# Arithmetical rank and cohomological dimension of generalized binomial edge
ideals

Anargyros Katsabekis  Department of Mathematics, Bilkent University, 06800
Ankara, Turkey<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a connected and simple graph on the vertex set $[n]$. To the graph
$G$ one can associate the generalized binomial edge ideal $J_{m}(G)$ in the
polynomial ring $R=K[x_{ij}:i\in[m],j\in[n]]$. We provide a lower bound for
the cohomological dimension of $J_{m}(G)$. We also study when $J_{m}(G)$ is a
cohomologically complete intersection. Finally, we show that the arithmetical
rank of $J_{2}(G)$ equals the projective dimension of $R/J_{2}(G)$ in several
cases.

###### Key words and phrases:

Arithmetical rank; Cohomological dimension; Generalized binomial edge ideals

###### 1991 Mathematics Subject Classification:

13F20, 13D45, 14M12, 05C25

##  1\. Introduction

Let $I$ be an ideal in a polynomial ring $R$. A basic problem in Commutative
Algebra asks to compute the smallest integer $s$ for which there exist
polynomials $f_{1},\ldots,f_{s}$ in $I$ such that ${\rm rad}(I)={\rm
rad}(f_{1},\ldots,f_{s})$. This integer is called the arithmetical rank of $I$
and will be denoted by ${\rm ara}(I)$. A lower bound for ${\rm ara}(I)$ is
provided by the so-called cohomological dimension of $I$, which is defined as
follows: ${\rm cd}(I)={\rm max}\\{i\in\mathbb{Z}:H_{I}^{i}(R)\neq 0\\}$, where
$H_{I}^{i}(R)$ denotes the $i$-th local cohomology module of $R$ with support
in $I$. In this paper we study the above problem for a special class of
polynomial ideals, namely the generalized binomial edge ideals of graphs.

Let $m\geq 2$, $n\geq 2$ be integers and $R=K[x_{ij}:i\in[m],j\in[n]]$ be the
polynomial ring over a field $K$ in $mn$ variables $x_{ij}$, where
$i=1,\ldots,m$ and $j=1,\ldots,n$. Let $G$ be a connected and simple graph on
the vertex set $[n]$. The generalized binomial edge ideal $J_{m}(G)$ of $G$ is
generated by all the $2$-minors of the $m\times n$ matrix $X=(x_{ij})$ of the
form $[k,l|i,j]=x_{ki}x_{lj}-x_{kj}x_{li}$, where $1\leq k<l\leq m$ and
$\\{i,j\\}$ is an edge of $G$ with $i<j$. When $m=2$, $J_{m}(G)$ coincides
with the classical binomial edge ideal introduced independently in [9] and
[12].

By [13, Theorem 2], $J_{m}(G)$ has a squarefree initial ideal with respect to
a term order $<$. Thus $J_{m}(G)$ is a radical ideal. By [6, Corollary 2.7],
${\rm depth}(R/J_{m}(G))={\rm depth}(R/{\rm in}_{<}(J_{m}(G)))$, and therefore
${\rm pd}(R/J_{m}(G))={\rm pd}(R/{\rm in}_{<}(J_{m}(G)))$. Since ${\rm
in}_{<}(J_{m}(G))$ is generated by squarefree monomials, its cohomological
dimension equals the projective dimension (see [11]). So ${\rm
pd}(R/J_{m}(G))={\rm cd}({\rm in}_{<}(J_{m}(G)))$. By [6, Proposition 3.6],
the inequality ${\rm pd}(R/J_{m}(G))\leq{\rm cd}(J_{m}(G))$ holds.

For a generalized binomial edge ideal $J_{m}(G)$ the following inequalities
hold:

${\rm ht}(J_{m}(G))\leq{\rm pd}(R/J_{m}(G))\leq{\rm cd}(J_{m}(G))\leq{\rm
ara}(J_{m}(G))\leq\mu(J_{m}(G)).$

Here ${\rm ht}(J_{m}(G))$ denotes the height and $\mu(J_{m}(G))$ denotes the
minimal number of generators of $J_{m}(G)$. The ideal $J_{m}(G)$ is a set-
theoretic complete intersection if ${\rm ara}(J_{m}(G))={\rm ht}(J_{m}(G))$,
while $J_{m}(G)$ is a cohomologically complete intersection if ${\rm
cd}(J_{m}(G))={\rm ht}(J_{m}(G))$. Clearly if $J_{m}(G)$ is a set-theoretic
complete intersection, then it is also a cohomologically complete
intersection. It is worth mentioning that ${\rm ara}(J_{m}(G))=mn-3$ when $G$
is the complete graph on $[n]$, see [4, Theorem 2].

In Section 2 we determine all graphs $G$ for which either $\mu(J_{m}(G))={\rm
ht}(J_{m}(G))$ or $\mu(J_{m}(G))={\rm ht}(J_{m}(G))+1$, see Theorems 2.3, 2.4.
Additionally we provide (see Theorem 2.7) a lower bound for the cohomological
dimension of the ideal $P_{\emptyset}(G)\cap P_{T}(G)$, where
$P_{\emptyset}(G)$ and $P_{T}(G)$ are distinct minimal primes of $J_{m}(G)$.
We also show that, for a non-complete graph $G$ with vertex connectivity
$k(G)$, ${\rm cd}(J_{m}(G))\geq mn-m-n+k(G)$.

In Section 3 we prove that $J_{m}(G)$ is a cohomologically complete
intersection for $m\geq 3$ if and only if $G$ is the complete graph on $[n]$
and $K$ has positive characteristic. As a consequence, we derive that
$J_{m}(G)$ is not a set-theoretic complete intersection for every $m\geq 3$.
In fact there can be a large deviation between ${\rm ara}(J_{m}(G))$ and ${\rm
ht}(J_{m}(G))$, see Proposition 3.6. We also study the case $m=2$, namely
$J_{2}(G)$ is a binomial edge ideal. When $\mu(J_{2}(G))={\rm ht}(J_{2}(G))+1$
and $G$ is not a triangle, we show that $J_{2}(G)$ is a cohomologically
complete intersection if and only if $J_{2}(G)$ is Cohen-Macaulay. Finally we
provide classes of graphs such that ${\rm ara}(J_{2}(G))={\rm
pd}(R/J_{2}(G))$, see Theorems 3.7, 3.8, Corollary 3.9, and Theorems 3.15,
3.17.

The computations of this paper are performed by using CoCoA [5].

##  2\. Cohomological dimension

Let $G$ be a connected graph on the vertex set $V(G)=[n]$ and let
$T\subset[n]$. By $G\setminus T$ we denote the graph that results from
deleting all vertices in $T$ and their incident edges from $G$. Let $c(T)$ be
the number of connected components of $G\setminus T$. We say that $T$ is a cut
set of $G$ if either $T=\emptyset$ or $c(T\setminus\\{i\\})<c(T)$ for every
$i\in T$. We denote by $\mathcal{C}(G)$ the collection of cut sets of $G$.

Let $G_{1},\ldots,G_{c(T)}$ denote the connected components of $G\setminus T$
and also let $\tilde{G}_{i}$ denote the complete graph on the vertices of
$G_{i}$. Consider the ideal

$P_{T}(G)=\left(\bigcup_{i\in
T}\\{x_{1i},\ldots,x_{mi}\\}\right)+J_{m}(\tilde{G}_{1})+\cdots+J_{m}(\tilde{G}_{c(T)})$

of the polynomial ring $R$. Then $P_{T}(G)$ is a prime ideal for every
$T\subset[n]$. Furthermore $P_{T}(G)$ is a minimal prime of $J_{m}(G)$ if and
only if $T\in\mathcal{C}(G)$. By [13, Theorem 7],
$J_{m}(G)=\cap_{T\in\mathcal{C}(G)}P_{T}(G)$. Since ${\rm
ht}(P_{\emptyset}(G))=(m-1)(n-1)$, it holds that ${\rm dim}(R/J_{m}(G))\geq
m+n-1$.

######  Remark 2.1.

(1) If $J_{m}(G)$ is Cohen-Macaulay, then ${\rm dim}(R/J_{m}(G))=m+n-1$.
(2) If for a set $T\subsetneqq[n]$ we have that $c(T)=1$, then
$P_{\emptyset}(G)\subset P_{T}(G)$ and therefore $P_{T}(G)$ is not a minimal
prime of $J_{m}(G)$. So, if $P_{T}(G)$ is a minimal prime of $J_{m}(G)$, then
either $T=\emptyset$ or $c(T)\geq 2$.

######  Proposition 2.2.

Let $G$ be a connected graph on the vertex set $[n]$. Then

${\rm ht}(J_{m}(G))={\rm min}\\{(m-1)(n-c(T))+|T|:T\in\mathcal{C}_{G}\\}.$

Proof. Let $T\in\mathcal{C}_{G}$ and $|V(\tilde{G}_{i})|=n_{i}$ for $1\leq
i\leq c(T)$. We have that

${\rm ht}(P_{T}(G))={\rm ht}\left(\bigcup_{i\in
T}\\{x_{1i},\ldots,x_{mi}\\}\right)+\sum_{i=1}^{c(T)}{\rm
ht}(J_{m}(\tilde{G}_{i}))=$
$m|T|+\sum_{i=1}^{c(T)}(m-1)(n_{i}-1)=m|T|+(m-1)\sum_{i=1}^{c(T)}(n_{i}-1).$

It holds that

$\sum_{i=1}^{c(T)}(n_{i}-1)=\sum_{i=1}^{c(T)}n_{i}-\sum_{i=1}^{c(T)}1=n-|T|-c(T),$

so ${\rm ht}(P_{T}(G))=(m-1)(n-c(T))+|T|$ and therefore

${\rm ht}(J_{m}(G))={\rm min}\\{(m-1)(n-c(T))+|T|:T\in\mathcal{C}_{G}\\}.\ \
\square$

We say that $J_{m}(G)$ is a complete intersection if $\mu(J_{m}(G))={\rm
ht}(J_{m}(G))$, while $J_{m}(G)$ is an almost complete intersection if
$\mu(J_{m}(G))={\rm ht}(J_{m}(G))+1$.

######  Theorem 2.3.

Let $G$ be a connected graph on the vertex set $[n]$. Then $J_{m}(G)$ is a
complete intersection if and only if $m=2$ and $G$ is a path graph.

