the set of edges of all relation trees of a given Cohen-
Macaulay monomial ideal $I$ of codimension 2, the Taylor graph of $I$. Then
each of the relation trees is a spanning tree of the Taylor graph. The natural
question arises whether the set of relation trees of $I$ is precisely the set
of spanning trees of $G$. We show by an example that this is not the case in
general. On the other hand, we prove in Theorem 2.5 that each relation tree of
$I$ is a spanning tree of $G$, if $I$ has a linear resolution. In order to
obtain a complete description of all possible relation trees when $I$ has a
linear resolution, it is therefore required to find all possible Taylor graphs
of such ideals. This is done in Theorem 2.6, where it is shown that a finite
connected simple graph is the Taylor graph of Cohen-Macaulay monomial ideal of
codimension $2$ with linear resolution, if and only if $G$ is chordal and any
two maximal cliques of $G$ have at most one vertex in common.

##  1\. On the structure of Cohen-Macaulay monomial ideals of codimension 2

In [1, Remark 6.3] the following observation was made regarding the structure
of a codimension 2 Cohen-Macaulay monomial ideal $I$: let

$\\{u_{1},u_{2},...,u_{m+1}\\}$

the unique minimal set of monomial generators of $I$. Consider the Taylor
complex of the sequence $u_{1},u_{2},...,u_{m+1}$

$...\rightarrow\bigoplus_{i=1}^{m+1\choose 2}Se_{i}\wedge
e_{j}\overset{\varphi_{2}}{\rightarrow}\bigoplus_{i=1}^{m+1}Se_{i}\overset{\varphi_{1}}{\rightarrow}S$

The matrix corresponding to $\varphi_{2}$ is of size ${m+1\choose 2}\times
m+1$ whose rows correspond to Taylor relation (cf. [4]), namely to the
relations

$e_{i}\wedge e_{j}\mapsto u_{ji}e_{j}-u_{ij}e_{i}$

where $i<j$ and $u_{ji}=u_{i}/\gcd(u_{i},u_{j})$,
$u_{ij}=u_{j}/\gcd(u_{i},u_{j})$.

Let $U=\operatorname{Ker}(\varphi_{1})$; then the Taylor relations form a
homogeneous system of generators of $U$. Since
$\operatorname{proj\,dim}S/I=2$, it follows that $U$ is free of rank $m$. In
particular $U$ is minimally generated by $m$ elements. Applying the graded
Nakayama Lemma (cf. [2] or [6, Lemma 1.2.6]), a minimal system of graded
generators of $U$ can be chosen among the Taylor relations. We then obtain a
minimal graded free resolution

$0\rightarrow S^{m}\overset{A}{\rightarrow}S^{m+1}\rightarrow S\rightarrow
S/I\rightarrow 0$

of $S/I$, where $A$ is a matrix whose rows correspond to Taylor relations. Any
such matrix will be called a Hilbert–Burch matrix of $I$

Notice that each row of $A$ has exactly two nonzero entries. We obtain a graph
$\Gamma$ on the vertex set $[m+1]=\\{1,\ldots,m+1\\}$ from the matrix $A$ as
follows: we say that $\\{i,j\\}$ is an edge of $\Gamma$, if and only if there
is a row of $A$ whose nonzero entries are the $i$th and $j$th components.

We claim that every column of $A$ has a nonzero entry. In fact, if this would
not be the case, say, the $k$th column of $A$ has all entries zero, then the
relation $u_{k+1,k}e_{k+1}-u_{k,k+1}e_{k}\in U$ could not be written as a
linear combination of the minimal graded homogeneous generators of $U$. This
shows that $\Gamma$ has no isolated vertex. On the other hand, since the
number of vertices of $\Gamma$ is $m+1$ and the number of edges of $\Gamma$ is
$m$, we see that $\Gamma$ is a tree, which is called a relation tree of $I$.
The set of all relation trees of $I$ will be denoted by $\mathcal{T}(I)$.

Conversely, given a tree $\Gamma$ on the vertex set $[m+1]$ with $m\geq 2$, we
are going to construct a codimension 2 Cohen-Macaulay monomial ideal $I$ for
which $\Gamma$ is a relation tree. We assign to $\Gamma$ an
$m\times(m+1)$-matrix $A(\Gamma)=(a_{ij})$ whose entries are either $0$ or
indeterminates. The matrix $A(\Gamma)$ is defined as follows: let $E(\Gamma)$
be the set of edges of $\Gamma$. Since $\Gamma$ is a tree, there are exactly
$m$ edges. We choose an arbitrary order of the edges of $\Gamma$, and assign
to the $k$th edge $\\{i,j\\}\in E(\Gamma)$ the $k$th row of $A(\Gamma)$ by

(4) $\displaystyle a_{kl}=\left\\{\begin{array}[]{ll}-x_{ij}&\text{ if
$l=i$,}\\\ x_{ji}&\mbox{ if $l=j$,}\\\ 0&\mbox{ otherwise.}\end{array}\right.$

For example if $\Gamma$ is the tree with edges $\\{1,2\\}$, $\\{2,3\\}$ and
$\\{2,4\\}$. Then we obtain the matrix

$A(\Gamma)=\begin{pmatrix}-x_{12}&x_{21}&0&0\\\ 0&-x_{23}&x_{32}&0\\\
0&-x_{24}&0&x_{42}\end{pmatrix}$

######  Definition 1.1.

Let $\Gamma$ be a tree on the vertex set $[m+1]$ and $i,j$ be two distinct
vertices of $\Gamma$. Then there exists a unique path from $i$ to $j$ denoted
by $i\rightarrow j$, in other words a sequence of numbers
$i=i_{0},i_{1},i_{2},\ldots,i_{k-1},i_{k}=j$ such that $\\{i_{l},i_{l+1}\\}\in
E(\Gamma)$ for $l=0,\ldots,k-1$. We set

$b(i,j)=i_{1}\quad\text{and}\quad e(i,j)=i_{k-1}$

######  Proposition 1.2.

Let $v_{j}$ be the minor of $A(\Gamma)$ which is obtained by omitting the
$j$th column of $A(\Gamma)$. Then $v_{j}=\pm\prod_{i=1\atop i\neq
j}^{m+1}x_{ib(i,j)}$ for $j=1,2,...,m+1$

###### Proof.

We prove the assertion by using induction on the number of edges of $\Gamma$.
If $|E(\Gamma)|=~{}1$, then

$A(\Gamma)=(-x_{12},x_{21})$

Therefore, $v_{1}=x_{21}$ and $v_{2}=-x_{12}$, as required.

Now assume that the assertion is true for $|E(\Gamma)|=m-1\geq 1$. Since
$\Gamma$ is a tree, there exists a free vertex of $\Gamma$, that is, a vertex
which belongs to exactly one edge. Such an edge of $\Gamma$ is called a leaf.
We may assume the edge $\\{m,m+1\\}$ is a leaf and that $m+1$ is a free vertex
of $\Gamma$. The tree which is obtained from $\Gamma$ by removing the leaf
$\\{m,m+1\\}$ will be denoted by $\Gamma^{\prime}$. By our induction
hypothesis the minors $v_{1}^{\prime},\ldots,v_{m}^{\prime}$ of
$\Gamma^{\prime}$ have the desired form. We may assume that the edge
$\\{m,m+1\\}$ is the last in the order of edges. Then $(m-1)\times m$ matrix
$A(\Gamma^{\prime})$ is obtained from the $m\times(m+1)$-matrix $A(\Gamma)$ by
removing the last row

$R_{m}=(0,\ldots,0,-x_{m,m+1},x_{m+1,m})$

and the last column

$\begin{pmatrix}0\\\ \vdots\\\ 0\\\ x_{m+1,m}\end{pmatrix}$

It follows that the minors $v_{1},\ldots,v_{m+1}$ of $A(\Gamma)$ are given by

(5) $\displaystyle v_{j}=x_{m+1,m}v_{j}^{\prime}\quad\text{for}\quad
j=1,\ldots,m,\quad\text{and}\quad v_{m+1}=x_{m,m+1}v_{m}^{\prime}.$

Therefore, our induction hypothesis implies that

$v_{j}=x_{m+1,m}v_{j}^{\prime}=\pm x_{m+1,m}\prod_{i=1,\;i\neq
j}^{m}x_{i,b(i,j)}=\pm\prod_{i=1,\;i\neq j}^{m+1}x_{i,b(i,j)}$

for $j=1,\ldots,m$, and

$v_{m+1}=x_{m,m+1}v_{m}^{\prime}=\pm
x_{m,m+1}\prod_{i=1}^{m-1}x_{i,b(i,m)}=\pm x_{m,b(m,m+1)}\prod_{i\neq
i=1}^{m-1}x_{i,b(i,m+1)},$

because $b(i,m)=b(i,m+1)$ for all $i\leq m$. So this implies that

$v_{m+1}=\pm\prod_{i=1,\;i\neq m+1}^{m+1}x_{i,b(i,m+1)},$

as desired.

For a tree $\Gamma$ on the vertex set $[m+1]$ we denote by $I(\Gamma)$ the
ideal generated by the minors $v_{1},\ldots,v_{m+1}$ of $A(\Gamma)$ and call
it the generic monomial ideal attached to the tree $\Gamma$.

######  Corollary 1.3.

The ideal $I(\Gamma)$ is a Cohen–Macaulay ideal of codimension $2$.

###### Proof.

The greatest common divisors of the monomial generators $v_{j}$ of $I(\Gamma)$
is one. This can easily be seen by the formulas (5) in the proof of
Proposition 1.2. The assertion follows then from [2, Theorem 1.4.17].

The generic ideal $I(\Gamma)$ has the following nice primary decomposition:

######  Proposition 1.4.

$I(\Gamma)=\bigcap_{1\leq i<j\leq m+1}(x_{ib(i,j)},x_{je(i,j)})$.

