# Steiner Configurations ideals: containment and colouring

Edoardo Ballico  Dipartimento di Matematica
Via Sommarive, 14 - 38123 Povo (TN), Italy<EMAIL_ADDRESS>,
Giuseppe Favacchio  Dipartimento di Matematica e Informatica
Viale A. Doria, 6 - 95100 - Catania, Italy<EMAIL_ADDRESS>www.dmi.unict.it/ gfavacchio ,  Elena Guardo  Dipartimento di Matematica e
Informatica
Viale A. Doria, 6 - 95100 - Catania, Italy<EMAIL_ADDRESS>www.dmi.unict.it/guardo ,  Lorenzo Milazzo${\dagger}$ Dipartimento di
Matematica e Informatica
Viale A. Doria, 6 - 95100 - Catania, Italy<EMAIL_ADDRESS>www.dmi.unict.it/milazzo and  Abu Chackalamannil Thomas  Department of
Mathematics,
Tulane University, New Orleans, LA, U.S.A 70118<EMAIL_ADDRESS>
###### Abstract.

Given a homogeneous ideal $I\subseteq k[x_{0},\dots,x_{n}]$, the Containment
problem studies the relation between symbolic and regular powers of $I$, that
is, it asks for which pair $m,r\in\mathbb{N}$, $I^{(m)}\subseteq I^{r}$ holds.
In the last years, several conjectures have been posed on this problem,
creating an active area of current interests and ongoing investigations. In
this paper, we investigated the Stable Harbourne Conjecture and the Stable
Harbourne – Huneke Conjecture and we show that they hold for the defining
ideal of a Complement of a Steiner configuration of points in
$\mathbb{P}^{n}_{k}$. We can also show that the ideal of a Complement of a
Steiner Configuration of points has expected resurgence, that is, its
resurgence is strictly less than its big height, and it also satisfies
Chudnovsky and Demailly’s Conjectures. Moreover, given a hypergraph $H$, we
also study the relation between its colourability and the failure of the
containment problem for the cover ideal associated to $H$. We apply these
results in the case that $H$ is a Steiner System.

###### Key words and phrases:

monomial ideals; ideals of points; symbolic powers of ideals; Waldschmidt
constant; Steiner systems

###### 2010 Mathematics Subject Classification:

13F55, 13F20,14G50, 51E10,94B27

${\dagger}$ Deceased, March 4, 2019.

Last updated: Jan 15, 2021

##  1\. Introduction

In this paper we continue the study of Steiner configurations of points and
their invariants, such as Hilbert Function, Betti numbers, Waldschmidt
constant, regularity, resurgence found in [3]. We will focus on the
Containment problem and we will show that the Stable Harbourne Conjecture and
the Stable Harbourne – Huneke Conjecture hold for the defining ideal of a
Complement of a Steiner configuration of points in
$\mathbb{P}^{n}_{k}:=\mathbb{P}^{n}$. As pointed out in Remarks 2.5 and 2.6 in
[3] in the language of Algebraic Geometry/Commutative Algebra, Steiner
configurations of points and their Complement are special subsets of star
configurations.

First, we give an overview on the Containment problem to introduce the related
conjectures. Then we devote Section 2 to recall notation, definitions and
known results for a Steiner configuration of points and its Complement that we
will use to prove the results of this paper. Let $I$ be a homogeneous ideal in
the standard graded polynomial ring $R:=k[x_{0},\ldots,x_{n}]$, where $k$ is a
field. Given an integer $m$, we denote by $I^{m}$ the regular power of the
ideal $I$. The $m$-th symbolic power of $I$ is defined as

$I^{(m)}=\bigcap_{\mathfrak{p}\in Ass(I)}(I^{m}R_{\mathfrak{p}}\cap R)$

where $Ass(I)$ denotes the set of associated primes of $I$ and
$R_{\mathfrak{p}}$ is the localization of $R$ at a prime ideal
${\mathfrak{p}}$.

If $I$ is a radical ideal (this includes for instance squarefree monomial
ideals and ideals of finite sets of points) then

$I^{(m)}=\bigcap_{\mathfrak{p}\in Ass(I)}{\mathfrak{p}}^{m}.$

Symbolic powers of ideals play a significant role in the famous Zariski-Nagata
Theorem (see [48, 54]). If $R$ is a polynomial ring over an algebraically
closed field $k$, then $I^{(m)}$ consists precisely of those functions which
vanish on the algebraic variety defined by $I$ with multiplicity at least $m$.
It is easy to show from the definition that $I^{r}\subseteq I^{(m)}$ if and
only if $r\geq m$. The reverse inclusion $I^{(m)}\subseteq I^{r}$ motivates
the following question.

######  Question 1.1.

(Containment problem) Given a homogeneous ideal $I\subseteq
k[x_{0},\dots,x_{n}]$, for which pairs $m,r\in\mathbb{N}$, does
$I^{(m)}\subseteq I^{r}$ hold?

One of the initial works that introduce Question 1.1 is [51]. The problem is
still open in general and in the last couple of decades it was extensively
studied for several classes of ideals, in particular for ideals defining
finite sets of points in projective and multiprojective spaces, see [8, 9, 17,
13, 24, 25, 26, 35, 36, 37, 46] just to cite some among all the known results.
Containment problems are useful in giving lower bounds to nonzero homogeneous
forms vanishing through a finite set of points with a fixed multiplicity.