Proof. Since $G$ is a connected graph, it has at least $n-1$ edges and
therefore $\mu(J_{m}(G))\geq\frac{m(m-1)}{2}(n-1)$. If $J_{m}(G)$ is a
complete intersection, then it is unmixed. Since ${\rm
ht}(P_{\emptyset}(G))=(m-1)(n-1)$, we get that ${\rm
ht}(J_{m}(G))=(m-1)(n-1)$. Thus $\mu(J_{m}(G))=(m-1)(n-1)$, so
$(m-1)\geq\frac{m(m-1)}{2}$ and therefore $m\leq 2$. Consequently $m=2$. By
[14, Theorem 1], $J_{m}(G)$ is a complete intersection if and only if $G$ is a
path graph. $\square$

######  Theorem 2.4.

Let $G$ be a connected graph on the vertex set $[n]$, where $n\geq 3$. Then
$J_{m}(G)$ is an almost complete intersection if and only if $m=2$ and either

  1. (1)

$G$ is a tree which is not a path obtained by adding an edge between two
vertices of two paths or

  2. (2)

$G$ is not a tree obtained by adding an edge between two vertices of a path or
by attaching a path to each vertex of a triangle.

Proof. The graph $G$ is connected, so it has at least $n-1$ edges, thus
$\mu(J_{m}(G))\geq\frac{m(m-1)}{2}(n-1)$. If $J_{m}(G)$ is an almost complete
intersection, then $\mu(J_{m}(G))={\rm ht}(J_{m}(G))+1$ and therefore ${\rm
ht}(J_{m}(G))\geq\frac{m(m-1)}{2}(n-1)-1$. But ${\rm
ht}(J_{m}(G))\leq(m-1)(n-1)$, so $\frac{m(m-1)}{2}(n-1)-1\leq(m-1)(n-1)$. Thus
$(m-1)(n-1)(m-2)\leq 2$ from which we deduce that $m\leq 2$, and therefore
$m=2$. By [10, Theorems 4.3 and 4.4], $J_{m}(G)$ is an almost complete
intersection if and only if $m=2$ and either

  1. (1)

$G$ is a tree which is not a path obtained by adding an edge between two
vertices of two paths or

  2. (2)

$G$ is not a tree obtained by adding an edge between two vertices of a path or
by attaching a path to each vertex of a triangle. $\square$

######  Remark 2.5.

Let $G$ be a connected graph on the vertex set $[2]$, namely $G$ consists of
the edge $\\{1,2\\}$. We have that $\mu(J_{m}(G))=\frac{m(m-1)}{2}$ and also
${\rm ht}(J_{m}(G))=m-1$. For $m=3$ the ideal $J_{m}(G)$ is an almost complete
intersection, while $\mu(J_{m}(G))>{\rm ht}(J_{m}(G))+1$ for every integer
$m>3$.

Given a connected graph $G$ on the vertex set $[n]$, we say that $G$ is
$k$-vertex-connected if $k<n$ and $G\setminus T$ is connected for every subset
$T$ of $[n]$ with $|T|<k$. The vertex connectivity of $G$ is the maximum
integer $k$ such that $G$ is $k$-vertex-connected. We shall denote by $k(G)$
the vertex connectivity of the graph $G$.

######  Proposition 2.6.

Let $G$ be a connected graph on the vertex set $[n]$ and $P_{T}(G)$ be a
minimal prime of $J_{m}(G)$, where $T\neq\emptyset$. Then ${\rm
ht}(P_{\emptyset}(G)+P_{T}(G))\geq(m-1)(n-1)+k(G)$.

Proof. The sum $P_{\emptyset}(G)+P_{T}(G)$ is generated by
$\\{x_{ki}x_{lj}-x_{kj}x_{li}|1\leq k<l\leq m\ \textrm{and}\
i,j\in[n]\setminus T\\}\cup\\{x_{1i},\ldots,x_{mi}|i\in T\\}$, so

${\rm ht}(P_{\emptyset}(G)+P_{T}(G))=(m-1)(n-|T|-1)+m|T|=(m-1)(n-1)+|T|.$

It remains to show that $|T|\geq k(G)$. Note that $|T|\geq 1$. So if $k(G)=1$,
then clearly $|T|\geq k(G)$. Suppose that $k(G)\geq 2$. By Remark 2.1,
$c(T)\geq 2$ and therefore the graph $G\setminus T$ is disconnected. If
$|T|<k(G)$, then the graph $G\setminus T$ is connected, a contradiction. Thus
$|T|\geq k(G)$. $\square$

######  Theorem 2.7.

Let $K$ be a field of any characteristic and $P_{T}(G)$ be a minimal prime of
$J_{m}(G)$, where $T\neq\emptyset$. Then

${\rm cd}(P_{\emptyset}(G)\cap P_{T}(G))\geq mn-m-n+|T|.$

Proof. Recall that ${\rm ht}(P_{\emptyset}(G)+P_{T}(G))=(m-1)(n-1)+|T|$. Since
$|T|\geq 1$, we get ${\rm ht}(P_{\emptyset}(G))<{\rm
ht}(P_{\emptyset}(G)+P_{T}(G))$. Also ${\rm ht}(P_{T}(G))=(m-1)(n-c(T))+|T|$
and $c(T)\geq 2$, so ${\rm ht}(P_{T}(G))<{\rm ht}(P_{\emptyset}(G)+P_{T}(G))$.
By [16, Proposition 1.1.2], ${\rm cd}(P_{\emptyset}(G)\cap P_{T}(G))\geq{\rm
ht}(P_{\emptyset}(G)+P_{T}(G))-1$ and therefore ${\rm cd}(P_{\emptyset}(G)\cap
P_{T}(G))\geq(m-1)(n-1)+|T|-1$. Thus ${\rm cd}(P_{\emptyset}(G)\cap
P_{T}(G))\geq mn-m-n+|T|$. $\square$

######  Example 2.8.

Let $m=2$ and $G$ be a cycle with the edge set

$E(G)=\\{\\{1,2\\},\\{2,3\\},\\{3,4\\},\\{1,4\\}\\}.$

Let $T=\\{2,4\\}$, then the ideal $P_{T}(G)=(x_{12},x_{22},x_{14},x_{24})$ is
a minimal prime of $J_{2}(G)$. Also $P_{\emptyset}(G)\cap
P_{T}(G)=J_{2}(G)+(x_{12}x_{24}-x_{14}x_{22})$. By Theorem 2.7, ${\rm
cd}(P_{\emptyset}(G)\cap P_{T}(G))\geq 4$. Let

$I=(x_{11}x_{22}-x_{12}x_{21}+x_{13}x_{24}-x_{14}x_{23},x_{12}x_{23}-x_{13}x_{22},x_{11}x_{24}-x_{14}x_{21},x_{12}x_{24}-x_{14}x_{22}).$

Then $P_{\emptyset}(G)\cap P_{T}(G)={\rm rad}(I)$ since
$(x_{11}x_{22}-x_{12}x_{21})^{2}\in I$ and $(x_{13}x_{24}-x_{14}x_{23})^{2}\in
I$. So ${\rm ara}(P_{\emptyset}(G)\cap P_{T}(G))\leq 4$ and therefore ${\rm
cd}(P_{\emptyset}(G)\cap P_{T}(G))\leq 4$. Thus ${\rm cd}(P_{\emptyset}(G)\cap
P_{T}(G))=4$ and also ${\rm ara}(P_{\emptyset}(G)\cap P_{T}(G))=4$.

Let $\mathcal{K}_{n}$ denotes the complete graph on the vertex set $[n]$.

######  Remark 2.9.

It holds that

${\rm cd}(J_{m}(\mathcal{K}_{n}))=\begin{cases}(m-1)(n-1),&{\rm if}\ \ {\rm
char}(K)>0\\\ mn-3,&{\rm if}\ \ {\rm char}(K)=0.\end{cases}$

######  Proposition 2.10.

Let $G$ be a path graph on the vertex set $[n]$, where $n\geq 3$, with the
edge set $E(G)=\\{\\{1,2\\},\\{2,3\\},\ldots,\\{n-1,n\\}\\}$ and
$T=\\{2,3,\ldots,n-1\\}$.

  1. (1)

If $K$ is a field of positive characteristic, then ${\rm
cd}(P_{\emptyset}(G)\cap P_{T}(G))=mn-m-2$.

  2. (2)

If $K$ is a field of characteristic $0$, then $mn-m-2\leq{\rm
cd}(P_{\emptyset}(G)\cap P_{T}(G))\leq mn-3$.

Proof. Let $I=P_{\emptyset}(G)\cap P_{T}(G)$. Since $|T|=n-2$, we get ${\rm
cd}(I)\geq m(n-1)-2$. Notice that
$P_{T}(G)=\left(\cup_{i=2}^{n-1}\\{x_{1i},\ldots,x_{mi}\\}\right)$. Let
$M=P_{\emptyset}(G)+P_{T}(G)$, then $M=P_{T}(G)+J_{m}(\mathcal{K}_{2})$, where
$\mathcal{K}_{2}$ is the complete graph on the vertex set $\\{1,n\\}$. By [3,
Lemma 2.4], ${\rm cd}(M)=m(n-2)+{\rm cd}(J_{m}(\mathcal{K}_{2}))$. We
distinguish the following cases: (1) ${\rm char}(K)>0$. Since ${\rm
cd}(P_{\emptyset}(G))=(m-1)(n-1)$, ${\rm cd}(M)=m(n-1)-1$ and ${\rm
cd}(P_{T}(G))=m(n-2)$, we deduce from the Mayer-Vietoris sequence

$\cdots\rightarrow H_{M}^{i}(R)\rightarrow H_{P_{\emptyset}(G)}^{i}(R)\oplus
H_{P_{T}(G)}^{i}(R)\rightarrow H_{I}^{i}(R)\rightarrow
H_{M}^{i+1}(R)\rightarrow\cdots$

that ${\rm cd}(I)\leq m(n-1)-2$ and therefore ${\rm cd}(I)=m(n-1)-2$.
(2) ${\rm char}(K)=0$. Since ${\rm cd}(P_{\emptyset}(G))=mn-3$, ${\rm
cd}(M)=mn-3$ and ${\rm cd}(P_{T}(G))=m(n-2)$, we deduce from the Mayer-
Vietoris sequence

$\cdots\rightarrow H_{M}^{i}(R)\rightarrow H_{P_{\emptyset}(G)}^{i}(R)\oplus
H_{P_{T}(G)}^{i}(R)\rightarrow H_{I}^{i}(R)\rightarrow
H_{M}^{i+1}(R)\rightarrow\cdots$

that ${\rm cd}(I)\leq mn-3$. $\square$

######  Theorem 2.11.

Let $G$ be a connected graph on the vertex set $[n]$ and $K$ be a field of any
characteristic. If $G$ is not the complete graph, then