###### Proof.

We prove the assertion by using induction on the number of edges of $\Gamma$.
For $|E(\Gamma)|=1$ we have,

$A(\Gamma)=(-x_{12},x_{21}).$

with $v_{1}=x_{21}$ , $v_{2}=-x_{12}$. Therefore
$I(\Gamma)=(x_{21},x_{12})=(x_{1b(1,2)},x_{2e(1,2)})$. Now assume that
assertion is true if $|E(\Gamma)|=m-1\geq 1$. Since $\Gamma$ is a tree, there
exists a free vertex of $\Gamma$, that is, a vertex which belongs to exactly
one edge. Such an edge of $\Gamma$ is called a leaf. We may assume the
$\\{m,m+1\\}$ is a leaf and that $m+1$ is a free vertex of $\Gamma$. The tree
which is obtained from $\Gamma$ by removing the leaf $\\{m,m+1\\}$ will be
denoted by $\Gamma^{\prime}$. So then for $A(\Gamma^{\prime})$ we have

$I(\Gamma^{\prime})=(v^{\prime}_{1},v^{\prime}_{2},...,v^{\prime}_{m})=\bigcap_{1\leq
i<j\leq m}(x_{ib(i,j)},x_{je(i,j)}).$

We may assume that the edge $\\{m,m+1\\}$ is the last in the order of edges.
Then $(m-1)\times m$ matrix $A(\Gamma^{\prime})$ is obtained from the
$m\times(m+1)$-matrix $A(\Gamma)$ by deleting the last row

$R_{m}=(0,\ldots,0,-x_{m,m+1},x_{m+1,m})$

and the last column

$\begin{pmatrix}0\\\ \vdots\\\ 0\\\ x_{m+1,m}\end{pmatrix}$

It follows that the minors $v_{1},\ldots,v_{m+1}$ of $A(\Gamma)$ are given by

$v_{j}=x_{m+1,m}v_{j}^{\prime}\quad\text{for}\quad
j=1,\ldots,m,\quad\text{and}\quad v_{m+1}=x_{m,m+1}v_{m}^{\prime}.$

Hence

$I(\Gamma)=(v_{1},v_{2},...,v_{m+1}).$

On the other hand, by using the induction hypothesis and the fact that
$e(i,m+1)=m$ for all $i\leq m$, we get

$\displaystyle\bigcap_{1\leq i<j\leq m+1}(x_{ib(i,j)},x_{je(i,j)})$
$\displaystyle=$ $\displaystyle\bigcap_{1\leq i<j\leq
m}(x_{ib(i,j)},x_{je(i,j)})\cap\bigcap_{i=1}^{m}(x_{ib(i,m+1)},x_{m+1,e(i,m+1)})$
$\displaystyle=$
$\displaystyle(v^{\prime}_{1},v^{\prime}_{2},...,v^{\prime}_{m})\cap\bigcap_{i=1}^{m}(x_{ib(i,m+1)},x_{m+1,m})$
$\displaystyle=$
$\displaystyle(v^{\prime}_{1},v^{\prime}_{2},...,v^{\prime}_{m})\cap(\prod_{i=1}^{m}x_{ib(i,m+1)},x_{m+1,m})$
$\displaystyle=$
$\displaystyle(v^{\prime}_{1},v^{\prime}_{2},...,v^{\prime}_{m})\cap(x_{m,m+1}\prod_{i=1}^{m-1}x_{ib(i,m+1)},x_{m+1,m})$
$\displaystyle=$
$\displaystyle(v^{\prime}_{1},v^{\prime}_{2},...,v^{\prime}_{m})\cap(x_{m,m+1}v^{\prime}_{m},x_{m+1,m}).$

Observing that $\gcd(v_{i}^{\prime},x_{m+1,m})=1$ it follows that

$\displaystyle(v^{\prime}_{1},v^{\prime}_{2},...,v^{\prime}_{m})$
$\displaystyle\cap$ $\displaystyle(x_{m,m+1}v^{\prime}_{m},x_{m+1,m})$
$\displaystyle=$
$\displaystyle(x_{m+1,m}v^{\prime}_{1},x_{m+1,m}v^{\prime}_{2},...,x_{m+1,m}v^{\prime}_{m},x_{m,m+1}v^{\prime}_{m})$
$\displaystyle=$ $\displaystyle(v_{1},v_{2},...,v_{m},v_{m+1})=I(\Gamma).$

Hence

$I(\Gamma)=\bigcap_{1\leq i<j\leq m+1}(x_{ib(i,j)},x_{je(i,j)}),$

as desired.

As an application of Proposition 1.2, Corollary 1.3 and Proposition 1.4 we
obtain the following characterization of Cohen-Macaulay monomial ideals of
codimension 2.

######  Theorem 1.5.

(a) Let $I\subset S=K[x_{1},x_{2},...,x_{n}]$ be a Cohen-Macaulay monomial
ideal of codimension $2$ generated by $m+1$ elements. Then there exists a tree
$\Gamma$ with $m+1$ vertices and for each edge $\\{i,j\\}$ of $\Gamma$ there
exists a monomials $u_{ij}$ and $u_{ji}$ in $S$ such that

  * (i)

$\gcd(u_{ib(i,j)},u_{je(i,j)})=1$ for all $i<j$, and

  * (ii)

$I=(\prod_{i=2}^{m+1}u_{ib(i,1)},\ldots,\prod_{i=1\atop i\neq
j}^{m+1}u_{ib(i,j)},\ldots,\prod_{i=1}^{m}u_{ib(i,m+1)})$

(b) Conversely, if $\Gamma$ is a tree with $[m+1]$ vertices and for each
$\\{i,j\\}\in E(\Gamma)$ we are given monomials $u_{ij}$ and $u_{ji}$ in $S$
satisfying (a)(i). Then the ideal defined in (a)(ii) is Cohen-Macaulay of
codimension $2$.

###### Proof.

(a) (ii) Let $A$ be an $m\times m+1$ matrix of Taylor relations which
generated the relation module of $U$ of $I$, and let $\Gamma$ be the
corresponding relation tree. We apply the Hilbert–Burch Theorem ([2, 1.4.17])
according to which the ideal $I$ is generated by the maximal minors of $A$.
The matrix $A$ is obtained from $A(\Gamma)$ by the substitution:

$x_{ij}\mapsto u_{ij}.$

Therefore statement (ii) follows from Proposition 1.2.

Now we shall prove assertion (i). For this we use Proposition 1.4 which says
that

$I(\Gamma)=\bigcap_{1\leq i<j\leq m+1}(x_{ib(i,j)},x_{je(i,j)}).$

Applying the substitution map introduced in the proof of (ii) we obtain

(6) $\displaystyle I\subseteq\bigcap_{i<j}(u_{ib(i,j)},u_{je(i,j)}).$

Suppose $\gcd(u_{ib(i,j)},u_{je(i,j)})\neq 1$ for some $i$ and $j$. Then it
follows from (6) that $I$ is contained in a principal ideal. This is a
contradiction, because $\operatorname{height}I=2$.

(b) Let $\Gamma$ be a tree with vertex set $[m+1]$ and $m$ edges. For each
$\\{i,j\\}\in E(\Gamma)$ we have monomials $u_{ij},u_{ji}\in S$ satisfying
condition (a)(i). Let $A$ be the matrix obtained from $A(\Gamma)$ by the
substitutions $x_{ij}\mapsto u_{ij}$, and let $I$ be the ideal generated by
the maximal minors of $A$. It follows from Proposition 1.2 that
$I=(v_{1},\ldots,v_{m+1})$ where $v_{j}=\prod_{i=1\atop i\neq
j}^{m+1}u_{ib(i,j)}$.

First we shall prove that

$\gcd(v_{1},v_{2},...,v_{m+1})=1.$

We shall prove this by induction on the number of edges of $\Gamma$. The
assertion is trivial if $\Gamma$ has only one edge. Now let $|E(\Gamma)|=m>1$
and assume that the assertion is true for any tree with $m-1$ edges.

We may assume that $(m,m+1)$ is a leaf of $\Gamma$. Let $\Gamma^{\prime}$ be
the tree obtained from $\Gamma$ by removing the edge $\\{m,m+1\\}$. The matrix
$A(\Gamma^{\prime})$ is obtained from $A(\Gamma)$ by removing the row
$(0,\ldots,-x_{m,m+1},x_{m+1,m})$ and the column

$\begin{pmatrix}0\\\ \vdots\\\ 0\\\ x_{m+1,m}\end{pmatrix}.$

Let $A^{\prime}$ be the matrix obtained from $A(\Gamma^{\prime})$ by the
substitutions $x_{ij}\mapsto u_{ij}$, and let
$I^{\prime}=(v_{1}^{\prime},\ldots,v_{m}^{\prime})$ be the ideal of maximal
minors of $A^{\prime}$ where, up to sign, $v_{j}^{\prime}$ is the $j$th
maximal minor of $A^{\prime}$. Expanding the matrix $A$ we see that

$v_{j}=\pm v^{\prime}_{j}u_{m+1,m}\quad\text{for}\quad
j=1,2,\ldots,m\quad\text{and}\quad v_{m+1}=\pm v^{\prime}_{m}u_{m,m+1}.$

Therefore

$\gcd(v_{1},v_{2},\ldots,v_{m},v_{m+1})=\gcd(v^{\prime}_{1}u_{m+1,m},v^{\prime}_{2}u_{m+1,m},...,v^{\prime}_{m}u_{m+1,m},v_{m+1}).$

By induction hypothesis we have
$\gcd(v^{\prime}_{1},v^{\prime}_{2},...,v^{\prime}_{m})=1$, so that

$\gcd(v^{\prime}_{1}u_{m+1,m},v^{\prime}_{2}u_{m+1,m},...,v^{\prime}_{m}u_{m+1,m})=u_{m+1,m}.$

Hence it is enough to prove that

$\gcd(u_{m+1,m},v_{m+1})=1.$

Note that $u_{m+1,m}=u_{m+1,e(i,m+1)}$ for all $i$, and
$v_{m+1}=\prod_{i=1}^{m}u_{ib(i,m+1)}$. Therefore

$\gcd(u_{m+1,m},v_{m+1})=\gcd(u_{m+1,e(i,m+1)},\prod_{i=1\atop}^{m}u_{ib(i,m+1)})=1,$

since by our hypothesis (a)(i) we have
$\gcd(u_{m+1,e(i,m+1)},u_{ib(i,m+1)})=1$ for all $i$.