It is of great interest to study the ideals of fat points. Given distinct
points $P_{1},\dots,P_{s}\in\mathbb{P}^{n}$ and nonnegative integers $m_{i}$
(not all $0$), let $Z=m_{1}p_{1}+\dots+m_{s}p_{s}$ denote the scheme (called a
fat point scheme) defined by the ideal
$I_{Z}=\cap_{i=1}^{s}(I_{P_{i}}^{m_{i}})\subseteq k[\mathbb{P}^{n}]$, where
$I_{P_{i}}$ is the ideal generated by all homogeneous polynomials vanishing at
$P_{i}$. Symbolic powers of $I_{Z}$ take the form
$I_{Z}^{(m)}=I_{mZ}=\cap_{i=1}^{s}I_{P_{i}}^{mm_{i}}$. We say that $Z$ is
reduced if $I_{Z}$ is a radical ideal.

The Containment problem also helps us to bound certain useful invariants like
Waldschmidt constant, $\widehat{\alpha}(I)$, of an ideal $I$ defined as

$\widehat{\alpha}(I)=\lim_{m\to\infty}\frac{\alpha(I^{(m)})}{m},$

where $\alpha(I)$ is the minimum integer $d$ such that $I_{d}\neq(0)$, that
is, it is the least degree of a minimal generator of $I$. This limit exists
and was first defined by Waldschmidt [53] for ideals of finite sets of points
in the context of complex analysis; specifically, in our language, the problem
was to determine the minimal degree of a hypersurface that passed through a
collection of points with prescribed multiplicities.

The following slight different version of Question 1.1 was introduced in [39].
Recall that the big height of an ideal $I$ refers to the maximum of all the
heights of its associated prime ideals.

######  Conjecture 1.2.

Let $Z\subset\mathbb{P}^{n}$ be a fat point scheme and $I:=I_{Z}$ the ideal
defining $Z$. Let $\mathcal{M}=(x_{0},\dots,x_{n})$ be the graded maximal
ideal. Then $I^{(rn)}\subseteq\mathcal{M}^{r(n-1)}I^{r}$ holds for all $r>0$.

B. Harbourne conjectured in [4]:

######  Conjecture 1.3.

Given a nonzero, proper, homogeneous, radical ideal $I\subseteq
k[x_{0},\dots,x_{n}]$ with big height $h$,

$I^{(hr-h+1)}\subseteq I^{r}$

for all $r\geq 1.$

A counterexample to the above conjecture was initially found in [23].

A celebrated result of [24, 41, 46] is shown in the next theorem.

######  Theorem 1.4.

Let $R$ be a regular ring and $I$ a radical ideal in $R$. Then for all
$n\in\mathbb{N}$,

$I^{(hn)}\subseteq I^{n},$

whenever $h$ is the big height of $I$.

One could hope to sharpen the containment by reducing the symbolic power on
the left hand side by a constant or increasing the ordinary power on the right
hand side by a fixed constant. This motivates us to look at stable versions of
Conjectures 2.1 and 4.1 in [39], respectively.

######  Conjecture 1.5.

(Stable Harbourne Conjecture) Given a nonzero, proper, homogeneous, radical
ideal $I\subseteq k[x_{0},\dots,x_{n}]$ with big height $h$, then

$I^{(hr-h+1)}\subseteq I^{r}$

for all $r\gg 0.$

######  Conjecture 1.6.

(Stable Harbourne–Huneke Conjecture) Let $I\subseteq k[x_{0},\dots,x_{n}]$ be
a homogeneous radical ideal of big height $h$. Let
$\mathcal{M}=(x_{0},\dots,x_{n})$ be the graded maximal ideal. Then for $r\gg
0$,

  1. (1)

$I^{(hr)}\subseteq\mathcal{M}^{r(h-1)}I^{r}$

  2. (2)

$I^{(hr-h+1)}\subseteq\mathcal{M}^{(r-1)(h-1)}I^{r}$.

In the study of finding the least degree of minimal generators of an ideal
$I$, Chudnovsky made the following:

######  Conjecture 1.7.

(Chudnovsky’s Conjecture). Suppose that $k$ is an algebraically closed field
of characteristic $0$. Let $I$ be the defining ideal of a set of points
$X\subseteq\mathbb{P}^{n}_{k}$. Then, for all $h>1$,

$\frac{\alpha(I^{(h)})}{h}\geq\frac{\alpha(I)+n-1}{n}.$

A generalization of Chudnovsky’s Conjecture is the following:

######  Conjecture 1.8.

(Demailly’s Conjecture). Suppose that $k$ is an algebraically closed field of
characteristic $0$. Let $I$ be the defining ideal of a set of points
$X\subseteq\mathbb{P}^{n}_{k}$ and let $m\in\mathbb{N}$ be any integer. Then,
for all $h>1$,

$\frac{\alpha(I^{(h)})}{h}\geq\frac{\alpha(I^{(m)})+n-1}{m+n-1}.$

Two recent preprints, [5, 6], focus on the Containment problem and related
conjectures. In the first one, the authors show that Chudnovsky’s Conjecture
holds for sufficiently many general points and to prove it they show that one
of the containments conjectured by Harbourne and Huneke holds eventually,
meaning for large powers (see Theorem 4.6 in [5]). They also show other
related results, for example, that general sets of points have expected
resurgence and thus satisfy the Stable Harbourne Conjecture.

In the second preprint, the authors show that Demailly’s Conjecture (which is
a generalization of Chudnovsky’s) also holds for sufficiently many general
points, for star configurations (in general, not just points) and for generic
determinantal ideals.

In this paper we prove that the Stable Harbourne Conjecture and the Stable
Harbourne–Huneke Conjecture hold for ideals defining the Complement of a
Steiner Configuration of points in $\mathbb{P}^{n}$ that are special subsets
of star configurations and, hence, far from being general. We will give more
details in Section 3.