${\rm cd}(J_{m}(G))\geq mn-m-n+k(G).$

Proof. Let $\mathcal{C}_{G}=\emptyset\cup\\{T_{1},\ldots,T_{s}\\}$ be the cut
sets of $G$, where $T_{i}\neq\emptyset$ for every $1\leq i\leq s$. Since $G$
is not the complete graph, the ideal $J_{m}(G)$ is not prime. We can write
$J_{m}(G)=P_{\emptyset}(G)\cap M$, where $M=\cap_{i=1}^{s}P_{T_{i}}(G)$ . It
holds that ${\rm ht}(M)\leq{\rm ht}(P_{T_{i}}(G))$ and also ${\rm
ht}(P_{T_{i}}(G))<{\rm ht}(P_{\emptyset}(G)+P_{T_{i}}(G))$, for every $1\leq
i\leq s$. There is $j\in\\{1,\ldots,s\\}$ such that ${\rm
ht}(P_{\emptyset}(G)+P_{T_{j}}(G))\leq{\rm ht}(P_{\emptyset}(G)+M)$.
Consequently ${\rm ht}(M)<{\rm ht}(P_{\emptyset}(G)+M)$. Also ${\rm
ht}(P_{\emptyset}(G)+P_{T_{j}}(G))>{\rm ht}(P_{\emptyset}(G)$, so ${\rm
ht}(P_{\emptyset}(G)+M)>{\rm ht}(P_{\emptyset}(G)$. By [16, Proposition
1.1.2], ${\rm cd}(J_{m}(G))\geq{\rm ht}(P_{\emptyset}(G)+M)-1$. From
Proposition 2.6 we have that ${\rm
ht}(P_{\emptyset}(G)+P_{T_{j}}(G))\geq(m-1)(n-1)+k(G)$, so ${\rm
cd}(J_{m}(G))\geq mn-m-n+k(G)$. $\square$

A complete bipartite graph of the form $\mathcal{K}_{1,n-1}$ is a star graph
with $n$ vertices.

######  Proposition 2.12.

Let $G$ be a star graph on the vertex set $[n]$, where $n\geq 3$.

  1. (1)

If $K$ is a field of positive characteristic, then ${\rm
cd}(J_{m}(G))=(m-1)(n-1)$.

  2. (2)

If $K$ is a field of characteristic $0$, then

$(m-1)(n-1)\leq{\rm cd}(J_{m}(G))\leq mn-3.$

Proof. The vertex connectivity of $G$ is $k(G)=1$, so ${\rm
cd}(J_{m}(G))\geq(m-1)(n-1)$. Let $i$ be the vertex of $G$ with degree $n-1$
and $T=\\{i\\}$. Then $J_{m}(G)=P_{\emptyset}(G)\cap P_{T}(G)$, where
$P_{T}(G)=(x_{1i},x_{2i},\ldots,x_{mi})$. Let $M=P_{\emptyset}(G)+P_{T}(G)$,
then $M=(x_{1i},x_{2i},\ldots,x_{mi})+J_{m}(\mathcal{K}_{n-1})$, where
$\mathcal{K}_{n-1}$ is the complete graph on the vertex set
$[n]\setminus\\{i\\}$. By [3, Lemma 2.4], ${\rm cd}(M)=m+{\rm
cd}(J_{m}(\mathcal{K}_{n-1}))$. So

${\rm cd}(M)=\begin{cases}(m-1)(n-1)+1,&{\rm if}\ \ {\rm char}(K)>0\\\
mn-3,&{\rm if}\ \ {\rm char}(K)=0.\end{cases}$

We distinguish the following cases: (1) ${\rm char}(K)>0$. Since ${\rm
cd}(P_{\emptyset}(G))=(m-1)(n-1)$, ${\rm cd}(M)=(m-1)(n-1)+1$ and ${\rm
cd}(P_{T}(G))=m$, we deduce from the Mayer-Vietoris sequence

$\cdots\rightarrow H_{M}^{i}(R)\rightarrow H_{P_{\emptyset}(G)}^{i}(R)\oplus
H_{P_{T}(G)}^{i}(R)\rightarrow H_{J_{m}(G)}^{i}(R)\rightarrow
H_{M}^{i+1}(R)\rightarrow\cdots$

that ${\rm cd}(J_{m}(G))\leq(m-1)(n-1)$, and therefore ${\rm
cd}(J_{m}(G))=(m-1)(n-1)$.
(2) ${\rm char}(K)=0$. Since ${\rm cd}(P_{\emptyset}(G))=mn-3$, ${\rm
cd}(M)=mn-3$ and ${\rm cd}(P_{T}(G))=m$, we deduce from the Mayer-Vietoris
sequence

$\cdots\rightarrow H_{M}^{i}(R)\rightarrow H_{P_{\emptyset}(G)}^{i}(R)\oplus
H_{P_{T}(G)}^{i}(R)\rightarrow H_{J_{m}(G)}^{i}(R)\rightarrow
H_{M}^{i+1}(R)\rightarrow\cdots$

that ${\rm cd}(J_{m}(G))\leq mn-3$. $\square$

##  3\. Arithmetical rank

In this section first we determine all graphs $G$ for which $J_{m}(G)$ is a
cohomologically complete intersection for $m\geq 3$.

######  Proposition 3.1.

Let $G$ be a connected graph on the vertex set $[n]$. If $J_{m}(G)$ is a
cohomologically complete intersection, then $J_{m}(G)$ is Cohen-Macaulay.

Proof. Suppose that ${\rm cd}(J_{m}(G))={\rm ht}(J_{m}(G))$, so ${\rm
pd}(R/J_{m}(G))={\rm ht}(J_{m}(G))$. Using the Auslander-Buchsbaum formula we
get ${\rm depth}(R/J_{m}(G))={\rm dim}(R/J_{m}(G))$, thus $J_{m}(G)$ is Cohen-
Macaulay. $\square$

######  Remark 3.2.

If $J_{m}(G)$ is a set-theoretic complete intersection, then $J_{m}(G)$ is
also Cohen-Macaulay.

######  Theorem 3.3.

Let $G$ be a connected graph on the vertex set $[n]$ and $m\geq 3$ be an
integer. Then $J_{m}(G)$ is a cohomologically complete intersection if and
only if $G$ is the complete graph and ${\rm char}(K)>0$.

Proof. If $G$ is the complete graph on the vertex set $[n]$ and ${\rm
char}(K)>0$, then ${\rm cd}(J_{m}(G))={\rm ht}(J_{m}(G))=(m-1)(n-1)$.
Conversely, if $J_{m}(G)$ is a cohomologically complete intersection, then
$J_{m}(G)$ is Cohen-Macaulay from Proposition 3.1. By [1, Corollary 4.3], $G$
is the complete graph on $[n]$. $\square$

######  Theorem 3.4.

Let $G$ be a connected graph on the vertex set $[n]$ and $m\geq 3$ be an
integer. Then the ideal $J_{m}(G)$ is not a set-theoretic complete
intersection.

Proof. Suppose that $J_{m}(G)$ is a set-theoretic complete intersection. By
Theorem 3.3, $G$ is the complete graph on the vertex set $[n]$ and ${\rm
char}(K)>0$. Notice that ${\rm ht}(J_{m}(G))=(m-1)(n-1)$. Thus ${\rm
ara}(J_{m}(G))=mn-3>{\rm ht}(J_{m}(G))$ a contradiction to the fact that
$J_{m}(G)$ is a set-theoretic complete intersection. $\square$

A maximum clique of a graph $G$ is a complete subgraph of $G$ that includes
the largest possible number of vertices.

######  Proposition 3.5.

Let $G$ be a connected graph on the vertex set $[n]$ with $q$ edges and let
$\mathcal{K}_{r}$ be a maximum clique of $G$, where $r\geq 2$. Then

${\rm ara}(J_{m}(G))\leq\frac{m(m-1)}{2}\left(q-\frac{r(r-1)}{2}\right)+rm-3.$

Proof. We have that $J_{m}(G)=J_{m}(\mathcal{K}_{r})+J_{m}(G^{\prime})$, where
$G^{\prime}$ is a graph with $q-\frac{r(r-1)}{2}$ edges. Since ${\rm
ara}(J_{m}(\mathcal{K}_{r}))=rm-3$, we have that

${\rm ara}(J_{m}(G))\leq\frac{m(m-1)}{2}\left(q-\frac{r(r-1)}{2}\right)+rm-3.\
\ \square$

######  Proposition 3.6.

Let $G$ be a star graph on the vertex set $[3]$ and $r\geq 1$ be an integer.
Then for every integer $m\geq r+3$ it holds that

${\rm ara}(J_{m}(G))>{\rm ht}(J_{m}(G))+r.$

Proof. By Proposition 2.12, ${\rm cd}(J_{m}(G))\geq 2m-2$ and therefore ${\rm
ara}(J_{m}(G))\geq 2m-2$. Also ${\rm ht}(J_{m}(G))=m$. For every $m\geq r+3$
we have that $2m-2>m+r$, so ${\rm ara}(J_{m}(G))>{\rm ht}(J_{m}(G))+r$.
$\square$

Next we prove that in various cases ${\rm ara}(J_{2}(G))={\rm
pd}(R/J_{2}(G))$.
Let $G_{1}=(V(G_{1}),E(G_{1}))$, $G_{2}=(V(G_{2}),E(G_{2}))$ be graphs such
that $G_{1}\cap G_{2}$ is a complete graph. The new graph $G=G_{1}\cup G_{2}$
with the vertex set $V(G)=V(G_{1})\cup V(G_{2})$ and edge set
$E(G)=E(G_{1})\cup E(G_{2})$ is called the clique sum of $G_{1}$ and $G_{2}$
in $G_{1}\cap G_{2}$. We write $G=G_{1}\bigoplus_{\hat{v}}G_{2}$ to indicate
that $G$ is the clique sum of $G_{1}$ and $G_{2}$ and that $V(G_{1})\cap
V(G_{2})=\hat{v}$.

The diamond graph is the graph obtained from $\mathcal{K}_{4}$ by deleting one
edge. Let

$f_{i,j}:=x_{1i}x_{2j}-x_{1j}x_{2i}.$

######  Theorem 3.7.

Let $G$ be a connected graph on the vertex set $[n]$ with $n\geq 6$. Suppose
that $G$ has a diamond subgraph $D$ and also

$G=D\bigoplus_{v_{1}}T_{1}\bigoplus_{v_{2}}\cdots\bigoplus_{v_{s}}T_{s},$

where $\\{v_{1},\ldots,v_{s}\\}\subset V(D)$, $s\geq 2$, $v_{h}$ are pairwise
distinct and $T_{h}$ are trees. Then ${\rm ara}(J_{2}(G))={\rm
pd}(R/J_{2}(G))=n-1$.