The Hilbert–Burch Theorem [2, 1.4.17] then implies that $I$ is a perfect ideal
of codimension $2$, and hence a Cohen–Macaulay ideal.

##  2\. The possible sets of relation trees attached to Cohen-Macaulay
monomial ideals of codimension 2

In this section we want to study set $\mathcal{T}(I)$ of all relation trees of
a Cohen–Macaulay monomial ideal of codimension 2. In general one may have more
than just one Hilbert–Burch matrix for an ideal $I$, and consequently more
than one relation trees. For example the ideal
$I=(x_{4}x_{5}x_{6},x_{1}x_{5}x_{6},x_{1}x_{2}x_{6},x_{1}x_{2}x_{3}x_{5})\subset
S=K[x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}]$ has the following two Hilbert–Burch
matrices

$A_{1}=\begin{pmatrix}-x_{1}&x_{4}&0&0\\\ 0&-x_{2}&x_{5}&0\\\
0&0&-x_{3}x_{5}&x_{6}\end{pmatrix},$

or

$A_{2}=\begin{pmatrix}-x_{1}&x_{4}&0&0\\\ 0&-x_{2}&x_{5}&0\\\
0&-x_{2}x_{3}&0&x_{6}\end{pmatrix}.$

The corresponding relation trees are $\Gamma_{1}$ and $\Gamma_{2}$ with
$E(\Gamma_{1})=\\{\\{1,2\\},\\{2,3\\},\\{3,4\\}\\}$ and
$E(\Gamma_{2})=\\{\\{1,2\\},\\{2,3\\},\\{2,4\\}\\}$.

However in the generic case we have

######  Proposition 2.1.

Let $\Gamma$ be a tree on the vertex set $[m+1]$ and let $I(\Gamma)$ be the
generic monomial ideal attached to $\Gamma$. Then
${\mathcal{T}}(I(\Gamma))=\\{\Gamma\\}$.

Recall that $I(\Gamma)$ is the ideal of maximal minors of the matrix
$A(\Gamma)$ defined in (4). Up to signs the minors of $A(\Gamma)$ are the
monomials $v_{i}=\prod_{r=1\atop r\neq i}^{m+1}x_{rb(r,i)}$, see Proposition
1.2.

For the proof of Proposition 2.1 we shall need

######  Lemma 2.2.

Let $\Gamma$ be a tree, then $\\{i,j\\}$ is an edge of $\Gamma$ if and only if

$\operatorname{lcm}(v_{i},v_{j})=v_{j}x_{ji}=v_{i}x_{ij}.$

###### Proof.

Let $\\{i,j\\}$ be an edge of $\Gamma$ and suppose that $i<j$. Note that

(7) $\displaystyle b(k,i)=b(k,j)$

for all $k$ which are different from $i$ and $j$, because if the path from $k$
to $i$ is $k=k_{0},k_{1},\ldots,k_{l}=i$, then the path from $k$ to $j$ will
be $k=k_{0},k_{1},\ldots,k_{l-1}=j$ or $k=k_{0},k_{1},\ldots,k_{l-1},i,j$
since $\\{i,j\\}$ be an edge of $\Gamma$. Now using (4) we have

$v_{i}=\pm\prod_{r=1\atop r\neq i}^{m+1}x_{rb(r,i)}=\pm\prod_{r=1\atop r\neq
i,j}^{m+1}x_{rb(r,i)}x_{jb(j,i)}=\pm\prod_{r=1\atop r\neq
i,j}^{m+1}x_{rb(r,i)}x_{ji}=\pm\prod_{r=1\atop r\neq
i,j}^{m+1}x_{rb(r,j)}x_{ji}.$

Similarly $v_{j}=\pm\prod_{r=1\atop r\neq i,j}^{m+1}x_{rb(r,j)}x_{ij}$. Hence
$\operatorname{lcm}(v_{i},v_{j})=v_{j}x_{ji}=v_{i}x_{ij}$.

On the other hand, suppose that $\\{i,j\\}$ is not an edge of $\Gamma$, then
there exists a vertex, different from $i$ and $j$, say $k$, which belongs to
the path from $i$ to $j$. Therefore $b(k,i)\neq b(k,j)$, and hence
$x_{kb(k,i)}\neq x_{kb(k,j)}$. Since $x_{kb(k,i)}\mid v_{i}$ and since
$x_{kb(k,j)}\mid v_{j}$ we cannot have
$\operatorname{lcm}(v_{i},v_{j})=~{}v_{j}x_{ji}=v_{i}x_{ij}$.

###### Proof of Proposition 2.1.

Since all monomial generators of $I(\Gamma)$ are of degree $m$ and since, by
the Hilbert–Burch Theorem [2, 1.4.17], these generators are the maximal minors
of any of its Hilbert–Burch matrices, it follows that all Hilbert–Burch
matrices must be linear. However by Lemma 2.2 we have only $m$ linear Taylor
relations. Therefore there exists only one Hilbert–Burch matrix for $I$.

In contrast to the result stated in Proposition 2.1 we have

######  Proposition 2.3.

Let $I=(u_{1},\ldots,u_{m+1})$ be the monomial ideal in
$K[x_{1},\ldots,x_{m+1}]$ with $u_{i}=x_{1}\cdots x_{i-1}x_{i+1}\cdots
x_{m+1}$ for $i=1,\ldots,m+1$. Then ${\mathcal{T}}(I)$ is the set of all
possible trees on the vertex set $[m+1]$.

###### Proof.

Let $\Gamma$ be an arbitrary tree on the vertex set $[m+1]$. For the $k$th
edge $\\{i,j\\}$ of $\Gamma$ take the monomial generators $u_{i}$ and $u_{j}$
of $I$. Then we have the Taylor relation $x_{j}e_{j}-x_{i}e_{i}$. Let $A$ be
the $m\times m+1$-matrix whose rows $(0,\cdots,-x_{i},\cdots,x_{j},\cdots,0)$
correspond to the Taylor relations $x_{j}e_{j}-x_{i}e_{i}$ arising from the
edges of $\Gamma$. Observe that the generic matrix $A(\Gamma)$ is mapped to
$A$ by the substitutions $x_{ij}=x_{i}$. Moreover the maximal minor $\pm
v_{i}$ of $A(\Gamma)$ is mapped to $u_{i}$ for all $i$. Therefore the $u_{i}$
are the maximal minors of $A$ which shows that $A$ is the Hilbert–Burch matrix
of $I$.

In order to study the general nature of $\mathcal{T}(I)$ we introduce the
following concept. Let $\mathcal{S}$ be a finite set. Recall that a collection
$\mathcal{B}$ of subsets of $\mathcal{S}$ is said to be the set of bases of a
matroid, if all $B\in\mathcal{B}$ have the same cardinality and if the
following exchange property is satisfied:

For all $B_{1},B_{2}\in\mathcal{B}$ and $i\in B_{1}\setminus B_{2}$, there
exists $j\in B_{2}\setminus B_{1}$ such that
$(B_{1}\setminus\\{i\\})\cup\\{j\\}\in\mathcal{B}$.

A classical example is the following: let $K$ be a field, $V$ a $K$-vector
space and ${\mathcal{S}}=\\{v_{1},\ldots,v_{r}\\}$ any finite set of vectors
of $V$. Let $\mathcal{B}$ the set of subset $B$ of $\mathcal{S}$ with the
property that $B$ is a maximal set of linearly independent vectors in
$\mathcal{S}$. It easy to check and well known that $\mathcal{B}$ is the set
of bases of a matroid.

######  Proposition 2.4.

Let $I\subset S$ be a Cohen–Macaulay monomial ideal of codimension $2$. Then
${\mathcal{T}}(I)$ is the set of bases of a matroid.

###### Proof.

Let $I$ be minimally generated by the monomials $u_{1},\ldots,u_{m+1}$ and let

$0\longrightarrow G\longrightarrow F\longrightarrow I\longrightarrow 0$

be the graded minimal free $S$-resolution of $S/I$.

The set $\mathcal{S}$ of Taylor relations generate the first syzygy module $U$
of $I$ which is isomorphic to the free $S$-module $G$. Consider the graded
$K$-vector space $U/{\mathfrak{m}}U$ where
${\mathfrak{m}}=(x_{1},\ldots,x_{n})$ is the graded maximal ideal of $S$. Note
that $\dim_{K}U/{\mathfrak{m}}U=m$. Since the relations $r_{ij}$ generate $U$
it follows that their residue classes $\bar{r}_{ij}$ in the $K$-vector space
$U/{\mathfrak{m}}U$ form a system of generators of $U/{\mathfrak{m}}U$. By the
homogeneous version of Nakayama (see [2, 1.5.24]) it follows that a subset
$B=\\{r_{i_{1}j_{1}},\ldots,r_{i_{m}j_{m}}\\}$ of the Taylor relations
$\mathcal{S}$ is a minimal set of generators of $U$ (and hence establishes a
Hilbert–Burch matrix of $I$) if and only if
$\\{\bar{r}_{i_{1}j_{1}},\ldots,\bar{r}_{i_{m}j_{m}}\\}$ is a basis of the
$K$-vector space $U/{\mathfrak{m}}U$. The desired conclusion follows, since
the relation trees of $I$ correspond bijectively to the set of Hilbert–Burch
matrices of $I$.

Given a finite simple and connected graph $G$. A maximal subtree
$\Gamma\subset G$ is called a spanning tree. It is well-known and easy to see
that the set ${\mathcal{T}}(G)$ of spanning trees is the set of bases of a
matroid.