We remark that the least degree of a minimal generator of the ideal defining
the Complement of a Steiner Configuration of points in $\mathbb{P}^{n}$ is
strictly less than the least degree of a minimal generator of the ideal of a
star configurations (see Theorem 2.6 and also Proposition 2.9 in [29]). So, it
is worth investigating whether the Containment problem and its related
conjectures hold for the Complement of a Steiner Configuration of points in
$\mathbb{P}^{n}$.

In [3] the authors constructed a squarefree monomial ideal $J$ associated to a
set $X$ of points in $\mathbb{P}^{n}$ constructed from the Complement of a
Steiner system. The ideal $I_{X}$ defining the Complement of a Steiner system
is not a monomial ideal. But the authors proved that the symbolic powers of
$I_{X}$ and $J$ share the same homological invariants (see Proposition 3.6 in
[3]). This was possible because $J$ is the Stanley-Reisner ideal of a matroid,
so its symbolic powers define an arithmetically Cohen-Macaulay (ACM for short)
scheme which gives, after proper hyperplane sections, the scheme of fat points
supported on $X$. But we point out that the regular powers of $J$ are not
necessarily ACM anymore and we cannot relate them to squarefree monomial
ideals. Thus, the homological invariants of the regular powers of $J$ are not
immediately correlated to that of $I_{X}$.

In [17] the authors proved that the Chudnovsky’s Conjecture, the Harbourne’s
Conjecture and the Harbourne–Huneke containment conjectures hold for
squarefree monomial ideals.

As previously remarked, since the ideal $I_{X}$ defining the Complement of a
Steiner system is not a squarefree monomial ideal, we cannot recover the
Stable Harbourne Conjecture and the Stable Harbourne – Huneke Conjecture using
[17].

We also point out that the two above preprints [5, 6] do not compute the
Waldschmidt constant exactly for any class of ideals, they study lower bounds
for the Waldschmidt constant. And, since in [3] the authors found the exact
value of the Waldschmidt constant for the Complement of a Steiner
configurations of points, then Chudnovsky and Demailly’s Conjectures easily
follow for our class of ideals (see Section 3).

For other results on this topic we can also see [27, 47].

Another tool useful to measure the non–containment among symbolic and ordinary
powers of ideals is the notion of resurgence $\rho(I)$ of an ideal $I$,
introduced in [9] that gives some notion of how small the ratio $m/r$ can be
and still be sure to have $I^{(m)}\subseteq I^{r}$.

######  Definition 1.9.

Let $I$ be a nonzero, proper ideal in a commutative ring $R$, the resurgence
of the ideal $I$ is given by

$\rho(I)=\sup\left\\{\frac{m}{r}\quad|\quad I^{(m)}\nsubseteq I^{r}\right\\}.$

It always satisfies $\rho(I)\geq 1$. The groundbreaking results of [24, 41,
46] show that $\rho(I)\leq h$, where $h$ is the big height of the radical
ideal $I$. This motivates us to ask whether $\rho(I)$ can strictly be less
than its big height and which are some of the interesting consequences.
Although there are few cases where the resurgence has been computed, in
general, it is extremely difficult to estimate the exact value for $\rho(I)$.
The reader can look at [22] for the first examples where the resurgence and
the asymptotic resurgence are not equal. An asymptotic version of the
resurgence was introduced in the paper [35].

######  Definition 1.10.

For a nonzero, proper homogeneous ideal $I\subseteq k[x_{0},\dots,x_{n}]$, the
asymptotic resurgence $\rho_{a}(I)$ is defined as follows:

$\rho_{a}(I)=\sup\left\\{\frac{m}{r}\quad|\quad I^{(mt)}\nsubseteq
I^{rt},\quad\text{for all}\quad t\gg 0\right\\}.$

It is clear from the definition that $1\leq\rho_{a}(I)\leq\rho(I)$. As pointed
out in [40], DiPasquale, Francisco, Mermin, Schweig showed that
$\rho_{a}(I)=sup\\{m/r:~{}I^{(m)}\nsubseteq\overline{I^{r}}\\}$, where
$\overline{I^{r}}$ is the integral closure of $I^{r}$ (see also [20] Corollary
4.14) .

In this paper we study the containment properties of the ideal defining a
Complement of a Steiner configuration of points in $\mathbb{P}^{n}$. Section 2
is devoted to recall notation, definitions and known results from [3] that we
will use in the next sections. The main result of Section 3 is Theorem 3.5
where we prove that an ideal defining the Complement of a Steiner
Configuration of points in $\mathbb{P}^{n}$ satisfies both the Stable
Harbourne Conjecture and the Stable Harbourne–Huneke Conjecture. In Lemma 3.2,
we give a criterion for when the resurgence number can be computed in finite
number of steps. This result improves the bounds found in Corollary 4.8 in
[3]. We also point out that Lemma 3.2 is similar to results from [19, 20]. As
a consequence, in Corollary 3.6 we show that the ideal of a Complement of a
Steiner Configuration of points has expected resurgence, that is, its
resurgence is strictly less than its big height (see [34] for the first
definition). Moreover, using Theorem 2.6, Corollaries 2.7 and 2.8, we show
that the ideal of a Complement of a Steiner Configuration of points satisfies
Chudnovsky and Demailly’s Conjectures (see Corollary 2.8, Corollary 3.7 and
Corollary 3.8).

Finally, in Section 4, given a hypergraph ${H}$, we also study the relation
between its colourability and the failure of the Containment problem for the
cover ideal associated to $H$. The ideas come from the paper [28] where the
authors start to study the natural one-to-one correspondence between
squarefree monomial ideals and finite simple hypergraphs via the cover ideal
construction.

There exists an extensive literature on the subject of colourings both from
Design Theory and Algebraic Geometry/Commutative Algebra point of view. Among
all, we make use of [10, 28, 38, 31] as some referring texts for preliminaries
on hypergraph theory and associated primes and for an algebraic method to
compute the chromatic number, respectively.