Proof. By [2, Theorems 3.19 and 3.20], ${\rm pd}(R/J_{2}(G))\geq n-1$ and
therefore ${\rm ara}(J_{2}(G))\geq n-1$. Let $r_{h}$ be the number of edges of
the tree $T_{h}$ for $1\leq h\leq s$. The graph $G$ has
$n=4+\sum_{h=1}^{s}r_{i}$ vertices. Let $\\{v_{1},v_{5}\\}$ be an edge of
$T_{1}$ and $\\{v_{2},v_{6}\\}$ be an edge of $T_{2}$. Consider the subgraph
$G_{1}=D\cup\\{v_{1},v_{5}\\}\cup\\{v_{2},v_{6}\\}$ of $G$. We will show that
${\rm ara}(J_{2}(G_{1}))\leq 5$. We distinguish the following cases:
(1)
$E(D)=\\{\\{v_{1},v_{2}\\},\\{v_{2},v_{3}\\},\\{v_{3},v_{4}\\},\\{v_{1},v_{4}\\},\\{v_{1},v_{3}\\}\\}$.
Let

$I=(f_{v_{1},v_{3}}+f_{v_{2},v_{6}},f_{v_{1},v_{5}}+f_{v_{3},v_{4}},f_{v_{1},v_{2}},f_{v_{1},v_{4}},f_{v_{2},v_{3}}).$

Then $J_{2}(G_{1})={\rm rad}(I)$, since $f_{v_{1},v_{3}}^{2}\in I$,
$f_{v_{2},v_{6}}^{2}\in I$, $f_{v_{1},v_{5}}^{4}\in I$ and
$f_{v_{3},v_{4}}^{4}\in I$.
(2)
$E(D)=\\{\\{v_{1},v_{2}\\},\\{v_{2},v_{3}\\},\\{v_{3},v_{4}\\},\\{v_{1},v_{4}\\},\\{v_{2},v_{4}\\}\\}$.
Let

$I=(f_{v_{1},v_{5}}+f_{v_{2},v_{4}},f_{v_{2},v_{6}}+f_{v_{3},v_{4}},f_{v_{1},v_{2}},f_{v_{1},v_{4}},f_{v_{2},v_{3}}).$

Then $J_{2}(G_{1})={\rm rad}(I)$, since $f_{v_{1},v_{5}}^{2}\in I$,
$f_{v_{2},v_{4}}^{2}\in I$, $f_{v_{2},v_{6}}^{4}\in I$ and
$f_{v_{3},v_{4}}^{4}\in I$.
Let $G_{2}=T_{1}-\\{v_{1},v_{5}\\}$ and $G_{3}=T_{2}-\\{v_{2},v_{6}\\}$ be
subgraphs of $G$. We have that ${\rm ara}(J_{2}(G_{2}))\leq r_{1}-1$ and ${\rm
ara}(J_{2}(G_{3}))\leq r_{2}-1$. Thus ${\rm ara}(J_{2}(G))\leq
5+\sum_{h=1}^{s}r_{i}-2$ and therefore ${\rm
ara}(J_{2}(G))=3+\sum_{h=1}^{s}r_{i}$. $\square$

######  Theorem 3.8.

Let $G$ be a connected graph on the vertex set $[n]$ with $n\geq 6$. Suppose
that $G$ has a complete subgraph $\mathcal{K}_{4}$ and also

$G=\mathcal{K}_{4}\bigoplus_{v_{1}}T_{1}\bigoplus_{v_{2}}\cdots\bigoplus_{v_{s}}T_{s},$

where $\\{v_{1},\ldots,v_{s}\\}\subset V(\mathcal{K}_{4})$, $s\geq 2$, $v_{h}$
are pairwise distinct and $T_{h}$ are trees. Then ${\rm ara}(J_{2}(G))={\rm
pd}(R/J_{2}(G))=n-1$.

Proof. By [8, Theorem 1.1], ${\rm depth}(R/J_{2}(G))=n+1$ and therefore ${\rm
pd}(R/J_{2}(G))=n-1$. Thus ${\rm ara}(J_{2}(G))\geq n-1$. Let $r_{h}$ be the
number of edges of the tree $T_{h}$ for $1\leq h\leq s$. The graph $G$ has
$n=4+\sum_{h=1}^{s}r_{i}$ vertices. Let $\\{v_{1},\ldots,v_{4}\\}$ be the
vertex set of $\mathcal{K}_{4}$ and let
$G_{1}=\mathcal{K}_{4}\cup\\{v_{1},v_{5}\\}\cup\\{v_{2},v_{6}\\}$,
$G_{2}=T_{1}-\\{v_{1},v_{5}\\}$ and $G_{3}=T_{2}-\\{v_{2},v_{6}\\}$ be
subgraphs of $G$. Consider the ideal

$I=(f_{v_{1},v_{3}}+f_{v_{2},v_{6}},f_{v_{1},v_{5}}+f_{v_{3},v_{4}},f_{v_{1},v_{4}}+f_{v_{2},v_{3}},f_{v_{1},v_{2}},f_{v_{2},v_{4}}).$

We have that $J_{2}(G_{1})={\rm rad}(I)$, since $f_{v_{1},v_{3}}^{3}\in I$,
$f_{v_{2},v_{6}}^{3}\in I$, $f_{v_{1},v_{5}}^{5}\in I$,
$f_{v_{3},v_{4}}^{5}\in I$, $f_{v_{1},v_{4}}^{2}\in I$ and
$f_{v_{2},v_{3}}^{2}\in I$. Consequently ${\rm ara}(J_{2}(G_{1}))\leq 5$. Also
${\rm ara}(J_{2}(G_{2}))\leq r_{1}-1$ and ${\rm ara}(J_{2}(G_{3}))\leq
r_{2}-1$. Thus ${\rm ara}(J_{2}(G))\leq 5+\sum_{h=1}^{s}r_{i}-2$ and therefore
${\rm ara}(J_{2}(G))=3+\sum_{h=1}^{s}r_{i}$. $\square$

######  Corollary 3.9.

Let $G$ be a connected graph on the vertex set $[n]$ with $n\geq 6$. Suppose
that $G$ has a complete subgraph $\mathcal{K}_{4}$ and also

$G=\mathcal{K}_{4}\bigoplus_{v_{1}}T_{1}\bigoplus_{v_{2}}\cdots\bigoplus_{v_{s}}T_{s},$

where $\\{v_{1},\ldots,v_{s}\\}\subset V(\mathcal{K}_{4})$, $s\geq 2$, $v_{h}$
are pairwise distinct and $T_{h}$ are paths. Then $J_{2}(G)$ is a set-
theoretic complete intersection.

Proof. By [8, Theorem 1.1], $J_{2}(G)$ is Cohen-Macaulay, so ${\rm
dim}(R/J_{2}(G))=n+1$ and therefore ${\rm ht}(J_{2}(G))=n-1$. By Theorem 3.8,
${\rm ara}(J_{2}(G))={\rm ht}(J_{2}(G))$. $\square$

######  Theorem 3.10.

Let $G$ be a connected graph which is not a triangle and let $K$ be a field of
any characteristic. Suppose that $J_{2}(G)$ is an almost complete intersection
binomial edge ideal. Then $J_{2}(G)$ is a cohomologically complete
intersection if and only if $J_{2}(G)$ is Cohen-Macaulay.

Proof. The implication $\Rightarrow$ is obvious from Proposition 3.1.
Conversely suppose that $J_{2}(G)$ is Cohen-Macaulay. By [14, Theorem 2],
there are two cases.

  1. (1)

G is a graph on the vertex set
$\\{u_{1},\ldots,u_{r},v_{1},\ldots,v_{s},w_{1},\ldots,w_{t}\\}$ with $r\geq
2$, $s\geq 2$, $t\geq 2$ and edge set

$E(G)=\\{\\{u_{i},u_{i+1}\\}:i=1,\ldots,r-1\\}\cup\\{\\{v_{i},v_{i+1}\\}:i=1,\ldots,s-1\\}\cup$

$\ \ \ \ \ \ \ \ \
\cup\\{\\{w_{i},w_{i+1}\\}:i=1,\ldots,t-1\\}\cup\\{\\{u_{1},v_{1}\\},\\{u_{1},w_{1}\\},\\{v_{1},w_{1}\\}\\}$.
Then ${\rm ht}(J_{2}(G))=r+s+t-1$. Let $S$ be the subgraph of $G$ with the
edge set
$E(S)=\\{\\{u_{1},v_{1}\\},\\{u_{1},u_{2}\\},\\{v_{1},w_{1}\\},\\{u_{1},w_{1}\\}\\}$.
Let $I=(f_{u_{1},u_{2}}+f_{v_{1},w_{1}},f_{u_{1},v_{1}},f_{u_{1},w_{1}})$,
then $J_{2}(S)={\rm rad}(I)$ since $f_{u_{1},u_{2}}^{2}\in I$ and
$f_{v_{1},w_{1}}^{2}\in I$. So ${\rm ara}(J_{2}(S))\leq 3$ and also ${\rm
ara}(J_{2}(G))\leq 3+r+s+t-4$. Thus ${\rm ara}(J_{2}(G))=r+s+t-1$, so
$J_{2}(G)$ is a set-theoretic complete intersection.