Here we are interested in the spanning trees of the graph $G(I)$ whose set of
edges is given by with

$E(G(I))=\bigcup_{\Gamma\in\mathcal{T}(I)}E(\Gamma).$

We call $G(I)$ the Taylor graph of $I$. Obviously we have
${\mathcal{T}}(I)\subset{\mathcal{T}}(G(I))$. The question arises whether
${\mathcal{T}}(I)={\mathcal{T}}(G(I))$? Unfortunately this is not always the
case as the example at the beginning of this section shows. Indeed, in this
example, ${\mathcal{T}}(I)=\\{\Gamma_{1},\Gamma_{2}\\}$ with
$E(\Gamma_{1})=\\{\\{1,2\\},\\{2,3\\},\\{3,4\\}\\}$ and
$E(\Gamma_{2})=\\{\\{1,2\\},\\{2,3\\},\\{2,4\\}\\}$, so that
$E(G_{I})=\\{\\{1,2\\},\\{2,3\\},\\{2,4\\},\\{3,4\\}\\}$. This graph has the
spanning trees $\Gamma_{1},\Gamma_{2}$ and $\Gamma_{3}$ with
$E(\Gamma_{3})=\\{\\{1,2\\},\\{2,4\\},\\{3,4\\}\\}$. If $\Gamma_{3}$ would be
a relation tree of $I$, then

$A=\begin{pmatrix}-x_{1}&x_{4}&0&0\\\ 0&-x_{2}x_{3}&0&x_{6}\\\
0&0&-x_{3}x_{5}&x_{6}\end{pmatrix}.$

would have to be a Hilbert–Burch matrix of $I$, which is not the case since
the ideal of maximal minors of $A$ is the ideal $x_{3}I$.

However we have

######  Theorem 2.5.

Let $I$ be Cohen–Macaulay monomial ideal of codimension $2$ with linear
resolution. Then ${\mathcal{T}}(I)={\mathcal{T}}(G(I))$.

###### Proof.

Since $I$ has a linear resolution, it follows that all Hilbert–Burch matrices
of $I$ are matrices with linear entries. Let $L=\\{r_{1},\ldots,r_{k}\\}$ be
the set of linear Taylor relations. We may assume that $r_{1},\ldots,r_{m}$
are the rows of a Hilbert–Burch matrix of $I$, in other words, that
$r_{1},\ldots,r_{m}$ is a basis of the first syzygy module $U$ of $I$.

We first claim that $r_{i_{1}},\ldots,r_{i_{m}}\in L$ is basis of $U$ if and
only if the relations $r_{i_{1}},\ldots,r_{i_{m}}$ are $K$-linear independent.
Obviously, the relations must be $K$-linear independent in order to form a
basis of the free $S$-module $U$. Conversely, assume that
$r_{i_{1}},\ldots,r_{i_{m}}$ are $K$-linear independent. Since each
$r_{i_{j}}$ belongs to $U$ we can write

$r_{i_{j}}=f_{1j}r_{1}+f_{2j}r_{2}+\ldots+f_{mj}r_{m}\quad\text{with}\quad
f_{lj}\in S.$

The presentation can be chosen such that all $f_{lj}$ are homogeneous and such
that $\deg f_{lj}r_{l}=\deg r_{i_{j}}=1$ for all $l$ and $j$. In other words,
$\deg f_{lj}=0$ for all $l$ and $j$. Therefore the $m\times m$-matrix
$F=(f_{lj})$ is a matrix with coefficients in $K$. Since, by assumption the
relations $r_{i_{1}},\ldots,r_{i_{m}}$ are $K$-linear independent, it follows
that $F$ is invertible. This implies that the relations $r_{1},\ldots,r_{m}$
are linear combinations of the relations $r_{i_{1}},\ldots,r_{i_{m}}$.
Therefore these relations generate $U$ as well, and in fact form a basis of
$U$, since $U$ is free of rank $m$.

Our considerations so far have shown, that the set of Hilbert–Burch matrices
of $I$ correspond bijectively to the maximal $K$-linear subsets of $L$. Each
$r_{i}\in L$ is a row vector with exactly two non-zero entries. We attach to
$r_{i}$ the edge $e_{i}=\\{k,l\\}$, if the two non-zero entries of $r_{i}$ are
at position $k$ and $l$, and claim that

$E(G(I))=\\{e_{1},\ldots,e_{k}\\}.$

Indeed, according to the definition of $G(I)$ an edge $e$ belongs to
$E(G(I))$, if there exists a relation tree $T$ of $I$ with $e\in E(T)$. This
is equivalent to say that there exist linearly independent
$r_{i_{1}},\ldots,r_{i_{m}}\in L$ such that $e=e_{i_{j}}$ for some $j$. Now
choose $e_{i}\in\\{e_{1},\ldots,e_{k}\\}$. Then $r_{i}$ can be completed to
maximal set $\\{r_{i},r_{i_{2}},\ldots,r_{im}\\}$ of $K$-linear elements in
$L$. This shows that $e_{i}\in E(G(I))$ for $i=1,\ldots,k$, so that
$\\{e_{1},\ldots,e_{k}\\}\subset E(G(I))$. The other inclusion is trivially
true.

In order to complete the proof of the theorem we need to show that each
spanning tree $T$ of $G(I)$ is a relation tree of $I$. Let
$e_{i_{1}},\ldots,e_{i_{m}}$ be the edges of the tree. To prove that $T$ is a
relation tree amounts to show the relations $r_{i_{1}},\ldots,r_{i_{m}}$ are
$K$-linearly independent.

A free vertex of $T$ is a vertex which belongs to exactly one edge. Since $T$
is a tree, it has at least one free vertex. Say, $1$ is this vertex and
$e_{i_{1}}$ is the edge to which the free vertex $1$ belongs. Removing the
edge $e_{i_{1}}$ from $T$ we obtain a tree $T^{\prime}$ on the vertex set
$\\{2,3,\ldots,m+1\\}$. After renumbering the vertices and edges if necessary,
we may assume that $2$ is a free vertex of $T^{\prime}$ and $e_{i_{2}}$ the
edge to which $2$ belongs. Proceeding in this way we get, after a suitable
renumbering of the vertices and edges of $T$, a free vertex ordering of the
edges, that is, for all $j=1,\ldots,r$ the edges
$e_{i_{j}},e_{i_{j+1}},\ldots,e_{i_{m}}$ is the set of edges of a tree for
which $j$ is a free vertex belonging to $e_{i_{j}}$. Since renumbering of
vertices and of edges of $T$ means for the corresponding matrix of relations
simply permutation of the rows and columns, the rank of relation matrix is
unchanged. However in this new ordering, if we skip the last column of the
$m\times m+1$ relation matrix we obtain an upper triangular $m\times m$ matrix
with non-zero entries on the diagonal. This shows that the relations
$r_{i_{1}},\ldots,r_{i_{m}}$ are $K$-linearly independent, as desired.

Finally we will describe all the possible Taylor graphs of a Cohen–Macaulay
monomial ideal of codimension $2$ with linear resolution. Then, together with
Theorem 2.5, we have a complete description of all possible relation trees for
such ideals.

Let $G$ be finite connected simple graph on the vertex set $[n]$. Recall that
a subset $C$ of $[n]$ is called a clique of $G$ if for all $i$ and $j$
belonging to $C$ with $i\neq j$ one has $\\{i,j\\}\in E(G)$. The set of all
cliques $\Delta(G)$ is a simplicial complex, called the clique complex of $G$.

######  Theorem 2.6.

Let $G$ be finite connected simple graph. Then the following are equivalent:

  1. (a)

$G$ is a Taylor graph of a Cohen–Macaulay monomial ideal of codimension $2$
with linear resolution.

  2. (b)

$G$ is a chordal graph with the property that any two distinct maximal cliques
have at most one vertex in common.

###### Proof.

(a)${}\Rightarrow{}$(b): Let $I$ be generated by $m$ monomials and $G=G(I)$,
and let $C$ be a cycle of $G$. We first show that the restriction $G^{\prime}$
of $G$ to $C$ is a complete graph, that is, we show that for any two distinct
vertices $i,j\in C$ it follows that $\\{i,j\\}\in E(G)$. In particular, this
will imply that $G$ is chordal.

For simplicity we may assume that $E(C)=\\{e_{1},\ldots,e_{k}\\}$ with $k\geq
3$ and $e_{i}=\\{i,i+1\\}$ for $i=1,\ldots,k-1$ and $e_{k}=\\{k,1\\}$. Let
$r_{1},\ldots,r_{k}$ be the corresponding relations. Let $\varepsilon_{i}\in
K^{m-1}$, $i=1,\ldots m-1$ be the canonical basis vectors of $K^{m-1}$. Then
$r_{i}=-a_{i}\varepsilon_{i}+b_{i}\varepsilon_{i+1}$ for $i=1,\ldots,k-1$ and
$r_{k}=-b_{k}\varepsilon_{1}+a_{k}\varepsilon_{k}$, where $a_{i}$ and $b_{i}$
belong to $\\{x_{1},\ldots,x_{n}\\}$. Assume that $r_{1},\ldots,r_{k}$ are
$K$-linearly independent. Then $r_{1},\ldots,r_{k}$ can be completed to
$K$-basis $r_{1},\ldots,r_{m}$ of $L$. (Here we use the notation introduced in
the proof of Theorem 2.5.) Let $\Gamma$ be the tree corresponding to
$r_{1},\ldots,r_{m}$. Then $C$ is a subgraph of $\Gamma$, which is a
contradiction. Thus we see that the relations $r_{1},\ldots,r_{k}$ are
$K$-linearly dependent which implies at once that $a_{1}=b_{k}$ and
$a_{i}=b_{i-1}$ for $i=2,\ldots,k$. Hence we have
$r_{1}+\cdots+r_{i}=-a_{1}\varepsilon_{1}+b_{i}\varepsilon_{i+1}$ for
$i=1,\ldots,k-1$. This implies that $\\{1,i\\}$ is an edge of $G$ for
$i=2,\ldots k$. By symmetry, also the other edges $\\{i,j\\}$ with $2\leq
i<j\leq k$ belong to $G$.