Most of the existing papers are devoted to the case of weak colourings (or
vertex colourings), i.e. colourings where the colours are assigned to the
elements in such a way that no hyperedge is monochromatic (i.e. no hyperedge
has all its elements assigned the same colour). The reader can see [10] or
Chapter 3 in [31] for other different types of colouring a hypergraph, such as
strong vertex colouring, vertex equicolouring, good colouring of $H$.

In this paper we use the case of weak colouring to get result on the
Containment problem since it is the one commonly used in Combinatorial
Commutative Algebra. The main result of this section is Theorem 4.8 that more
generally predicts the failure of the containement for squarefree monomial
ideals based on the definition of coverability (see Definition 4.4). We apply
these results in the case that $H$ is a Steiner System.

We end the paper recalling some open questions posed in [3] and that are still
under investigations and posing new ones as possible further research
problems.

##  2\. Notation, definitions and known results for Ideals of a Steiner
configuration of points and its Complement

In this section we recall the main results from [3], where the authors studied
the homological properties of ideals constructed from Steiner systems,
especially in the zero dimensional case of $\mathbb{P}^{n}$.

A Steiner system $(V,B)$ of type $S(t,n,v)$ is a collection $B$ of $n$-subsets
(blocks) of a $v$-set $V$ such that each $t$-tuple of $V$ is contained in a
unique block in $B$. The elements in $V$ are called vertices or points and
those of $B$ are called blocks. In particular, a Steiner triple system of
order $v$, $STS(v)$, is a collection of triples ($3$-subsets) of $V$, such
that each unordered pair of elements is contained in precisely one block, and
a Steiner quadruple system of order $v$, $SQS(v)$, is a collection of
quadruples (4-subsets) of $V$ such that each triple is found in precisely one
block.

The existence of a Steiner system strongly depends on the parameters
$(t,n,v)$. If a Steiner system $(V,B)$ of type $S(t,n,v)$ exists, then

$|B|=\frac{{\binom{v}{t}}}{{\binom{n}{t}}}.$

We use [14] and [16] as main references for all the background on design
theory.

We recall the most known example.

######  Example 2.1.

One of the simplest and most known example of Steiner system is the Fano
Plane. It is unique up to isomorphism and it is a Steiner system $S(2,3,7)$
with block set

$B:=\\{\\{1,2,3\\},\\{3,4,5\\},\\{3,6,7\\},\\{1,4,7\\},\\{2,4,6\\},\\{2,5,7\\},\\{1,5,6\\}\\}.$

For the ease of the reader, we recall some definitions and results from [3].

Let $V:=\\{1,\ldots,v\\}$ and $\mathcal{H}:=\\{H_{1},\ldots H_{v}\\}$ be a
collection of distinct hyperplanes of $\mathbb{P}^{n}$, where $n\leq v$. Say
$H_{j}$ defined by the linear forms $\ell_{j}$ for $j=1,\dots,n$. Assume that
any $n$ hyperplanes in $\mathcal{H}$ meet properly, i.e., they meet in a
point. There is a natural way to associate a point in $\mathbb{P}^{n}$ to a
subset of $n$ elements of $V$. For
$\sigma:=\\{\sigma_{1},\ldots,\sigma_{n}\\}\subseteq V$, we denote by
$P_{\mathcal{H},\sigma}$ the point obtained by intersecting the hyperplanes
$H_{\sigma_{1}},\ldots,H_{\sigma_{n}}.$ Then, the ideal
$I_{P_{\mathcal{H},\sigma}}=(\ell_{\sigma_{1}},\ldots,\ell_{\sigma_{n}})\subseteq
k[\mathbb{P}^{n}]$ is the vanishing ideal of the point
$P_{\mathcal{H},\sigma}$.

######  Definition 2.2.

Let $Y$ be a collection of subsets of $V$ containing $n$ elements, and
$\mathcal{H}$ a set of hyperplanes meeting properly. We define the following
set of points in $\mathbb{P}^{n}$ with respect to $\mathcal{H}$

$X_{\mathcal{H},Y}:=\bigcup_{\sigma\in Y}~{}P_{\mathcal{H},\sigma}$

and its defining ideal

$I_{X_{\mathcal{H},Y}}:=\bigcap_{\sigma\in Y}~{}I_{P_{\mathcal{H},\sigma}}.$

Denoted by $C_{(n,v)}$ the set containing all the subsets of $V$ with $n$
elements the above definition applied to a Steiner system gives us two
different sets of points.

######  Definition 2.3.

Let $(V,B)$ be a Steiner system of type $S(t,n,v)$ with $t<v\leq n$. We
associate to $B$ the following set of points in $\mathbb{P}^{n}$

$X_{\mathcal{H},B}:=\bigcup_{\sigma\in B}~{}P_{\mathcal{H},\sigma}$

and its defining ideal

$I_{X_{\mathcal{H},B}}:=\bigcap_{\sigma\in B}~{}I_{P_{\mathcal{H},\sigma}}.$

We call $X_{\mathcal{H},B}$ the Steiner configuration of points associated to
the Steiner system $(V,B)$ of type $S(t,n,v)$ with respect to $\mathcal{H}$
(or just $X_{B}$ if there is no ambiguity).

######  Definition 2.4.

Let $(V,B)$ be a Steiner system of type $S(t,n,v)$ with $t<n\leq v$. We
associate to $C_{(n,v)}\setminus B$ the following set of points in
$\mathbb{P}^{n}$

$X_{\mathcal{H},C_{(n,v)}\setminus B}:=\bigcup_{\sigma\in C_{(n,v)}\setminus
B}~{}P_{\mathcal{H},\sigma}$

and its defining ideal

$I_{X_{\mathcal{H},C_{(n,v)}\setminus B}}:=\bigcap_{\sigma\in
C_{(n,v)}\setminus B}~{}I_{P_{\mathcal{H},\sigma}}.$

We call $X_{\mathcal{H},C_{(v,n)}\setminus B}$ the Complement of a Steiner
configuration of points with respect to $\mathcal{H}$ (or C-Steiner
$X_{\text{C}}$ if there is no ambiguity).