  2. (2)

G is a graph on the vertex set $\\{u_{1},\ldots,u_{r},v_{1},\ldots,v_{s}\\}$
with $r\geq 3$, $s\geq 3$ and edge set

$E(G)=\\{\\{u_{i},u_{i+1}\\}:i=1,\ldots,r-1\\}\cup\\{\\{v_{i},v_{i+1}\\}:i=1,\ldots,s-1\\}\cup$

$\ \ \ \ \ \ \ \ \ \cup\\{\\{u_{1},v_{1}\\},\\{u_{2},v_{2}\\}\\}$.
Then ${\rm ht}(J_{2}(G))=r+s-1$. Let $G_{1}$ be the subgraph of $G$ on the
vertex set $\\{u_{1},u_{2},u_{3},v_{1},v_{2},v_{3}\\}$ with edges
$\\{u_{1},u_{2}\\}$, $\\{u_{2},v_{2}\\}$, $\\{v_{1},v_{2}\\}$,
$\\{u_{1},v_{1}\\}$, $\\{u_{2},u_{3}\\}$ and $\\{v_{2},v_{3}\\}$. Let
$T_{1}=(\\{u_{3},u_{4}\\},\\{u_{4},u_{5}\\},\ldots,\\{u_{r-1},u_{r}\\})$ and
$T_{2}=(\\{v_{3},v_{4}\\},\\{v_{4},v_{5}\\},\ldots,\\{v_{s-1},v_{s}\\})$. The
ideals $J_{2}(T_{1})$ and $J_{2}(T_{2})$ are complete intersection, so ${\rm
cd}(J_{2}(T_{1}))=r-3$ and ${\rm cd}(J_{2}(T_{2}))=s-3$. Then
$G=G_{1}\bigoplus_{u_{3}}T_{1}\bigoplus_{v_{3}}T_{2}$. Let $G_{2}$ be the
graph on the vertex set $\\{u_{1},u_{2},u_{3},v_{1},v_{2},v_{3}\\}$ with edges
$\\{u_{1},u_{2}\\},$ $\\{u_{2},v_{2}\\}$, $\\{v_{1},v_{2}\\}$,
$\\{u_{1},v_{1}\\}$, $\\{u_{1},v_{2}\\}$, $\\{u_{2},u_{3}\\}$ and
$\\{v_{2},v_{3}\\}$. By Theorem 3.7, ${\rm cd}(J_{2}(G_{2}))=5$. Let $G_{3}$
be the graph on the vertex set $\\{u_{1},u_{2},u_{3},v_{1},v_{2},v_{3}\\}$
with edges $\\{u_{1},u_{2}\\}$, $\\{u_{2},v_{2}\\}$, $\\{v_{1},v_{2}\\}$,
$\\{u_{1},v_{1}\\}$, $\\{u_{2},v_{1}\\}$, $\\{u_{2},u_{3}\\}$ and
$\\{v_{2},v_{3}\\}$. By Theorem 3.7, ${\rm cd}(J_{2}(G_{3}))=5$. Also

$J_{2}(G_{2})=P_{\emptyset}(G)\cap P_{\\{u_{2}\\}}(G)\cap
P_{\\{v_{2}\\}}(G)\cap P_{\\{u_{1},v_{2}\\}}(G)\cap P_{\\{u_{2},v_{2}\\}}(G)$

and

$J_{2}(G_{3})=P_{\emptyset}(G)\cap P_{\\{u_{2}\\}}(G)\cap
P_{\\{v_{2}\\}}(G)\cap P_{\\{u_{2},v_{1}\\}}(G)\cap P_{\\{u_{2},v_{2}\\}}(G).$

Thus $J_{2}(G_{1})=J_{2}(G_{2})\cap J_{2}(G_{3})$, since
$\mathcal{C}_{G}=\\{\emptyset,\\{u_{2}\\},\\{v_{2}\\},\\{u_{1},v_{2}\\},\\{u_{2},v_{1}\\},$
$\\{u_{2},v_{2}\\}\\}$. Let $G_{4}$ be the graph on the vertex set
$\\{u_{1},u_{2},u_{3},v_{1},v_{2},v_{3}\\}$ with edges $\\{u_{1},u_{2}\\}$,
$\\{u_{2},v_{2}\\}$, $\\{v_{1},v_{2}\\}$, $\\{u_{1},v_{1}\\}$,
$\\{u_{1},v_{2}\\}$, $\\{u_{2},v_{1}\\}$, $\\{u_{2},u_{3}\\}$ and
$\\{v_{2},v_{3}\\}$. By Theorem 3.8, ${\rm cd}(J_{2}(G_{4}))=5$. Notice that
$J_{2}(G_{4})=J_{2}(G_{2})+J_{2}(G_{3})$. Since ${\rm cd}(J_{2}(G_{2}))=5$,
${\rm cd}(J_{2}(G_{3}))=5$ and ${\rm cd}(J_{2}(G_{4}))=5$, we deduce from the
Mayer-Vietoris sequence

$\cdots\rightarrow H_{J_{2}(G_{4})}^{i}(R)\rightarrow
H_{J_{2}(G_{2})}^{i}(R)\oplus H_{J_{2}(G_{3})}^{i}(R)\rightarrow
H_{J_{2}(G_{1})}^{i}(R)\rightarrow H_{J_{2}(G_{4})}^{i+1}(R)\rightarrow\cdots$

that ${\rm cd}(J_{2}(G_{1}))\leq 5$. By [7, Corollary 2.2], ${\rm
cd}(J_{2}(G))\leq{\rm cd}(J_{2}(G_{1}))+{\rm cd}(J_{2}(T_{1}))+{\rm
cd}(J_{2}(T_{2}))$, thus ${\rm cd}(J_{2}(G))\leq r+s-1$ and therefore ${\rm
cd}(J_{2}(G))=r+s-1$. $\square$

Let $G$ and $G^{\prime}$ be two simple graphs with the vertex sets $[p]$ and
$[q]$, respectively. The join of $G$ and $G^{\prime}$, denoted by $G\ast
G^{\prime}$, is the graph with vertex set $[p]\cup[q]$ and the edge set
$E(G\ast G^{\prime})=E(G)\cup
E(G^{\prime})\cup\\{\\{i,j\\}|i\in[p],j\in[q]\\}$.

######  Example 3.11.

Let $G$ be the graph on the vertex set $[2]$ with only one edge $\\{1,2\\}$
and $G^{\prime}$ be the graph on the vertex set $\\{3,4,5\\}$ with connected
components $\\{3,4\\}$ and $\\{5\\}$. Then $G\ast G^{\prime}$ is the graph on
the vertex set $[5]$ with edges $E(G\ast
G^{\prime})=\\{\\{1,2\\},\\{1,3\\},\\{1,4\\},\\{1,5\\},\\{2,3\\},\\{2,4\\},\\{2,5\\},\\{3,4\\}\\}$.
We have that ${\rm pd}(R/J_{2}(G\ast G^{\prime}))=5$ and therefore ${\rm
ara}(J_{2}(G\ast G^{\prime}))\geq 5$. Let

$I=(f_{1,3}+f_{2,5},f_{1,4}+f_{2,3},f_{1,5}+f_{3,4},f_{1,2},f_{2,4}).$

Then $J_{2}(G\ast G^{\prime})={\rm rad}(I)$, since $f_{1,3}^{3}\in I$,
$f_{2,5}^{3}\in I$, $f_{1,4}^{2}\in I$, $f_{2,3}^{2}\in I$, $f_{1,5}^{5}\in I$
and $f_{3,4}^{5}\in I$. So ${\rm ara}(J_{2}(G\ast G^{\prime}))\leq 5$ and
therefore ${\rm ara}(J_{2}(G\ast G^{\prime}))=5$.

######  Proposition 3.12.

Let $G$ be a graph on the vertex set $[p]$ with $r\geq 1$ edges and
$G^{\prime}$ be a graph on the vertex set $[q]$ with $t\geq 1$ edges, where
$q\geq 3$. Then

${\rm ara}(J_{2}(G\ast G^{\prime}))\leq pq+r+t-3.$

Proof. Suppose that $\\{1,2\\}$ and $\\{3,4\\}$ are edges of $G$ and
$G^{\prime}$, respectively. Also assume that $5$ is a vertex of $G^{\prime}$.
Consider the subgraph $S$ of $G\ast G^{\prime}$ with the edge set
$E(S)=\\{\\{1,2\\},\\{1,3\\},\\{1,4\\},\\{1,5\\},\\{2,3\\},\\{2,4\\},\\{2,5\\},\\{3,4\\}\\}$.
By Example 3.11, ${\rm ara}(J_{2}(S))=5$. Since the graph $G\ast G^{\prime}$
has $pq+r+t$ edges, we get ${\rm ara}(J_{2}(G\ast G^{\prime}))\leq pq+r+t-3$.
$\square$

We concentrate now on graphs of the form $G\ast 2\mathcal{K}_{1}$, where $G$
is a graph on the vertex set $[p]$ and $2\mathcal{K}_{1}$ is the graph
consisting of two isolated vertices. By [15, Theorem 5.3], ${\rm
depth}(R/J_{2}(G\ast 2\mathcal{K}_{1}))=4$, so using the Auslander-Buchsbaum
formula we get ${\rm pd}(R/J_{2}(G\ast 2\mathcal{K}_{1}))=2p$. Thus
$2p\leq{\rm ara}(J_{2}(G\ast 2\mathcal{K}_{1}))$.

A null graph is a graph in which there are no edges between its vertices.

######  Proposition 3.13.

Let $G$ be a null graph on the vertex set $[p]$ where $p\geq 2$. Then ${\rm
ara}(J_{2}(G\ast 2\mathcal{K}_{1}))=2p$.

Proof. The graph $G\ast 2\mathcal{K}_{1}$ is complete bipartite with $p+2$
vertices. Also $\mu(J_{2}(G\ast 2\mathcal{K}_{1}))=2p$, so ${\rm
ara}(J_{2}(G\ast 2\mathcal{K}_{1}))\leq 2p$ and therefore ${\rm
ara}(J_{2}(G\ast 2\mathcal{K}_{1}))=2p$. $\square$

######  Remark 3.14.

For the complete bipartite graph $\mathcal{K}_{2,p}$ with $p+2$ vertices we
have that ${\rm ara}(J_{2}(\mathcal{K}_{2,p}))=2p$.

######  Theorem 3.15.

For every even positive integer $n$ there is a graph $G$ such that ${\rm
depth}(R/J_{2}(G))=4$ and ${\rm ara}(J_{2}(G))={\rm pd}(R/J_{2}(G))=n$.

Proof. Let $n=2p$, where $p\geq 1$ is an integer. Consider the complete
bipartite graph $\mathcal{K}_{2,p}$ with $p+2$ vertices. By Remark 3.14, ${\rm
ara}(J_{2}(\mathcal{K}_{2,p}))=2p$. $\square$

Given a connected and simple graph $G$ on the vertex set $[n]$ with edge set
$E(G)=\\{e_{1},\ldots,e_{r}\\}$, we write $\\{e_{1},\ldots,e_{r}\\}\ast
2\mathcal{K}_{1}:=G\ast 2\mathcal{K}_{1}$.

######  Example 3.16.

Let $G$ be a graph on the vertex set $[p]$ with exactly one edge
$e=\\{1,2\\}$, where $p\geq 2$. Then $J_{2}(G\ast
2\mathcal{K}_{1})=J_{2}(\\{e\\}\ast
2\mathcal{K}_{1})+J_{2}(\mathcal{K}_{2,p-2})$, where $\mathcal{K}_{2,p-2}$ is
a complete bipartite graph. Notice that $J_{2}(\mathcal{K}_{2,p-2})=(0)$ if
$p=2$. Let $\\{i,j\\}$ be the vertex set of $2\mathcal{K}_{1}$ and consider
the ideal $I=(f_{1,i}+f_{2,j},f_{1,2},f_{1,j},f_{2,i})$. Then
$J_{2}(\\{e\\}\ast 2\mathcal{K}_{1})={\rm rad}(I)$ since $f_{1,i}^{2}\in I$
and $f_{2,j}^{2}\in I$. Thus ${\rm ara}(J_{2}(\\{e\\}\ast
2\mathcal{K}_{1}))\leq 4$, so ${\rm ara}(J_{2}(G\ast 2\mathcal{K}_{1}))\leq
2p$ since ${\rm ara}(J_{2}(\mathcal{K}_{2,p-2}))=2p-4$. Consequently ${\rm
ara}(J_{2}(G\ast 2\mathcal{K}_{1}))=2p$.