Now let $G_{1}$ and $G_{2}$ be two distinct maximal cliques of $G$, and assume
that they have two vertices in common, say, the vertices $i$ and $j$. Let
$k\in G_{1}\setminus\\{i,j\\}$ and $l\in G_{2}\setminus\\{i,j\\}$. Then the
graph $C$ with edges $\\{i,k\\},\\{k,j\\},\\{j,l\\},\\{l,i\\}$ is a cycle in
$G$. Therefore, by what we have shown, it follows that $\\{k,\l\\}$ is an edge
of $G$. Thus for any two vertices $k,l\in V(G_{1})\cup V(G_{2})$ it follows
that $\\{k,l\\}\in E(G)$, contradicting the fact that $G_{1}$ and $G_{2}$ are
distinct maximal cliques of $G$.

(b)${}\Rightarrow{}$(a): Let $C_{1},\ldots,C_{r}$ be the maximal cliques of
the chordal graph $G$, and let $\Delta(G)$ be the clique complex of $G$. Then
the $C_{i}$ are the facets of $\Delta(G)$. One version of Dirac’s theorem [3]
says that $\Delta(G)$ is a quasi-forest, see [5]. This means, that there is an
order of the facets, say, $C_{1},C_{2},\ldots,C_{r}$ such that for each $i$
there is a $j<i$ with the property that $C_{k}\cap C_{i}\subset C_{j}\cap
C_{i}$ for all $k<i$. Given this order, then our hypothesis (b) implies that
for each $i=2,\ldots,r$ there exists a vertex $k_{i}\in C_{i}$ such $C_{i}\cap
C_{i-1}=\\{k_{i}\\}$ and $C_{i}\cap C_{j}=\\{k_{i}\\}$ for all $j<i$ with
$C_{i}\cap C_{j}\neq\emptyset$. The following example illustrates the
situation. Let $G$ be the graph on the vertex set $[7]$ with edges
$\\{1,2\\},\\{1,3\\},\\{2,3\\},\\{3,4\\},\\{3,5\\},\\{4,5\\},\\{5,6\\},\\{5,7\\}$.

Then $G$ is a connected simple graph satisfying the condition in (b). The
maximal cliques of $G$ ordered as above are $C_{1}=\\{1,2,3\\}$,
$C_{2}=\\{3,4,5\\}$, $C_{3}=\\{5,6\\}$ and $C_{4}=\\{5,7\\}$ and intersection
vertices are $k_{2}=3$, $k_{3}=5$ and $k_{4}=5$.

After having fixed the order of the cliques, we may assume that the vertices
of $G$ are labeled as follows: if $|C_{1}\cup\cdots\cup C_{i}|=s_{i}$, then
$C_{1}\cup\cdots\cup C_{i}=\\{1,2,\ldots,s_{i}\\}$. In other words,
$C_{1}=\\{1,\ldots,s_{1}\\}$ and
$C_{i}\setminus\\{k_{i}\\}=\\{s_{i-1}+1,\ldots,s_{i}\\}$ for $i>1$. The
vertices on the graph in Figure 1 are labeled in this way. Now we let
$\Gamma\subset G$ be the spanning tree of $G$ whose edges are $\\{j,k_{2}\\}$
with $j\in C_{1}$ and $j\neq k_{2}$, and for $i=1,\ldots,r$ the edges
$\\{j,k_{i}\\}$ with $j\in C_{i}$ and $j\neq k_{i}$. In our example the edges
of $\Gamma$ are $\\{1,3\\}$, $\\{2,3\\}$, $\\{3,4\\}$,$\\{3,5\\}$, $\\{5,6\\}$
and $\\{5,7\\}$.

Let $m+1=s_{r}$. Then $m+1$ is the number of vertices of $G$. We now assign to
$\Gamma$ the following $m\times m+1$-matrix $A$ whose rows $r_{e}$ correspond
to the edges $e$ of $\Gamma$ as follows: we set
$r_{e}=-x_{1j}\varepsilon_{j}+x_{1k_{2}}\varepsilon_{k_{2}}$ for
$e=\\{j,k_{2}\\}$ and $j\in C_{1}$ with $j\neq k_{2}$, and we set
$r_{e}=-x_{ij}\varepsilon_{j}+x_{ik_{i}}\varepsilon_{k_{i}}$ for
$e=\\{j,k_{i}\\}$ and $j\in C_{i}$ with $j\neq k_{i}$ and $i>1$. Here
$\varepsilon_{i}$ denotes the $i$th canonical unit vector in
${\mathbb{R}}^{m+1}$.

The rows $r_{e}$ can be naturally ordered according to the size of $j$ in the
edge $e=\\{j,k_{i}\\}$. Thus in our example we obtain the matrix

$\begin{pmatrix}-x_{11}&0&x_{13}&0&0&0&0\\\ 0&-x_{12}&x_{13}&0&0&0&0\\\
0&0&x_{23}&-x_{24}&0&0&0\\\ 0&0&x_{23}&0&-x_{25}&0&0\\\
0&0&0&0&x_{35}&-x_{36}&0\\\ 0&0&0&0&x_{45}&0&-x_{47}\\\ \end{pmatrix}$

Our next goal is to show that our matrix $A$ is a Hilbert–Burch matrix. We
apply Theorem 1.5. Tor each edge $\\{i,j\\}\in\Gamma$ the monomials $u_{ij}$
and $u_{ji}$ are, according to the choice of $A$, the following:

$u_{jk_{2}}=-x_{1j},\quad u_{k_{2}j}=x_{1k_{2}}\quad\text{for}\quad j<k_{2},$

and for $i=2,\ldots,r$

$u_{k_{i}j}=x_{ik_{i}},\quad u_{jk_{i}}=-x_{ij}\quad\text{for}\quad
k_{i}<j,\quad j\in C_{i}.$

According to Theorem 1.5(b) we have to show that
$\gcd(u_{ib(i,j)},u_{je(i,j)})=1$ for all $i<j$. Assume first that
$i,j\not\in\\{k_{2},\ldots,k_{r}\\}$. Then $u_{ib(i,j)}=-x_{ti}$ for $i\in
C_{t}$ and $x_{je(i,j)}=-x{sj}$ for$j\in C_{s}$. Thus in this case
$\gcd(u_{ib(i,j)},u_{je(i,j)})=1$. In the second case let
$i\not\in\\{k_{2},\ldots,k_{r}\\}$ and $j\in\\{k_{2},\ldots,k_{r}\\}$, let say
$j=k_{s}$. Then $b(i,j)=k_{t}$ for $i\in C_{t}$ and so $u_{ib(i,j)}=-x_{ti}$.
Suppose $\\{i,j\\}$ is not an edge then $e(i,j)=b(j,i)=b(k_{s},i)$ is either
$k_{s+1}$ or $k_{s-1}$. Then $u_{je(i,j)}$ is either $-x_{(s)j}$ or
$-x_{(s-1)j}$. On the other hand, if $\\{i,j\\}$ is an edge, then $e(i,j)=i$,
and so $u_{je(i,j)}=u_{ji}=u_{k_{s}i}=x_{sj}$. Thus in this case, too,
$\gcd(u_{ib(i,j)},u_{je(i,j)})=1$. Finally assume that
$i,j\in\\{k_{2},\ldots,k_{r}\\}$, and let
$i=k_{s_{1}},k_{s_{1}+1},\ldots,k_{s_{2}}=j$ be the path from $i$ to $j$. Then
$b(i,j)=k_{s_{1}+1}$ and $e(i,j)=k_{s_{2}-1}$ so $u_{ib(i,j)}=x_{s_{1}i}$ and
$u_{je(i,j)}=-x_{(s_{2}-1)j}$. Thus again in this case we have
$\gcd(u_{ib(i,j)},u_{je(i,j)})=1$. Thus in all cases
$\gcd(u_{ib(i,j)},u_{je(i,j)})=1$ for all $i<j$, as desired.

Let $I$ be the codimension $2$ ideal whose relation matrix is $A$, and let
$\\{i,j\\}$ be any edge of $G$. It remains to be shown that there exits a
relation tree $\Gamma^{\prime}$ with $\\{i,j\\}\in E(\Gamma^{\prime})$. If
$\\{i,j\\}\in E(\Gamma)$, we are done. Now assume that $\\{i,j\\}\not\in
E(\Gamma)$. We may assume that $\\{i,j\\}\in C_{t}$. Let $s=t$ if $t>1$, and
$s=2$ if $t=1$. We replace the row
$-x_{tj}\varepsilon_{j}+x_{tk_{s}}\varepsilon_{k_{s}}$ of $A$ by the
difference of the rows

$-x_{ti}\varepsilon_{i}+x_{tj}\varepsilon_{j}=(-x_{ti}\varepsilon_{i}+x_{tk_{s}}\varepsilon_{k_{s}})-(-x_{tj}\varepsilon_{j}+x_{tk_{s}}\varepsilon_{k_{s}}),$

and leave all the other rows of $A$ unchanged. The new matrix $A^{\prime}$ is
again a relation matrix of $I$ and the tree $\Gamma^{\prime}$ corresponding to
$A^{\prime}$ is obtained from $\Gamma$ by removing the edge $\\{j,k_{s}\\}$
and adding the edge $\\{i,j\\}$. This completes the proof of the theorem.

## References

  * [1] W. Bruns, J. Herzog. On multigraded resolutions, Math. Proc. Cambridge Phil. Soc. 118, 234-251 (1995).
  * [2] W. Bruns, J. Herzog. Cohen Macaulay rings, Revised Edition, Cambridge, 1996.
  * [3] G. A. Dirac. On rigid circuit graphs. Abh. Math. Sem. Univ. Hamburg, 38, 71 - 76 (1961).
  * [4] D. Eisenbud. Commutative algebra; with a view towards algebraic geometry, Graduate Texts Math., Springer. 1995.
  * [5] J. Herzog, T. Hibi and X. Zheng. Dirac’s theorem on chordal graphs and Alexander duality. European J. Comb. 25(7), 949–960 (2004).
  * [6] R. H. Villareal. Monomial algebras, Dekker, New York, 2001


November, 2022

# Weyl entropy behaviour and Naked Singularities111Based on presentation to be
given at the 32nd meeting of the Indian Association for General Relativity and
Gravitation, IISER Kolkata.