As pointed out in [3], Remarks 2.5 and 2.6 a Steiner configuration of points
and its Complement are subschemes of a star configuration of ${\binom{v}{n}}$
points in $\mathbb{P}^{n}$ (see [11, 12, 29, 30] just to cite some reference
on star configurations).

We also have

$\deg X_{\mathcal{H},C_{(n,v)}\setminus
B}={\binom{v}{t}}-|B|={\binom{v}{t}}-\dfrac{{\binom{v}{t}}}{{\binom{n}{t}}}.$

We recall the most known construction of Steiner Configuration of points and
its Complement.

######  Example 2.5.

Consider the Steiner configuration associated to $(V,B)$ of type $S(2,3,7)$ as
in Example 2.1. Take $\mathcal{H}:=\\{H_{1},\ldots,H_{7}\\}$ a collection of
$7$ distinct hyperplanes $H_{i}$ in $\mathbb{P}^{3}$ defined by a linear form
$\ell_{i}$ for $i=1,\dots,7$, respectively, with the property that any $3$ of
them meet in a point $P_{\mathcal{H},\sigma}=H_{\sigma_{1}}\cap
H_{\sigma_{2}}\cap H_{\sigma_{3}}$, where
$\sigma=\\{\sigma_{1},\sigma_{2},\sigma_{3}\\}\in B$. We get that
$X_{\mathcal{H},C_{(3,7)}}$ is a star configuration of ${\binom{7}{3}}=35$
points in $\mathbb{P}^{3}$, $X_{\mathcal{H},B}:=\cup_{\sigma\in
B}~{}\\{P_{\mathcal{H},\sigma}\\}$ is a Steiner configuration consisting of
$7$ points in $\mathbb{P}^{3}$ and $X_{\mathcal{H},C_{(3,7)}\setminus B}$ is a
C-Steiner configuration consisting of ${\binom{7}{3}}-7=28$ points in
$\mathbb{P}^{3}.$ Their defining ideals are respectively,

$I_{X_{\mathcal{H},B}}:=\cap_{\sigma\in
B}~{}I_{P_{\mathcal{H},\sigma}}\textrm{ and
}I_{X_{\mathcal{H},C_{(3,7)}\setminus B}}:=\cap_{\sigma\in C_{(3,7)}\setminus
B}~{}I_{P_{\mathcal{H},\sigma}}.$

In [3] the authors constructed a squarefree monomial ideal $J$ associated to a
set $X_{\mathcal{H},C}$ of points in $\mathbb{P}^{n}$ constructed from the
Complement of a Steiner system. The ideal $I_{X_{\mathcal{H},C}}$ defining the
Complement of a Steiner system is not a monomial ideal. But the authors proved
that the symbolic powers of $I_{X_{\mathcal{H},C}}$ and $J$ share the same
homological invariants (see Proposition 3.6 in [3]).

The following results give the least degree of a minimal generator and the
regularity and the Waldschmidt constant of an ideal defining the Complement of
a Steiner configuration of points, respectively.

######  Theorem 2.6 ([3], Theorem 3.9).

Let $(V,B)$ be a Steiner system of type $S(t,n,v)$. Set
$I_{X_{\mathcal{H},C}}:=I_{X_{C}}$ the ideal defining the Complement of the
Steiner configuration of points associated to $S(t,n,v)$. Then

  * i)

$\alpha(I_{X_{C}})=v-n$;

  * ii)

$\alpha(I_{X_{C}}^{(q)})=v-n+q$ for $2\leq q<n$;

  * iii)

$\alpha(I_{X_{C}}^{(m)})=\alpha(I_{X_{C}}^{(q)})+pv$, where $m=pn+q$ and
$0\leq q<n$ and $\alpha(I_{X_{C}}^{(n)})=\alpha(I_{X_{C}}^{(0)})+v=v$.

######  Corollary 2.7 ([3], Corollary 4.2).

Let $\text{reg}(I_{X_{C}})$ be the regularity of a Complement of a Steiner
configuration. Then $\text{reg}(I_{X_{C}})=\alpha(I_{X_{C}})+1=v-n+1$.

######  Corollary 2.8 ([3], Corollary 3.12).

If $(V,B)$ is a Steiner system of type $S(t,n,v)$, then the Waldschmidt
constant of its Complement is $\widehat{\alpha}(I)=\frac{v}{n}$.

##  3\.  Asymptotic resurgence and Stable Harbourne Conjecture

Containment problems have been of interest among commutative algebraists and
algebraic geometers. In the last decade, several conjectures related to this
problem have been posed creating an active area of current interests and
outgoing investigations.

A celebrated result of [24, 41, 46] is that $I^{(hn)}\subseteq I^{n}$ for all
$n\in\mathbb{N}$, whenever $h$ is the big height of $I$. One could hope to
sharpen the containment by reducing the symbolic power on the left hand side
by a constant or increasing the ordinary power on the right hand side by a
fixed constant. This motivates us to look at the Stable Harbourne Conjecture
and the Stable Harbourne – Huneke Conjecture and study which class of ideals
satisfies them. Here, we prove that the ideal defining a Complement of a
Steiner configurations of points satisfies both conjectures. We need to recall
some known results.