######  Theorem 3.17.

Let $G$ be a graph on the vertex set $[p]$ which has at most $3$ edges, where
$p\geq 2$. Then ${\rm ara}(J_{2}(G\ast 2\mathcal{K}_{1}))={\rm
pd}(R/J_{2}(G))=2p$.

Proof. It is enough to show that ${\rm ara}(J_{2}(G\ast 2\mathcal{K}_{1}))\leq
2p$. Let $\\{i,j\\}$ be the vertex set of $2\mathcal{K}_{1}$. We distinguish
the following cases:

  1. (1)

$G$ is a null graph. By Proposition 3.13, ${\rm ara}(J_{2}(G\ast
2\mathcal{K}_{1}))=2p$.

  2. (2)

$G$ has exactly one edge $e$. By Example 3.16, ${\rm ara}(J_{2}(G\ast
2\mathcal{K}_{1}))=2p$.

  3. (3)

$G$ has exactly two edges $e_{1}$ and $e_{2}$. There are two cases: (i)
$e_{1}\cap e_{2}=\emptyset$. Then $J_{2}(G\ast
2\mathcal{K}_{1})=J_{2}(\\{e_{1}\\}\ast
2\mathcal{K}_{1})+J_{2}(\\{e_{2}\\}\ast
2\mathcal{K}_{1})+J_{2}(\mathcal{K}_{2,p-4})$. Also ${\rm
ara}(J_{2}(\\{e_{1}\\}\ast 2\mathcal{K}_{1}))=4$, ${\rm
ara}(J_{2}(\\{e_{2}\\}\ast 2\mathcal{K}_{1}))=4$ and ${\rm
ara}(J_{2}(\mathcal{K}_{2,p-4})=2p-8$. Thus ${\rm ara}(J_{2}(G\ast
2\mathcal{K}_{1}))\leq 2p$.
(ii) $e_{1}\cap e_{2}\neq\emptyset$. Let $e_{1}=\\{1,2\\}$ and
$e_{2}=\\{2,3\\}$. Then $J_{2}(G\ast
2\mathcal{K}_{1})=J_{2}(\\{e_{1},e_{2}\\}\ast
2\mathcal{K}_{1})+J_{2}(\mathcal{K}_{2,p-3})$. Let
$I=(f_{1,2},f_{2,3},f_{1,j},f_{3,i},f_{1,i}+f_{2,j},f_{2,i}+f_{3,j})$. We have
that $J_{2}(\\{e_{1},e_{2}\\}\ast 2\mathcal{K}_{1})={\rm rad}(I)$, since
$f_{1,i}^{2}\in I$, $f_{2,j}^{2}\in I$, $f_{2,i}^{2}\in I$ and $f_{3,j}^{2}\in
I$. So ${\rm ara}(J_{2}(\\{e_{1},e_{2}\\}\ast 2\mathcal{K}_{1}))\leq 6$ and
therefore ${\rm ara}(J_{2}(G\ast 2\mathcal{K}_{1}))\leq 2p$, since ${\rm
ara}(J_{2}(\mathcal{K}_{2,p-3}))=2p-6$.

  4. (4)

$G$ has exactly three edges $e_{1}$, $e_{2}$ and $e_{3}$. There are five
cases: (i) $e_{r}\cap e_{s}=\emptyset$, for every $1\leq r<s\leq 3$. Then
$J_{2}(G\ast 2\mathcal{K}_{1})=J_{2}(\\{e_{1}\\}\ast
2\mathcal{K}_{1})+J_{2}(\\{e_{2}\\}\ast
2\mathcal{K}_{1})+J_{2}(\\{e_{3}\\}\ast
2\mathcal{K}_{1})+J_{2}(\mathcal{K}_{2,p-6})$. Also ${\rm
ara}(J_{2}(\\{e_{i}\\}\ast 2\mathcal{K}_{1}))=4$, for $1\leq i\leq 3$, and
${\rm ara}(J_{2}(\mathcal{K}_{2,p-6}))=2p-12$. Thus ${\rm ara}(J_{2}(G\ast
2\mathcal{K}_{1}))\leq 2p$.
(ii) $e_{1}\cap e_{2}\neq\emptyset$, $e_{1}\cap e_{3}=\emptyset$ and
$e_{2}\cap e_{3}=\emptyset$. Then $J_{2}(G\ast
2\mathcal{K}_{1})=J_{2}(\\{e_{1},e_{2}\\}\ast
2\mathcal{K}_{1})+J_{2}(\\{e_{3}\\}\ast
2\mathcal{K}_{1})+J_{2}(\mathcal{K}_{2,p-5})$. Also ${\rm
ara}(J_{2}(\\{e_{1},e_{2}\\}\ast 2\mathcal{K}_{1}))=6$, ${\rm
ara}(J_{2}(\\{e_{3}\\}\ast 2\mathcal{K}_{1}))=4$ and ${\rm
ara}(J_{2}(\mathcal{K}_{2,p-5}))=2p-10$. So ${\rm ara}(J_{2}(G\ast
2\mathcal{K}_{1}))\leq 2p$.
(iii) $\\{e_{1},e_{2},e_{3}\\}$ is a path in $G$. Let $e_{1}=\\{1,2\\}$,
$e_{2}=\\{2,3\\}$ and $e_{3}=\\{3,4\\}$. Then $J_{2}(G\ast
2\mathcal{K}_{1})=J_{2}(\\{e_{1},e_{2},e_{3}\\}\ast
2\mathcal{K}_{1})+J_{2}(\mathcal{K}_{2,p-4})$. Let
$I=(f_{1,2},f_{2,3},f_{3,4},f_{1,j},f_{4,i},f_{1,i}+f_{2,j},f_{2,i}+f_{3,j},f_{3,i}+f_{4,j})$.
We have that $J_{2}(\\{e_{1},e_{2},e_{3}\\}\ast 2\mathcal{K}_{1})={\rm
rad}(I)$, since $f_{1,i}^{2}\in I$, $f_{2,j}^{2}\in I$, $f_{2,i}^{4}\in I$,
$f_{3,j}^{4}\in I$, $f_{3,i}^{2}\in I$ and $f_{4,j}^{2}\in I$. So ${\rm
ara}(J_{2}(\\{e_{1},e_{2},e_{3}\\}\ast 2\mathcal{K}_{1}))\leq 8$ and therefore
${\rm ara}(J_{2}(G\ast 2\mathcal{K}_{1}))\leq 2p$, since ${\rm
ara}(J_{2}(\mathcal{K}_{2,p-4}))=2p-8$.
(iv) $e_{1}\cap e_{2}\cap e_{3}\neq\emptyset$. Let $e_{1}=\\{1,2\\}$,
$e_{2}=\\{1,3\\}$ and $e_{3}=\\{1,4\\}$. Then $J_{2}(G\ast
2\mathcal{K}_{1})=J_{2}(\\{e_{1},e_{2},e_{3}\\}\ast
2\mathcal{K}_{1})+J_{2}(\mathcal{K}_{2,p-4})$. Let
$I=(f_{1,2}+f_{3,j},f_{1,i}+f_{2,j},f_{1,j}+f_{4,i},f_{1,3},f_{1,4},f_{2,i},f_{3,i},f_{4,j})$.
We have that $J_{2}(\\{e_{1},e_{2},e_{3}\\}\ast 2\mathcal{K}_{1})={\rm
rad}(I)$, since $f_{1,2}^{2}\in I$, $f_{3,j}^{2}\in I$, $f_{1,i}^{3}\in I$,
$f_{2,j}^{3}\in I$, $f_{1,j}^{2}\in I$ and $f_{4,i}^{2}\in I$. So ${\rm
ara}(J_{2}(\\{e_{1},e_{2},e_{3}\\}\ast 2\mathcal{K}_{1}))\leq 8$ and therefore
${\rm ara}(J_{2}(G\ast 2\mathcal{K}_{1}))\leq 2p$, since ${\rm
ara}(J_{2}(\mathcal{K}_{2,p-4}))=2p-8$.
(v) $\\{e_{1},e_{2},e_{3}\\}$ is the edge set of a triangle. Let
$e_{1}=\\{1,2\\}$, $e_{2}=\\{2,3\\}$ and $e_{3}=\\{1,3\\}$. Then $J_{2}(G\ast
2\mathcal{K}_{1})=J_{2}(\\{e_{1},e_{2},e_{3}\\}\ast
2\mathcal{K}_{1})+J_{2}(\mathcal{K}_{2,p-3})$. Let
$I=(f_{1,2}+f_{3,i},f_{1,3}+f_{2,j},f_{1,j}+f_{2,i},f_{1,i},f_{2,3},f_{3,j})$.
We have that $J_{2}(\\{e_{1},e_{2},e_{3}\\}\ast 2\mathcal{K}_{1})={\rm
rad}(I)$, since $f_{1,2}^{3}\in I$, $f_{3,i}^{3}\in I$, $f_{1,3}^{2}\in I$,
$f_{2,j}^{2}\in I$, $f_{1,j}^{5}\in I$ and $f_{2,i}^{5}\in I$. So ${\rm
ara}(J_{2}(\\{e_{1},e_{2},e_{3}\\}\ast 2\mathcal{K}_{1}))\leq 6$ and therefore
${\rm ara}(J_{2}(G\ast 2\mathcal{K}_{1}))\leq 2p$, since ${\rm
ara}(J_{2}(\mathcal{K}_{2,p-3}))=2p-6$. $\square$

## References

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A 4-fold Categorical Equivalence

Ray Maresca

###### Abstract

In this note, we will illuminate some immediate consequences of work done by
Reineke in [4] that may prove to be useful in the study of elliptic curves. In
particular, we will construct an isomorphism between the category of smooth
projective curves with a category of quiver grassmannians. We will use this to
provide a 4-fold categorical equivalence between a category of quiver
grassmannians, smooth projective curves, compact Riemann surfaces and fields
of transcendence degree 1 over $\mathbb{C}$. We finish with noting that the
category of elliptic curves is isomorphic to a category of quiver
grassmannians, whence providing an analytic group structure to a class of
quiver grassmannians.