Vaibhav Kalvakota1222E-mail<EMAIL_ADDRESS>1Turito
Institute1Turito Institute Hyderabad Hyderabad 500081 500081 India
India

###### Abstract

In order to show that there exists a relation between the validity of the Weyl
entropy and the cosmic censorship hypothesis, we discuss the relation between
naked singularities and the gravitational entropy estimator using the Weyl
invariant. We look at the examples of Vaidya and Tolman-Bondi spacetimes in
asymptotically flat and de Sitter spacetime and look at Weyl entropy as a
possible indicator of the prohibition of naked singularities.
Keywords Naked singularities, Weyl curvature hypothesis, Gravitational
entropy, Cosmic censorship conjecture

##  1 Introduction

Following Penrose’s Weyl curvature hypothesis [1], there has been a growing
mathematical interest on the definition of gravitational entropy estimators
and their cosmological implications (for instance Gron and Hervick[2],
Chakraborty et al[3], Pizana et al[4] and Lima et al[5]). The foundation of
the Weyl hypothesis is that the Weyl tensor might be an estimator of the
gravitational entropy of the spacetime. In using the Weyl tensor to define the
gravitational entropy, there are three conditions to be met: (1) the Weyl
tensor must be zero at $t=0$ for a cosmology and increase monotonically in the
entire future evolution of the cosmology, (2) it must reduce to the
holographic principle form for a given spacetime when applicable and (3) it
must account for the anisotropies forming in the cosmology. Each of these
points can be elaborated as follows: (1) implies that the gravitational
entropy is monotonically increasing, accounting for structure formations in
the cosmology. It must be zero at the initial singularity accounting for the
perfect Friedmann-Robertson modelling of the spacetime, which has a metric
that is conformally flat. Due to this, the Weyl tensor vanishes, which can be
attributed to the lower anisotropies in the spacetime. Note that this does not
correspond to the matter entropy, which clearly is non-zero. (2) refers to the
nature of the gravitational entropy of spacetimes with a horizon, in which
cases the gravitational entropy must reduce to the holographic principle. For
black hole spacetimes this would be the Hawking-Bekenstein entropy, while for
cosmologies with a horizon this would be the Gibbons-Hawking entropy[9]. (3)
accounts for the relation between anisotropies and the gravitational entropy.
The natural choice for defining the estimator for Weyl entropy is the Weyl
invariant $\mathcal{W}\equiv
C_{\alpha\beta\gamma\delta}C^{\alpha\beta\gamma\delta}$, but due to isotropic
singularities it is necessary to add a prefactor of the Ricci scalar
$\textbf{Ric}\equiv R_{\alpha\beta}R^{\alpha\beta}$ or the Kretschmann scalar
$\mathcal{K}\equiv R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}$ .
However, for the sake of convenience, we will opt for the earlier choice of
using purely the Weyl invariant $\mathcal{W}$.

##  2 Naked Singularities and Weyl entropy

The relation between the possibility of formation is proposed to be found out
by finding the behaviour of the Weyl entropy in the case of a cosmological
initial singularity and naked singularities from gravitational collapse. This
is done by identifying the location of the singularity with respect to an
outgoing geodesic – for an initial singularity, this would be in the past,
while for a gravitational collapse resulting in a singularity with an event
horizon (assuming the validity of the cosmic censorship conjecture), the
singularity would lie in the future for a family of geodesics that are emitted
after the horizon forms – which is the definition of a trapped surface.

The key point used in this paper is to identify the nature of the Weyl scalar
based scalar $\mathcal{P}$ along an outgoing geodesic $x^{\mu}$. The Weyl
curvature hypothesis states that for $x^{\mu}$ originating from an initial
singularity, $\mathcal{P}(x^{\mu})$ must be diverging and monotonically
increasing. In the case when we assume the cosmic censorship conjecture to be
violated, a naked singularity will be found without a trapped surface,
implying that for an outgoing geodesic, the singularity lies in the past,
which is similar to the cosmological initial singularity. In the first case,
there is clearly a difference between the initial and the gravitational
collapse singularity, whereas in the second case this is not so. In the former
case, there is clearly a difference between the black hole singularity and the
initial singularity, since the Weyl entropies are not the same[7]. The black
hole Weyl entropy would be the Hawking-Bekenstein formula, which is non-zero.
We can check whether or not the two singularities are same in the latter case
by looking at the gravitational entropies of the singularities. Assuming the
cosmic censorship conjecture to be invalid, it would mean that a naked
singularity can form and light rays will not be trapped. In this case, the
similarity between the initial singularity and the naked singularity is clear,
since both would be past singularities for an outgoing geodesic. As discussed
previously by Barve and Singh[6], it is shown that for inhomogeneous
distributions such as in Tolman-Bondi and Vaidya spacetimes, naked
singularities conflict with the validity of the Weyl curvature hypothesis.
This nature of diverging Weyl entropy has been studied in Barve and Singh[7]
is found in the case of Lemaitré-Tolman-Bondi spacetimes. There are then two
possibilities – first, since the Weyl entropy must be zero at the initial
singularity333This is contradicted since the Weyl entropy diverges as you
approach the past singularity in a Tolman-Bondi spacetime, which implies that
the initial singularity must also have a higher Weyl entropy state than in
later times., it is possible that naked singularities don’t form, which would
validate the cosmic censorship hypothesis by restricting the formation of
naked singularities altogether. Second, it is possible that the Weyl curvature
hypothesis may not be valid in all cases, which would have many cosmological
implications.

There is also the possibility that the very fact that there is no horizon
could contribute to the invalidity of either cosmic censorship or the Weyl
hypothesis, since the area of the event horizon is related to the entropy of
the black hole.
The area of a black hole is related to the entropy of the black hole in the
form of the Hawking-Bekenstein entropy as proposed by Hawking and
Bekenstein[8]:

$S_{HB}=\frac{A_{BH}}{4G\hbar}$ (1)

Where $A$ is the area of the event horizon and $k_{B}=c=1$. In the case when
we are defining a black hole Weyl entropy, we require that $S_{W}=S_{HB}$, or
more broadly $S_{W}=S_{\textrm{holographic}}$ in cases when a cosmological
horizon exists, which would be the cosmological holographic principle
counterpart to $S_{HB}$, called the Gibbons-Hawking entropy [9]. When there
does not exist a horizon, there are two implications for the Weyl entropy –
firstly, there is no horizon based contribution to $S_{W}$ as $A_{BH}=0$, and
secondly, the Weyl entropy of the singularity diverges when defining the
entropy density as specified by [10]. That is to say that the Weyl entropy
when solely considering the naked singularity may be considered to be the
equivalent of the usual Weyl entropy estimator considering only the
singularity and no other contribution, which is divergent. This further
implies that for an outgoing geodesic in the case of a violation of the cosmic
censorship conjecture, the Weyl entropy vanishes following an outgoing
geodesic. This is clearly in conflict with the Weyl entropy strictly diverging
along an outgoing geodesic.

While we have not looked at other forms of gravitational entropy estimators
such as the Clifton-Ellis-Tavakol proposal [11] or spin coefficients proposal
[12], our interest in using the purely Weyl invariant based entropy is
supported by the fact that the ”strong” condition here is that the Weyl
invariant is a direct measure of gravitational entropy in some sense444Here,
by ”direct” we mean the proportionality $S_{g}\propto\mathcal{W}$. In other
proposals such as the CET proposal, the measurement is done using a different
approach which, while considering the Weyl tensor, are not purely independent
of other factors. Interested readers are directed to the works by Chakraborty
et al[3], Pizana et al[4] and Gregoris et al[13] for cosmological studies
using the CET proposal. – the hypothesis is simply that the Weyl entropy at
the cosmological initial singularity must be zero, and it must monotonically
grow, while also acting as the measure for other spacetimes where
gravitational entropy is of interest such as black hole solutions. In order to
capture these points, we will discuss the example of the Vaidya-de Sitter
solution to the field equations, where we capture the nature of the Weyl
invariant and discuss the conditions under which naked singularities pose a
problem.

##  3 Charged and Uncharged Vaidya solutions

The Vaidya[14] solutions to the Einstein field equations are spacetimes
characterized by imploding shells of null fluid. In the case of uncharged
Vaidya spacetime to the field equations, we have a self-similar spacetime
without an electric charge, while in the charged de Sitter case, we have an
electric charge and a non-zero cosmological constant. In these cases, we will
adopt the $(v,r,\theta,\phi)$ coordinates, where $v$ is the advanced Eddingtom
time coordinate.

###  3.1 Uncharged Vaidya solution

The Einstein field equations with a cosmological constant are given by

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+g_{\mu\nu}\Lambda=kT_{\mu\nu}$ (2)

In cases where we consider a de Sitter type spacetime, $\Lambda$ will not be
zero. The first case is that of uncharged advanced Vaidya metric, where the
electric charge and $\Lambda$ are both set to zero. Then, the Vaidya spacetime
is described by the line-element:

$\mathrm{d}s^{2}=-\left(1-\frac{2m(v)}{r}\right)\mathrm{d}v^{2}+2\mathrm{d}u\mathrm{d}v+r^{2}\mathrm{d}\Omega^{2}$
(3)

For this to be self-similar, we impose the condition of $m(v)=\chi_{0}v$, so
that $\chi_{0}$ is the infall rate of null dust[15]. For any value less than
or equal to $0.125$, a naked singularity is formed. The Weyl invariant
$\mathcal{W}$ is a function of $v$ and $r$, given by

$\mathcal{W}=\frac{12\chi^{2}v^{2}}{r^{6}}$

We define the tangent of an outgoing geodesic from the naked singularity as
$X=v/r$, with $X_{0}$ being the limiting value. Further, the Ricci scalar is
zero in this case, indicating a dominance of the Weyl tensor. We see that the
Weyl invariant diverges as $r\to 0$, which is a conflicting statement with the
Weyl curvature hypothesis, which states that the initial singularity must have
a lower entropy value that diverges with time[6]. Therefore, the similarity
between the naked singularity here and the initial singularity is in conflict
with the Weyl curvature hypothesis.