In [33], the Conjecture 1.5 is shown to hold

  1. (1)

if there exists $k>0$ such that $I^{(hk-h)}\subseteq I^{k}$;

  2. (2)

if $I^{(hk-h+1)}\subseteq I^{k}$ for some $k$ and $I^{(r+h)}\subseteq
II^{(r)}$ for all $r\geq k$;

  3. (3)

if the resurgence satisfies $\rho(I)<h$.

In particular, condition (2) gives a criterion for the Stable Harbourne
Conjecture (SHC for short) to hold. Namely, for a radical ideal of big height
$h$, if for all $k\geq 1$, it is $I^{(k+h)}\subseteq II^{(k)}$ and fix an
integer $C$ and $m$ such that $I^{(hm-C)}\subseteq I^{m}$ holds, then for all
$q\geq m$, we have

$I^{(hq-C)}=I^{(h(q-m)+hm-h+h-C)}\subseteq II^{(h(q-m-1)+hm-C)}\subseteq
I^{q-m}I^{(hm-C)}\subseteq I^{q-m}I^{m}=I^{q},$

that is, $I^{(hq-C)}\subseteq I^{q}$.

######  Theorem 3.1 (Theorem 2.5, [33]).

Let $R$ be a regular ring containing a field, and let $I$ be a radical ideal
in $R$ with big height $h$. If $I^{(h(m-1))}\subseteq I^{m}$ for some $m\geq
2$, then $I^{(h(k-1))}\subseteq I^{k}$ for all $k>>0$ (indeed for all $k\geq
hm$).

We have learned that Harbourne, Kettinger, Zimmitti in [40] and DiPasquale,
Drabkin in [19] proved independently that $\rho_{a}(I)<h$ if and only if
$\rho(I)<h$. As pointed out in [19] Remark 2.3, the next result is similar as
Proposition 4.1.3 of Denkert’s thesis [18], as Lemma 4.12 in [20] and as
Proposition 2.2 in [19].

For the ease of the reader, we adapt the proof in our case.

######  Lemma 3.2.

Let $I\subseteq k[x_{0},\dots,x_{n}]$ be a homogeneous radical ideal with
$big\ height(I)=h$, such that $\rho(I)>\rho_{a}(I)$. Suppose we have the
equality

$\rho_{a}(I)=\frac{hr_{1}-h}{r_{1}}$

for some $r_{1}>0$. Then $\rho(I)$ can be computed by taking the maximum of
finitely many $\frac{s}{r}$ with $I^{(s)}\nsubseteq I^{r}$.

###### Proof.

Using Briancon Skoda Theorem ([52], Corollary 13.3.4), we have that
$\overline{I^{r+n}}\subseteq I^{r}$, where $n+1$ is the number of variables in
the polynomial ring and $\overline{I}$ denotes the integral closure of the
ideal $I$. For $s,r\in\mathbb{N}$ such that $I^{(s)}\nsubseteq I^{r}$ then
$I^{(s)}\nsubseteq\overline{I^{r+n}}$. Using [20], Lemma 4.12, we get
$\frac{s}{r+n}<h(1-1/r_{1})=\rho_{a}(I)$, that is,

$\frac{s}{r}<(1+n/r)h(1-1/r_{1}).$

If $\rho(I)>\rho_{a}(I)$, applying [19] Proposition 2.2, then there exist
$s_{0},r_{0}$, such that $I^{(s_{0})}\nsubseteq I^{r_{0}}$ and

$\rho(I)\geq\frac{s_{0}}{r_{0}}\geq(1+\frac{n}{r})h(1-\frac{1}{r_{1}}),$

solving for $r$ gives us the inequality

$r\geq\frac{n}{\frac{s_{0}/r_{0}}{h(1-1/r_{1})}-1},$

so whenever $r\geq\frac{n}{\frac{s_{0}/r_{0}}{h(1-1/r_{1})}-1}$ and $s$ is
such that $I^{(s)}\nsubseteq I^{r}$, we have
$\frac{s}{r}<\frac{s_{0}}{r_{0}}$.

Hence, it suffices to look at

$r\leq\frac{n}{\frac{s_{0}/r_{0}}{h(1-1/r_{1})}-1}$

and $s\leq(r+n)h(1-\frac{1}{r_{1}})$. ∎

######  Corollary 3.3.

If the resurgence can be computed by taking the maximum of finitely many
ratios of the form $\frac{m}{r}$ for which $I^{(m)}\nsubseteq I^{r}$, then
$\rho(I)<h$.

###### Proof.

Suppose we have $\frac{a}{b}=h$, then $I^{(hb)}\nsubseteq I^{b}$ is a
contradiction since $I^{(hk)}\subseteq I^{k}$ for all $k\in\mathbb{N}$ (from
[24, 41, 46]). Hence, $\rho(I)<\frac{a}{b}=h$. ∎

The next proposition shows that Conjecture 3.1 in [7] holds for the Complement
of a Steiner Configuration of points.

######  Proposition 3.4.

Let $I\subset k[\mathbb{P}^{n}]$ be an ideal defining a Complement of a
Steiner Configuration of points and let $\mathcal{M}=(x_{0},\dots,x_{n})$ be
the homogeneous maximal ideal. Then $I^{(nr)}\subseteq\mathcal{M}^{rn}I^{r}$
holds for all $r\in\mathbb{N}.$