##  1 Introduction

It is well known that there is a three-fold equivalence between the categories
of compact Riemann surfaces, fields of transcendence degree 1 over
$\mathbb{C}$, and smooth projective curves [2] and [9]. A more recent
development is the notion of quiver grassmannians, first introduced by
Schofield in [8]. Since their introduction, they have become a popular topic
of research. It has been known that quiver grassmannians are projective
varieties, but just how much projective geometry is captured by quiver
grassmannians was unclear until the early 2010’s. A famous result of Hille
[3], Huisgen-Zimmermann, and Rieneke [4], is that all projective varieties can
be realized as quiver grassmannians for some wild acyclic quiver $Q$.
Actually, even more is true. Expanding on his work in [5] in which he proved
the result for a generalized Kronecker quiver, Ringel showed in [7], the
incredible result that given any wild quiver $Q$, we can realize all
projective varieties as the quiver grassmannian of a suitable
$Q$-representation. It may be interesting to ask, is there is a ‘best’ quiver
with which to study projective varieties, and if not, which quivers are
‘better’ in which circumstances? Another natural question to ask is, can we
restrict the quiver $Q$ and still get a similar result? Ringel showed in [6]
that the answer to this question is partially yes. Namley, Ringel showed that
for a (controlled) wild algebra, any projective variety can be realized as an
Auslander variety, but not necessarily as a quiver Grassmannian.

In this note, we will use the construction given by Rieneke in [4] to define a
functor from the category of smooth projective curves to a subcategory of
quiver grassmannians. We will show that this functor is an isomorphism of
categories, which will ultimately yield a four-fold categorical equivalence.
We finish this note with some immediate consequences regarding elliptic
curves.

##  2 Preliminaries

To establish the equivalence, we will first recall some definitions.

###  2.1 Projective Varieties

Following [9], let $\Bbbk$ denote a perfect field. We begin by recalling that
projective $n$-space over a field $\Bbbk$ is defined as
$\mathbb{P}^{n}_{\overline{\Bbbk}}=\mathbb{P}^{n}={\overline{\Bbbk}^{n+1}\over\sim}$
where $(z_{0},\dots,z_{n})\sim(z^{\prime}_{0},\dots,z^{\prime}_{n})$ if and
only if there exists $\lambda\in\overline{\Bbbk}^{*}$ such that $(\lambda
z_{0},\dots,\lambda z_{n})=(z^{\prime}_{0},\dots,z^{\prime}_{n})$ and
$\overline{\Bbbk}$ denotes the algebraic closure of $\Bbbk$. Denote by
$[z_{0},\dots,z_{n}]$ the class of $(z_{0},\dots,z_{n})$ under the
aforementioned quotient map. Let $R=\overline{\Bbbk}[x_{0},\dots,x_{n}]$ be
the polynomial ring in $n+1$ variables over $\overline{\Bbbk}$. A polynomial
$P\in R$ is homogeneous of degree $d$ if $P(\lambda x)=\lambda^{d}P(x)$ for
all $\lambda\in\overline{\Bbbk}^{*}$. An ideal $I\subset R$ is homogeneous if
$I$ is generated by homogeneous polynomials. A projective algebraic set is
some subset of $\mathbb{P}^{n}$ of the form
$V(I)=\\{[x_{0},\dots,x_{n}]\in\mathbb{P}^{n}:P(x)=0\,\,\text{for all
homogeneous}\,\,P\in I\subset R\\}$ where $I$ is a homogeneous ideal. A
projective algebraic variety is $V(I)$ for $I$ a prime homogeneous ideal of
$R$.

We define the field of rational functions of $\mathbb{P}^{N}$ by
$\overline{\Bbbk}(\mathbb{P}^{N})=\\{{f\over g}\\}$ where $f,g\in R$, $g\neq
0$ and both $f$ and $g$ are homogeneous of the same degree. The field of
rational functions of a projective variety $V\subset\mathbb{P}^{N}$ is defined
as $\overline{\Bbbk}(V)={\overline{\Bbbk}(\mathbb{P}^{N})\over\sim}$ where
${f_{1}\over g_{1}}\sim{f_{2}\over g_{2}}$ if and only if
$f_{1}g_{2}-f_{2}g_{1}\in I(V)$ where $I(V)=\\{\text{homogeneous}\,\,P\in
R:P(x)=0\,\,\text{for all}\,\,x\in\overline{\Bbbk}^{n+1}\\}$. We define a
rational map between a projective variety $V\subset\mathbb{P}^{N}$ and
$\mathbb{P}^{M}$ as the data of $M+1$ elements of $\overline{\Bbbk}(V)$. A
rational map $V\rightarrow V^{\prime}\subset\mathbb{P}^{M}$ is a rational map
$V\rightarrow\mathbb{P}^{M}$ such that $[f_{0},\dots,f_{M}](x)\in V^{\prime}$
for all $x\in V$ for which $[f_{0},\dots,f_{M}](x)$ is defined. A rational map
between varieties is called a morphism if it is defined everywhere.

The dimension of a projective variety is the transcendence degree of
$\overline{\Bbbk}(V)$ over $\overline{\Bbbk}$. Projective varieties
$V\subset\mathbb{P}^{2}$ of dimension one are called projective curves. A
projective variety is called non-singular, or smooth, if the dimension of its
tangent space equals its dimension at every point. For more on projective
algebraic geometry, see [9] and [2]. The following $3$-fold categorical
equivalence is well known.

######  Theorem 2.1.

The following three categories are equivalent:

  1. 1.

Compact connected Riemann Surfaces with holomorphic maps.

  2. 2.

Field extensions of transcendence degree one over $\mathbb{C}$ with field
morphisms.

  3. 3.

Smooth projective curves in $\mathbb{P}^{2}_{\mathbb{C}}$ with morphisms of
varieties. $\square$

###  2.2 Quiver Grassmannians

A quiver $Q$ is a directed graph. More formally, it is a $4$-tuple
$Q=(Q_{0},Q_{1},s,t)$ where $Q_{0}$ is the set of vertices, $Q_{1}$ is the set
of arrows, and $s$ and $t$ are maps that assign to each vertex a starting and
terminal point respectively. For a field $\Bbbk$ that is usually taken to be
algebraically closed but need not be, a representation $V$ of a quiver $Q$ is
an assignment of a $\Bbbk$-vector space $V_{i}$ for each $i\in Q_{0}$ and a
vector space morphism $\phi_{\alpha}:V_{i}\rightarrow V_{j}$ for each
$\alpha\in Q_{1}$ such that $s(\alpha)=i$ and $t(\alpha)=j$. A
subrepresentation $M=(M_{i},\psi_{\alpha})$ of a representation
$V=(V_{i},\phi_{\alpha})$ is a representation of $Q$ such that $M_{i}\subset
V_{i}$ is a sub vector space for all vertices $i$, $\psi_{\alpha}$ is the
restriction of $\phi_{\alpha}$ to $M_{s(\alpha)}$, and
$\psi_{\alpha}(M_{i})\subset M_{j}$ for all arrows $\alpha:i\rightarrow j\in
Q_{1}$. In other words, a subrepresentation is a collection of subspaces that
are compatible with the morphisms defining the parent representation. The
dimension vector of a representation is
$\textbf{dim}V=($dim$V_{1},\dots,$dim$V_{|Q_{0}|})$. We call a representation
$V$ finite dimensional if $V_{i}$ is a finite dimensional vector space for all
$i\in Q_{0}$.

$1$$2$$3$A quiver $\displaystyle
Q$$\overline{\Bbbk}$$\overline{\Bbbk}^{10}$$\overline{\Bbbk}^{6}$$\varphi_{0}$$\varphi_{1}$$\varphi_{2}$$f$A
representation $\displaystyle V$ of $\displaystyle Q$ with
dim$\displaystyle(V)=(1,10,6)$$0$$\overline{\Bbbk}$$\overline{\Bbbk}$$\varphi_{0}$$\varphi_{1}$$\varphi_{2}$$0$A
subrepresentation $\displaystyle M$ of $\displaystyle V$ with
dim$\displaystyle(M)=(0,1,1)$ Figure 1: The quiver key to Theorem 2.2

Given a quiver $Q$ and a representation $V$ of $Q$, the quiver grassmannian
$\text{Gr}_{\bm{e}}^{Q}(V)$ is the set of subrepresentations of $V$ with
dimension vector $\bm{e}$. The subrepresentation $M$ in Figure 1 is an element
of $\text{Gr}_{(0,1,1)}^{Q}(V)$. It is well known that quiver grassmannians
are projective varieties. For more on quiver grassmannians, see [1]. We also
have the following result of Hille, Huisgen-Zimmermann, and Reineke. The
wording below is consistent with Reineke’s in [4]:

######  Theorem 2.2.

Every projective variety is isomorphic to a quiver Grassmannian
$\text{Gr}^{Q}_{\bm{e}}(V)$ for an acyclic quiver Q with at most three
vertices, a Schurian representation V, and a thin dimension vector $\bm{e}$;
that is, $e_{i}\leq 1$ for all $i\in Q_{0}$. $\square$

##  3 The 4-fold Equivalence

Theorem 2.2 relies on the $d$-uple Veronese embedding. We will use essentially
the same idea to create a category of quiver Grassmannians equivalent to the
third category listed in Theorem 2.1.

Let $X\subset\mathbb{P}_{\overline{\Bbbk}}^{2}$ be a smooth projective curve.
Thus $X$ is defined as the vanishing locus of a homogeneous polynomial $P$ of
degree $d$ in three variables. Let
$\nu_{d}:\mathbb{P}_{\overline{\Bbbk}}^{2}\rightarrow\mathbb{P}_{\overline{\Bbbk}}^{\binom{d+2}{2}-1}$
denote the $d$-uple Veronese embedding, which is an isomorphism onto its image
since $\overline{\Bbbk}$ is a field of characteristic 0. Then by Reineke’s
result, Theorem 2.2, $\nu_{d}(X)=\text{Gr}_{(0,1,1)}(V)$ for $V$ a
representation of the quiver $Q$ in Figure 1 of dimension $(1,M,M^{\prime})$
where $M=\binom{d+2}{2}$ and $M^{\prime}=\binom{d+1}{2}$. On the top right of
Figure 1 is an example of $\nu_{3}(X)$ where $X$ is a projective curve defined
by the vanishing locus of a degree $d=3$ polynomial in three variables.