###  3.2 Charged Vaidya-de Sitter solution

Another case of Vaidya spacetimes is that of those with charge and $\Lambda$
being non-zero. Such a spacetime is a Vaidya-de Sitter spacetime with charge,
with the metric

$\mathrm{d}s^{2}=-\left(1-\frac{2m(v)}{r}+\frac{q^{2}(v)}{r^{2}}-\frac{\Lambda
r^{2}}{3}\right)\mathrm{d}v^{2}+2\mathrm{d}u\mathrm{d}v+r^{2}\mathrm{d}\Omega^{2}$
(4)

This spacetime represents inflow of null fluid with a non-zero charge in an
empty de Sitter region. Assuming the collapse takes places at $r=0$ with the
last infalling cloud at some time $v=t$, we find that there are three
conditions on $m(v)$ and $q(v)$ being proportional to $v$. The bounds for this
function are $v<0$, $v\in[0,t]$ and $v>t$ respectively [16]. Based on this, we
have conditions on $m(v)\propto v$ and $q^{2}(v)\propto v^{2}$, using which
the piecewise definitions of $m$ and $q$ can be found out. In this case, for
the interval $v\in[0,t]$, we define $m(v)=\lambda v$ for $\lambda>0$, while
after $v=t$, $m(v)$ is a constant $M$. A similar definition is used for
$q(v)$, where we define $q(v)=\theta v$ for $v\in[0,t]$ with $\theta>0$ and a
constant $\Theta$ for $v>t$. For $v<0$, both $m(v)$ and $q(v)$ are zero. The
factors $m(v)$ and $q(v)$ are determined in accordance to the energy
conditions, and for the bound $[0,t]$, this would be the charged Vaidya-de
Sitter spacetime. In order to see the nature of the Weyl invariant for this
spacetime, we start by defining the tangent to an outgoing geodesic by
$X=v/r$. Using these definitions, we can show that for null geodesics,

$\frac{\mathrm{d}r}{\mathrm{d}v}=\frac{1}{2}\left(1-2\lambda
X+\theta^{2}X^{2}-\frac{\Lambda r^{2}}{3}\right)$ (5)

This clearly has a singularity at $r,v=0$. In order to determine the nature of
the singularity, we would need to check if there are non-spacelike curves that
are escaping from the singularity. In order to do so, we let a limiting value
$X_{0}$ on $X$:

$X_{0}=\lim_{r,v\to 0}\frac{v}{r}$ (6)

By differentiating the numerator and denominator and using eq:cvdssc,
eq:cvdslv can be solved to yield

$\theta^{2}X_{0}^{3}-2\lambda X_{0}^{2}+X_{0}-2=0$ (7)

If this has a positive root, it implies that there exists a locally naked
singularity, which eq:cvdsr does due to the non-negative values of $\lambda$
and $\theta$. This can be seen by observing that eq:cvdsr is a cubic equation,
and therefore it must be true that $\theta>0$ and $\lambda>0$.

The Weyl invariant $\mathcal{W}$ is a function of $X$ and $r$, which can be
found out to be

$\mathcal{W}(X,r)=\frac{48}{r^{4}}\left(\lambda^{2}X^{2}-2\lambda\theta^{2}X^{3}+\theta^{4}X^{4}\right)$
(8)

Unlike the uncharged Vaidya spacetime case, this spacetime has a non-dominant
Weyl scalar over $\mathbf{Ric}$, which diverges at the same rate as the Weyl
invariant. In this case, equation eq:wicvds shows that the Weyl invariant
diverges at the singularity. Since this solution allows naked singularities
and has a diverging Weyl invariant at the singularity, it may be considered
that the Weyl entropy for an outgoing geodesic from the singularity has a
lower entropy state further in time than at the singularity. In contrast, an
outgoing geodesic from the cosmological initial singularity would yield a
diverging Weyl entropy. Therefore, this may be considered as a contradictory
statement to the Weyl curvature hypothesis assuming cosmic censorship is
invalid.

###  3.3 Charged Vaidya anti-de Sitter solution

If $\Lambda<0$, the metric eq:cvds represents a charged Vaidya solution in an
AdS background[17]. In this case too, a similar approach can be used to find
out the nature of the singularity for specific initial data[18] and the Weyl
invariant. By following the same approach as the previous case as illustrated
in section 3.2, we start by defining the limiting value $X_{0}$ as
$X_{0}=\lim_{r,v\to 0}\;\frac{v}{r}$. By differentiating the numerator and the
denominator, we can use the slope condition for this spacetime to get
eq:cvdsr, which in this case would have varying values of $\theta$ and
$\lambda$ with respect to $\Lambda$ that admit a positive root, which implies
the existence of a naked singularity.

##  4 Tolman-Bondi solutions

As further examples on the diverging Weyl entropy, consider the case of the
spherical dust collapse from the Tolman-Bondi solutions. We will discuss the
Tolman-Bondi dust collapse under asymptotically flat ($\Lambda=0$) and
asymptoticaly de Sitter ($\Lambda>0$) spacetimes. The general form of the
metric for such cases is given by

$\mathrm{d}s^{2}-\mathrm{d}t^{2}+e^{-2\chi(r)}\mathrm{d}r^{2}+R(r,t)\mathrm{d}\Omega^{2}\;\;\;.$
(9)

###  4.1 $\Lambda=0$

Another spacetime with a diverging Weyl entropy in a gravitational collapse is
that of Tolman-Bondi spacetime, defined by the metric

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+\frac{R^{\prime}(r,t)^{2}}{1+f(r)}\mathrm{d}r^{2}+R(r,t)^{2}\mathrm{d}\Omega^{2}$
(10)

where the prime denotes differentiation w.r.t. the comoving $r$ coordinate.
Defining the mass function for the cloud of dust as $m(r)$, the Weyl invariant
is of the form

$\mathcal{W}(r,t)=\frac{48}{R(r,t)^{4}}\left(\frac{m(r)}{R(r,t)}-\frac{m^{\prime}(r)}{3R^{\prime}(r,t)}\right)^{2}$
(11)

This clearly diverges at $R=0$ – for both ingoing and outgoing geodesics,
there is a divergent Weyl invariant. In this case, $\mathbf{Ric}$ diverges at
the same rate as that of $\mathcal{W}$. In order to find the condition for
naked singularities as seen before, we define the tangent $R=Xr^{\alpha}$ for
$\alpha>1$. For this spacetime too, we see that the Weyl invariant is a
function of $r$ such that at $r=0$ there is a naked singularity for the range
$\alpha\in[5/3,3]$. Clearly again, along an outgoing geodesic we observe a
vanishing Weyl invariant, which implies that the naked singularity cannot be
similar to the initial singularity, in conflict with the Weyl curvature
hypothesis[7].

###  4.2 $\Lambda>0$

This is the case of Tolman-Bondi dust scenario eq:gtb seen above in
asymptotically de Sitter spacetime. The stress energy momentum tensor is given
by

$T_{\mu\nu}=\rho(r,t)u_{\mu}u_{\nu}-g_{\mu\nu}\frac{\Lambda}{8\pi}$ (12)

And we have $m(r,t)=\frac{R}{2}\left(\Dot{R}^{2}-\frac{\Lambda
R^{2}}{3}+f\right)$ In this case the constraint on $R$ is that it must satisfy

$m^{\prime}=4\pi\rho R^{2}R^{\prime}$

The Kretschmann scalar $\mathcal{K}$ diverges, as does the Ricci scalar Ric.
The determination of a singularity’s nature being naked in this case too can
be seen by finding an outgoing null geodesic that has an origin at the
singularity. At $r=0$, we define $\rho(r,t=0)$, and by defining the expansion
as proposed by Barve and Singh[6], we find that there exist initial data555The
interested reader is directed to Goncalves[20] for a complete derivation on
the initial data and the locality of the singularity, where the expansion put
forward by Barve and Singh[7] is used. for which there exists at least one
outgoing null geodesic with a starting point at the singularity. Further, the
Weyl invariant is found to be similar to that of the $\Lambda=0$ case, and at
the singularity for specific initial data, $\mathcal{W}$ diverges to infinity.
We have discussed subcases of two general solutions to the field equations,
the Vaidya and the Tolman-Bondi collapse. In the next section, we will briefly
remark on the nature of gravitational entropy as a horizon based entropy.

##  5 Horizon based entropy

By definition, the gravitational entropy of a spacetime with a horizon must
reduce to its corresponding horizon based entropy, i.e.

$S_{\text{grav}}\thicksim A_{\text{horizon}}$

Then, we can base the definition of gravitational entropy on the horizon
entropy. This was proposed by Rudjord and Gron [10] and verified in several
literatures (such as the papers on black holes[19], wormholes[5], etc.) for
various solutions. However, in the case of Petrov type O spacetimes with a
cosmological horizon, it fails to recover the appropriate entropy, which is
the Gibbons-Hawking entropy. Since $S_{\text{grav}}\thicksim
C_{\alpha\beta\gamma\delta}C^{\alpha\beta\gamma\delta}$, the gravitational
entropy must be zero, indicating a purely thermodynamic entropy contribution.
This is not a serious problem – for instance, by use of spin coefficients as
proposed by Gregoris and Ong[12], it is possible to show that a horizon based
entropy can be defined for cases where $C_{\alpha\beta\gamma\delta}=0$.
Therefore, while the Weyl estimator works in some cases, it is possible that
there are other estimators that define gravitational entropy more
”strongly”666In the spin coefficients approach, this refers to the use of the
Weyl tensor implictly, since the ground for defining gravitational entropy is
no longer a direct quantification of the Weyl tensor but instead the use of
spin coefficients takes its place, which reduces to an appropriate form of the
Weyl tensor when it is non-zero; refer to sections II-IV in the paper by
Gregoris and Ong[12]. – in any case, since the horizon based entropy is
defined on the pretext of the spacetime having a horizon, it is clear that if
a gravitational collapse leads to a naked singularity, the Weyl entropy there
would face a divergence due to the singularity in the volume integral. This
can be shown by considering the direct Weyl approach[10], where we define the
volume integral as

$S_{grav}=k_{s}\int_{V}\;\frac{C_{\alpha\beta\gamma\delta}C^{\alpha\beta\gamma\delta}}{\mathcal{R}}dV$
(13)

Where $k_{s}$ is a constant and $\mathcal{R}$ is a curvature invariant
correction to the Weyl invariant due to rescaling for isotropic
singularities[21], which further also indicates the dominance of the Weyl
invariant over the Riemann tensor based invariants
$\mathcal{R}=\mathcal{K}\equiv
R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}$ or
$\mathcal{R}=\mathbf{Ric}\equiv R_{\alpha\beta}R^{\alpha\beta}$. Clearly, in
order to define the horizon entropy of a black hole, we must subtract the
singularity by defining a spherical element of radius $\epsilon\to 0$ and
subtracting this from the singularity. However, in the case of naked
singularities, no horizon is observed – then, a divergent Weyl invariant
directly implies a diverging gravitational entropy, which is the base of the
conflict between cosmic censorship and the Weyl curvature hypothesis. Since
the constraint to begin with on gravitational entropy is that it must be
independent of local matter fields $\mathbf{\Psi}$ on $(M,g)$, the Weyl
invariant is the appropriate choice. Further, the validity of cosmic
censorship, even if found to be true, does not guarantee the validity of the
Weyl curvature hypothesis in the Weyl entropy-cosmic censorship scheme. Since
the diverging nature of the Weyl invariant is geometrically intrinsic to
singularities, the validity of the Weyl curvature hypothesis with respect to
cosmic censorship remains to be answered.