###### Proof.

From Theorem 2.6, we have $\alpha(I^{(nr)})=rv$. From [3], Corollary 4.7, we
have $\omega(I^{r})=\alpha(I^{r})=r(v-n)$, where $\omega(I)$ is the maximum of
the generating degrees of the ideal $I$. Since $I^{(nr)}\subseteq I^{r}$ for
all $r\geq 1$, we have $\alpha(I^{(nr)})\geq r\omega(I)=r\alpha(I)$ and
$\alpha(I^{r})=r(v-n)$, so $\alpha(I^{(nr)})-r\omega(I)$ = $rv-r(v-n)=rn$.
Since every minimal generator of $I^{(nr)}$ is contained inside $I^{r}$ and
the difference between the degree of any nonzero homogeneous polynomial in
$I^{(nr)}$ and that of generators of $I^{r}$ is at least $rn$, we have that
$I^{(nr)}\subseteq\mathcal{M}^{rn}I^{r}$, the conclusion follows. ∎

We prove the main result of this section:

######  Theorem 3.5.

Let $I\subseteq k[x_{0},\dots,x_{n}]$ be the ideal defining the Complement of
a Steiner Configuration of points in $\mathbb{P}^{n}_{k}$. Then $I$ satisfies

  1. (1)

Stable Harbourne–Huneke Conjecture;

  2. (2)

Stable Harbourne Conjecture.

###### Proof.

(1) Consider the Steiner Configuration of points $S(t,n,v)$ in
$\mathbb{P}_{k}^{n}$ and $I:=I_{C}$ the ideal defining its Complement.

Using Theorem 2.6 , (iii), it is $\alpha(I^{(n(r-1))})=(r-1)v$. Using
Corollary 2.7, and choosing $r\gg 0$, such that

$(r-1)v\geq r\cdot\operatorname{reg}(I)=r(v-n+1)$

we get $I^{(n(r-1))}\subseteq I^{r}.$ Moreover, since
$\alpha(I^{(n(r-1))})-\alpha(I^{r})=v(r-1)-r(v-n)=rn-v$, we get
$I^{(n(r-1))}\subseteq\mathcal{M}^{rn-v}I^{r}.$ Using Euler’s Formula, we get

$\begin{array}[]{rcl}I^{(n(r-1)+1)}&\subseteq&\mathcal{M}I^{(n(r-1))}\subseteq\mathcal{M}^{rn-v+1}I^{r}=\mathcal{M}^{rn-
v-n+n+1}I^{r}=\mathcal{M}^{rn-n-(v-n)+1}I^{r}\\\ &\subseteq&\mathcal{M}^{rn-
n-r+1}I^{r}=\mathcal{M}^{(r-1)(n-1)}I^{r}.\end{array}$

(2) We have the containment $I^{(n(r-1))}\subseteq I^{r}$ for $r\gg 0$.
Let $k=nm+t$. From [44], we have $I^{(ns+a_{1}+\cdots+a_{s})}\subseteq
I^{(a_{1}+1)}I^{(a_{2}+1)}\cdots I^{(a_{s}+1)},$ letting
$s=n+t,a_{1}=a_{2}=\dots=a_{n}=nm-n-1,a_{n+1}=\dots=a_{n+t}=0$.

Let $k=nm+t=(v-1)n+t$ for $t\geq 0$ and let $s=n+t$ and $a_{1}=a_{2}=\cdots
a_{n}=nm-n-1=n(v-1)-n-1,a_{n+1}=\dots=a_{t}=0$. Therefore,

$I^{(n(n+t)+n(nm-n-1))}=I^{(n^{2}+nt+n^{2}m-n^{2}-n)}=I^{(nk-n)}\subseteq(I^{((n(v-1)-n))})^{n}I^{t}=I^{n(v-1)}I^{t}=I^{k}.$

Hence $I^{(nk-n)}\subseteq I^{k}$ for $k\gg 0$. ∎

As a consequence, we can show that the ideal of a Complement of a Steiner
Configuration of points has expected resurgence, that is, its resurgence is
strictly less than its big height (see [34]).

######  Corollary 3.6.

Let $I\subseteq k[x_{0},\dots,x_{n}]$ be the ideal defining the Complement of
a Steiner Configuration of points in $\mathbb{P}^{n}_{k}$. Then, $\rho(I)<n.$

###### Proof.

From Theorem 3.5, we have $\rho_{a}(I)<n$. Note that $\rho_{a}(I)\leq\rho(I)$.
If $\rho_{a}(I)=\rho(I)$, then clearly $\rho(I)<n$. On the other hand, if
$\rho_{a}(I)<\rho(I)$, then from Lemma 3.2 and Corollary 3.3, we conclude that
$\rho(I)<n.$ ∎

We give an alternative proof of Chudnovky’s Conjecture:

######  Corollary 3.7.

Let $I\subseteq k[x_{0},\dots,x_{n}]$ be the ideal defining the Complement of
a Steiner Configuration of points in $\mathbb{P}^{n}_{k}$. Then Chudnovky’s
Conjecture holds for $I$.

###### Proof.

From Theorem 2.6, item i) $\alpha(I)=v-n$ and from Theorem 2.8 it is
$\widehat{\alpha}(I)=\frac{v}{n}$. Then

$\widehat{\alpha}(I)\geq\frac{\alpha(I)+n-1}{n}\Leftrightarrow\frac{v}{n}\geq\frac{v-1}{n}.$

∎

######  Corollary 3.8.

Let $I\subseteq k[x_{0},\dots,x_{n}]$ be the ideal defining the Complement of
a Steiner Configuration of points in $\mathbb{P}^{n}_{k}$. Then Demailly’s
Conjecture holds for $I$.