The condition of being in the image of the $d$-uple Veronese embedding is
encoded into an $M^{\prime}\times 3$ matrix $A_{d}(x)$, whose $2\times 2$ sub
minors vanish. In particular, let $M_{2,d}$ be the set of tuples
$m=(m_{0},m_{1},m_{2})\in\mathbb{N}^{3}$ summing to $d$, so that $M$ is the
cardinality of $M_{2,d}$, and $\nu_{d}$ maps homogeneous coordinates
$[x_{0},x_{1},x_{2}]$ to $[\dots,x^{m},\dots]_{m\in M_{2,d}}$, where
$x^{m}=x_{0}^{m_{0}}x_{1}^{m_{1}}x_{2}^{m_{2}}$. We define the matrix
$A_{d}(x)$ with rows indexed by the $n\in M_{2,d-1}$ and columns indexed by
the $i=0,1,2$ with the $(2,i)$-th entry being $x_{n+e_{i}}$. Then
$x\in\nu_{d}(X)$ if and only if $P(x)=0$ and $A_{d}(x)$ has rank 1. We thus
define $V$ as in Figure 1 with $f=\nu_{d}(P)$ and $\varphi_{i}$ the $i$th
column of $A_{d}(x)$. Then $\nu_{d}(X)=\text{Gr}_{(0,1,1)}(V)$. For more on
this construction, see [4].

Fix the quiver $Q$ to be the one in Figure 1 and let $d\in\mathbb{Z}^{\geq
0}$. Let $V_{d}^{f}$ be a $\overline{\Bbbk}$-representation of $Q$ such that
$\textbf{dim}(V)=(1,M,M^{\prime})$, $\varphi_{i}$ is the $i$th column of
$A_{d}(x)$, and the preimage of the linear map $f$, denoted by
$\nu_{d}^{-1}(f)$, is irreducible as a homogeneous polynomial in
$\overline{\Bbbk}[x_{0},x_{1},x_{2}]$.

######  Definition 3.1.

Define $\mathcal{GR}_{(0,1,1)}(V)$ to be the category whose objects are
$\text{Gr}_{(0,1,1)}(V_{d}^{f})$ for all $d\in\mathbb{Z}^{\geq 0}$ and any $f$
such that $\nu_{d}^{-1}(f)$ is irreducible, and whose morphisms are those of
projective varieties. Let $\mathcal{GR}_{(0,1,1)}^{\text{sm}}(V)$ be the full
subcategory of $\mathcal{GR}_{(0,1,1)}(V)$ who’s objects are smooth.

######  Theorem 3.1.

The category of non-singular projective curves in
$\mathbb{P}^{2}_{\overline{\Bbbk}}$ with morphisms of varieties, (NPC) for
short, is isomorphic to $\mathcal{GR}_{(0,1,1)}^{\text{sm}}(V)$.

###### Proof.

We begin by constructing a functor
$\nu:\text{(NPC)}\rightarrow\mathcal{GR}_{(0,1,1)}^{\text{sm}}(V)$. For $X$ a
non-singular projective curve cut out by a homogeneous polynomial of degree
$d$, define $\nu(X):=\nu_{d}(X)$. Then $\nu(X)$ is non-singular since $X$ is
and $\nu(X)\cong X$. Moreover, $\nu(X)=\text{Gr}_{(0,1,1)}(V_{d}^{f})$ for $f$
the homogeneous polynomial of degree $d$ that cuts out $X$ by Reineke’s
result, Theorem 2.2. Thus $\nu(X)\in\mathcal{Gr}_{(0,1,1)}^{\text{sm}}(V_{d})$
and $\nu$ is well defined on objects. Given a morphism $\psi:X\rightarrow Y$
between two non-singular projective curves cut out by homogeneous polynomials
of degree $d$ and $d^{\prime}$ respectively, define
$\nu(\psi):\nu(X)\rightarrow\nu(Y)$ by
$\nu_{d^{\prime}}\circ\psi\circ\nu_{d}^{-1}$. Since $\nu(\psi)$ is a morphism
of varieties and $\nu$ preserves the identity and composition, $\nu$ defines a
functor.

It is well known that the $d$-uple Veronese embedding is an isomorphism onto
its image. Notice by construction, for any $d$, the image of $\nu_{d}$
restricted to projective curves cut out by a homogeneous polynomial of degree
$d$ is equal to the collection of $\text{Gr}_{(0,1,1)}(V_{d}^{f})$. This
allows us to define
$\nu^{-1}:\mathcal{GR}_{(0,1,1)}^{\text{sm}}(V)\rightarrow(\text{NPC})$
analogously to $\nu$, and these two functors are inverse. ∎

By taking $\Bbbk=\mathbb{C}$, an immediate consequence of Theorems 2.2 and 3.1
is the following 4-fold categorical equivalence.

######  Corollary 3.2.

The following four categories are equivalent:

  1. 1.

Compact connected Riemann Surfaces with holomorphic maps.

  2. 2.

Field extensions of transcendence degree one over $\mathbb{C}$ with field
morphisms.

  3. 3.

Smooth projective curves in $\mathbb{P}^{2}_{\overline{\mathbb{C}}}$ with
morphisms of varieties.

  4. 4.

$\mathcal{GR}_{(0,1,1)}^{\text{sm}}(V)$ where $V$ is a
$\mathbb{C}$-representation of $Q$. $\square$

Recall that by definition, elliptic curves are non-singular curves of genus
one; however, every such curve can be written as the locus in
$\mathbb{P}_{\overline{\Bbbk}}^{2}$ of a cubic equation with the base point on
the line at $\infty$ [9]. The next corollary follows from the fact that
elliptic curves are the vanishing locus of an irreducible homogeneous
polynomial of degree 3 and the fact that the functor $\nu$ restricts to an
isomorphism, namely $\nu_{3}$.

######  Corollary 3.3.

The category of elliptic curves in $\mathbb{P}_{\overline{\Bbbk}}^{2}$ with
morhpisms of varieties is equivalent to
$\mathcal{GR}_{(0,1,1)}^{\text{sm}}(V_{3})$, the full subcategory of
$\mathcal{GR}_{(0,1,1)}^{\text{sm}}(V)$ whose objects are
$\text{Gr}_{(0,1,1)}^{\text{sm}}(V_{3}^{f})$ for $f$ such that
$\nu_{3}^{-1}(f)$ is irreducible. ∎

######  Remark 3.1.

Since in Corollary 3.3 we do not take $\Bbbk=\mathbb{C}$, this equivalence
along with the linear-algebraic nature of representations of quivers may prove
to be useful in the study of rational points of elliptic curves. Moreover, one
may be able to use the moduli space of quiver grassmannians to study that of
elliptic curves and vice versa.

Corollary 3.3 also provides us with a way to remove the artificial imposition
of smoothness in Definition 3.1. In determining smoothness of
$\text{Gr}_{(0,1,1)}(V_{d}^{f})$, it suffices to check the Jacobian criterion
on the equations that cut out the quiver grassmannian; however, after
embedding into a higher dimensional projective space there can be several of
these equations and checking this criterion can quickly become computationally
expensive. We do however have the following proposition.

######  Proposition 3.4.

Suppose the characteristic of $\Bbbk$ is not 2 or 3 and consider a quiver
grassmannian $\text{GR}_{(0,1,1)}(V_{3}^{f})$. Then
$\text{GR}_{(0,1,1)}(V_{3}^{f})$ is smooth if and only if it is isomorphic to
$\text{GR}_{(0,1,1)}(V_{3}^{\xi})$ where $\xi=x_{7}-x_{0}-ax_{5}-bx_{9}$ and
$4a^{3}+27b^{2}\neq 0$.

###### Proof.

By definition, a curve in $\mathbb{P}_{\overline{\Bbbk}}^{2}$ cut out by a
degree 3 homogeneous polynomial is smooth if and only if it is an elliptic
curve. In the case of elliptic curves however, it is known that each curve can
be written in reduced Weierstrass form as $y^{2}=x^{3}+ax+b$ when the field is
not characteristic 2 or 3 [9]. Upon realizing this in homogeneous coordinates,
we get the equation $y^{2}z-x^{3}-axz^{2}-bz^{3}=0$. To attain the
corresponding quiver grassmannian, we analyze the 3-uple Veronese embedding:

$\nu_{3}(x,y,z)=(x^{3},x^{2}y,x^{2}z,xy^{2},xyz,xz^{2},y^{3},y^{2}z,yz^{2},z^{3}).$

Relabeling the $\mathbb{P}_{\overline{\Bbbk}}^{9}$ coordinates as
$(x_{0},x_{1},\dots,x_{9})$, any elliptic curve is the solution set to
$x_{7}-x_{0}-ax_{5}-bx_{9}=0$. Letting $\xi=x_{7}-x_{0}-ax_{5}-bx_{9}$, the
corresponding quiver grassmannian is $\text{Gr}_{(0,1,1)}(V_{3}^{\xi})$. Now
by Corollary 3.3, we have that $\text{Gr}_{(0,1,1)}(V_{3}^{\xi})$ is an object
of $\mathcal{GR}_{(0,1,1)}^{\text{sm}}(V_{3})$ if and only if $\xi$ is smooth,
which occurs if and only if $4a^{3}+27b^{2}\neq 0$ [9]. ∎

Using Corollary 3.3 we can also see that the objects of
$\text{GR}_{(0,1,1)}^{\text{sm}}(V_{3}^{f})$ can be endowed with a commutative
group structure inherited from that of elliptic curves. In the case
$\Bbbk=\mathbb{C}$, we can use Corollary 3.2 to further state that these
quiver grassmannians are also isomorphic to connected compact Riemann surfaces
of genus 1, hence complex tori.

######  Corollary 3.5.

Let $X$ be a connected compact Riemann surface. Then the following are
equivalent:

  1. 1.

$X$ has genus 1.

  2. 2.

$X$ has a structure of an analytic group.

  3. 3.

$X$ has a commutative analytic group structure.

  4. 4.

$X\cong{\mathbb{C}\over\mathbb{Z}l+\mathbb{Z}w}$

  5. 5.

$X\cong C_{F}$ where $C_{F}$ is an elliptic curve.

  6. 6.

$X\cong\text{Gr}_{(0,1,1)}^{\text{sm}}(V_{3}^{f})$ for some $f.\hfill\square$

##  4 References

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Cerulli Irelli, G. Three Lectures on Quiver Grassmannians, Preprint,
arXiv:2003.08265 [math.AG], 2020.

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Hartshorne, R. Algebraic Geometry, Springer, Volume 52, 1977.

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Hille, L. Moduli of representations, quiver Grassmannians and Hilbert schemes,
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Reineke, M. Every Projective Variety is a Quiver Grassmannian, Algebras and
Representation Theory, 16, 1313-1314, 2013.

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Ringel, C. The Eigenvector Variety of a Matrix Pencil, Linear Algebra Appl.,
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Ringel, C. Quiver Grassmannians and Auslander varieties for wild algebras, J.
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Ringel, C. Quiver Grassmannians for Wild Acyclic Quivers, Proc. AMS, 146, n.
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Schofield, A. General representations of quivers, Proc. London Math. Soc. (3)
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Silverman, Joseph H. The Arithmetic of Elliptic Curves, Springer, Volume 106,
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