We can note that the validity of the Weyl curvature hypothesis has an
interesting place in cosmology. In terms of cosmology, it would be preferable
that the Weyl curvature hypothesis is true – however, certain cases point
towards the contrary, such as Gregoris et al[13]. In addition to this, the
vanishing Weyl tensor for the de Sitter spacetime forces us to question the
Weyl curvature hypothesis. The cosmological no hair conjecture states that
certain cosmologies are such that in future states the expansion rate $a(t)$
becomes asymptotically de Sitter, i.e.

$\lim_{t\to\infty}a(t)_{M}=\exp{Ht}$ (14)

These certain cosmologies include the Bianchi spacetimes (excluding the $IX$
case)[26], under the requirement of the satisfaction of the strong energy
condition

$\left(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}tr(T_{\mu\nu})\right)\geq 0$

The cosmological no hair conjecture can be stated as: ”for a cosmology $(M,g)$
with a positive cosmological constant $\Lambda$ and spatial curvature $k\leq
0$, the future state would be asymptotically de Sitter”. Due to this, the
future state of the cosmology would have a Weyl curvature that is
asymptotically zero, conflicting with the Weyl curvature hypothesis. Due to
this, a firm choice of the validity of the Weyl curvature hypothesis cannot be
provided – the possible relation to cosmic censorship described previously is
based on the assumption that the Weyl curvature hypothesis is true and that
along all outgoing geodesics the Weyl curvature is strictly increasing – if
the Weyl curvature hypothesis is proved to be true, it would mean that cosmic
censorship would hold via the previously discussed relation.

##  6 Remarks

The validity of the Weyl curvature hypothesis would have implications on the
cosmic censorship conjecture in its weak form. The diverging nature of the
Weyl invariant poses a problem with the naked singularity after the collapse –
this results in two possible propositions: (1) that either the Weyl invariant
cannot be a suitable estimator of gravitational entropy directly, or (2) the
Weyl entropy in itself is not suitable for being related to the singularity.
However, the Weyl invariant has been found to be conveniently straightforward
for determining gravitational entropy – for example, for black hole solutions
it results in the correct entropy measure, i.e. the Hawking-Bekenstein
entropy. However, this comes at the price of being fully diverging in the case
of singularities without a horizon for entropy measure, which we would
ordinarily expect for curvature singularities. The second possibility, that
the Weyl entropy is not suitable in itself has not been considered because the
natural choice of validity in view of cosmology is that the Weyl curvature
hypothesis is true, i.e. the Weyl entropy along outgoing geodesics from the
initial singularity monotonically increases. Therefore, if cosmic censorship
is found invalid, it might imply that the Weyl entropy vanishing along
outgoing geodesics with past endpoint at the singularity formed by
gravitational collapse is naturally allowed. On the other hand, it could be
possible that Weyl entropy is allowed to be reducing along outgoing geodesics
originating from local singularities, i.e. singularities from which only one
set of null outgoing geodesics escape. However, in many cases this is not so –
global singularities also occur in the spacetime, which complicates the
situation further. It must be remarked that while the discussion in this paper
does not take further technicalities of the weak cosmic censorship, it must be
noted that under certain conditions global naked singularities do not occur –
refer to Christodoulou’s work[22, 23, 24] for a view on the nature of matter
content and the ”generic” nature of naked singularities to take into account
of conditions on the physical nature of matter in the spacetime such as the
satisfaction of the dominant energy condition, etc. It must be mentioned that
self-similar spacetimes pose a problem with determining the gravitational
entropy along homotheticities the form $\mathcal{W}/\mathbf{Ric}$ as it was
shown by Pelavas and Lake[25]. However, for the consideration of gravitational
entropy in this article, we have solely looked at the involvement of
$\mathcal{K}$ and $\mathbf{Ric}$ without the involvement of $\mathcal{R}$ in
the gravitational entropy estimator. Further, whether there exists a possible
relation between the strong cosmic censorship and the Weyl curvature
hypothesis is yet to be studied.

## References

  * [1] Penrose, R. General Relativity, an Einstein Centenary Survey ed S W Hawking and W Israel Cambridge: Cambridge University Press, 1979
  * [2] Grøn, Ø.; Hervik, S. Gravitational entropy and quantum cosmology. Class. Quant. Grav. 2001, 18, 601–618
  * [3] Chakraborty, S.; Guha, S.; Goswami, R. An investigation on gravitational entropy of cosmological models. Int, Journal Mod. Phys. D 2021, 30, 7, 2150051 [ arXiv:1912.01414 [gr-qc]]
  * [4] Pizana, F.; Sussman, R.; Hidalgo, J. Gravitational Entropy in Szekeres Class I Models.  arXiv:2205.02985 [gr-qc]
  * [5] Lima, R.; Nogales, J.; Pereira, S. Gravitational entropy of wormholes with exotic matter and in galactic halos. Int Journal Mod. Phys. D 2020, 29, 2, 2050015 [ arXiv:2003.08344 [gr-qc]]
  * [6] Barve, S.; Singh, T. Naked singularities and the Weyl Curvature Hypothesis. [arXiv:gr-qc/9705060]
  * [7] Barve, S.; Singh, T. Are naked singularites forbidden by the second law of thermodynamics? Mod. Phys. Lett. A 12 1997, 2415 [arXiv:gr-qc/9705057]
  * [8] Bekenstein, J. Generalized Second Law of Thermodynamics in Black-Hole Physics. Phys. Rev. D 1974, 9, 3292
  * [9] Gibbons, G.; Hawking, S. Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D 1977, 15, 2738
  * [10] Rudjord, Ø.; Grøn, Ø.; Hervick, S. The Weyl curvature conjecture and black hole entropy. Phys. Scr. 2008, 77, 055901 [arXiv:gr-qc/0607064]
  * [11] Clifton, T.; Ellis, G.; Tavakol, R. A Gravitational Entropy Proposal. Class. Quant. Grav. 2013, 30, 125009 [arXiv:1303.5612 [gr-qc]]
  * [12] Gregoris, D.; Ong, Y. Understanding gravitational entropy of black holes: A new proposal via curvature invariants. Phys. Rev. D 2022, 105, 104017 [ arXiv:2109.11968 [gr-qc]]
  * [13] Gregoris, D; Ong, Y; Wang, B. Thermodynamics of Shearing Massless Scalar Field Spacetimes is Inconsistent With the Weyl Curvature Hypothesis. Phys. Rev. D 2020, 102, 023539 [ arXiv:2004.10222 [gr-qc]]
  * [14] P. C. Vaidya, Proc. Indian Acad. Sci. A33, 264 (1951); Reprinted, Gen. Rel. Grav. 31, 119 (1999)
  * [15] Joshi, P.; Dwivedi, I. Naked singularities in non-self-similar gravitational collapse of radiation shells. Phys. Rev. D 1992, 45, 2147
  * [16] Wagh, S.; Maharaj, S. Naked Singularity of the Vaidya-deSitter Spacetime and Cosmic Censorship Hypothesis. Gen. Rel. Grav. 1999, 31 , 975
  * [17] Lemos, J. Collapsing shells of radiation in anti–de Sitter spacetimes and the hoop and cosmic censorship conjectures. Phys. Rev. D 1999, 59, 044020
  * [18] Ghosh, S. Charged null fluid collapse in anti–de Sitter spacetimes and naked singularities. Phys. Rev. D 2000, 62, 127505 [ arXiv:gr-qc/0106060]
  * [19] Li, N.; Li, X.; Song, S. An exploration of the black hole entropy via the Weyl tensor. Eur. Phys. J. C 2016, 76, 111 [arXiv:1510.09027 [gr-qc]]
  * [20] Goncalves, S. Naked singularities in Tolman–Bondi–de Sitter collapse. Phys. Rev. D 2001, 63, 064017
  * [21] Wainwright, J.; Anderson, P.J. Isotropic singularities and isotropization in a clas of Bianchi type $VIh$ cosmologies. Gen. Rel. Grav. 1984, 16, 609
  * [22] Christodoulou, D. Examples of naked singularity formation in the gravitational collapse of a scalar field. Annals of Mathematics, 1994, 3, 140, 607
  * [23] Christodoulou, D. The instability of naked singularities in the gravitational collapse of a scalar field. Annals of Mathematics, 1999, 1, 149 183
  * [24] Christodoulou, D. On the global initial value problem and the issue of singularities. Classical and Quantum Gravity, 1999, 12, 16, 12, 23
  * [25] Pelavas, N.; Lake, K. Measures of gravitational entropy: Self-similar spacetimes. Phys. Rev. D 2000, 62, 044009 [ arXiv:gr-qc/9811085]
  * [26] Wald, R. Asymptotic behavior of homogeneous cosmological models in the presence of a positive cosmological constant. Phys. Rev. D 1982, 28, 2118