###### Proof.

From Theorem 2.6, for $m=pn+q$ and for $2\leq q<n$ it is
$\alpha(I^{(m)})=pv+\alpha(I^{(q)})=pv+v-n+q$. From Corollary 2.8, we have
that $\widehat{\alpha}(I)=\frac{v}{n}$. Hence, whenever

  1. (1)

$m=pn+q$, with $2\leq q<n$, we have

$\frac{\alpha(I^{(m)})+n-1}{m+n-1}=\frac{pv+v-n+q+n-1}{pn+q+n-1}=\frac{(p+1)v+q-1}{(p+1)n+q-1}\leq\frac{v}{n}=\widehat{\alpha}(I)$

  2. (2)

$q=1$ and $m=np+1$, we have

$\frac{\alpha(I^{(m)})+n-1}{m+n-1}=\frac{pv+v-n+n-1}{pn+1+n-1}=\frac{(p+1)v-1}{(p+1)n}<\frac{v}{n}=\widehat{\alpha}(I)$

  3. (3)

$q=0$ and $m=np$, we have

$\frac{\alpha(I^{(m)})+n-1}{m+n-1}=\frac{pv+n-1}{pn+n-1}<\frac{v}{n}=\widehat{\alpha}(I).$

∎

######  Remark 3.9.

Chudnovsky’s Conjecture can be showed from Proposition 3.4. We have that
$I^{(nr)}\subseteq\mathcal{M}^{rn}I^{r}$. This gives us the inequality
$\alpha(I^{(nr)})\geq rn+r\alpha(I)\geq rn+r\alpha(I)-r$. Dividing both sides
by $nr$ and letting $r\rightarrow\infty$ gives

$\widehat{\alpha}(I)\geq\frac{\alpha(I)+n-1}{n}.$

##  4\. Containment and colouring

In this section we focus on the relation between the colourability of a
hypergraph ${H}$ and the failure of the containment problem for the cover
ideal associated to $H$. Then, we apply these results in the case that $H$ is
a Steiner System. There exists an extensive literature on the subject of
colourings both from Design Theory and Algebraic Geometry/Commutative Algebra
point of view. Among all, we make use of [10, 28, 38, 31] as some of referring
texts.

Most of the existing papers are devoted to the case of weak colourings (or
vertex colourings), i.e. colourings where the colours are assigned to the
elements in such a way that no hyperedge is monochromatic (i.e. no hyperedge
has all its elements assigned the same colour). The reader can see [10] or
Chapter 3 in [31] for other types of colouring a hypergraph, such as strong
vertex colouring, vertex equicolouring, good colouring of $H$.

In this paper we use the case of weak colouring to get results on Containment
problem.

We first recall some known definitions and results from [10] or [31], Chapter
2.

A hypergraph is a pair $H=(V,E)$, where $E=\\{x_{1},\dots,x_{n}\\}$ is a
finite nonempty set containing $n$ elements called vertices and
$E=\\{e_{i}\\}_{i\in I}$ ($I$ set of indices) is a family of subsets of $X$,
called edges, or otherwise hyperedges, such that for all $e\in
E,e\neq\emptyset$ and $\cup_{e\in E}\;e=X$.

A colouring of a hypergraph $H=(V,E)$ is a surjective mapping $c:V\rightarrow
C$ where $C$ is the set of colours. When $|C|=m$, then a proper $m$-colouring
of a hypergraph $H=(V,E)$ is a mapping $c:V\rightarrow\\{1,2,\dots,m\\}$ for
which every edge $e\in E$ has at least two vertices of different colours.

As for graphs, proper colourings generate partitions of the vertex set into a
number of stable (independent) non-empty subsets called colour classes, with
as many classes as the number of colours actually used.

Thus, we use an equivalent definition from [28, 38], used in Combinatorial
Algebraic Geometry/Commutative Algebra research, i.e,

######  Definition 4.1.

Let $H=(V,E)$ be a hypergraph. An $m$-colouring of $H$ is any partition of
$V=U_{1}\cup\cdots\cup U_{m}$ into $m$ disjoint sets such that for every $e\in
E$ we have $e\not\subseteq U_{j}$ for all $j=1,\dots,m$. The $U_{j}$’s are
called the colour classes.

The chromatic number of $H$, denoted by $\chi(H)$, is the minimum $m$ such
that $H$ has an $m$-colouring.

######  Definition 4.2.

A hypergraph $H=(V,E)$ is $m$-colourable if there exists a proper
$m$-colouring, i.e, if $\chi(H)\leq m$.

######  Definition 4.3.

We say $H$ is $m$-chromatic if it is $m$-colourable but not
$(m-1)$-colourable.

When $\chi(H)\leq 2$, the hypergraph $H$ is called bicolourable. (In parts of
the literature the term ‘bipartite’ is also used.)

######  Definition 4.4.

Let $H:=(V,E)$ be a hypergraph. For an integer $c,$ we say that $H$ is
$c$-coverable if there exists a partition $U_{1},U_{2},\dots,U_{c}$ of $V$
such that $e\cap U_{i}\neq\emptyset$ for each $i=1,\ldots,c$ and for each
$e\in E.$

######  Remark 4.5.

Note that, as an immediate consequence of the above definitions, if $H$ is
$c$-coverable, $c>1$, then $H$ is $c$-colourable.

######  Example 4.6.

Set $V:=\\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7}\\}$. Let $H$ be the set
of blocks of a $STS(7)$

(4.1)
$H:=\\{\\{x_{1},x_{2},x_{3}\\},\\{x_{1},x_{4},x_{5}\\},\\{x_{1},x_{6},x_{7}\\},\\{x_{2},x_{4},x_{6}\\},\\{x_{2},x_{5},x_{7}\\},\\{x_{3},x_{4},x_{7}\\},\\{x_{3},x_{5},x_{6}\\}\\}.$

Take, for instance, the partition
$\\{x_{1},x_{2},x_{5}\\},\\{x_{3},x_{4},x_{6}\\},\\{x_{7}\\}$ (see Figure 1).
$H$ is 3-colourable but it is not 3-coverable.

Notice also that no colouring of $H$ with two colours exists. Then
$\chi(H)=3$.

Figure 1.  The three colour classes of a $STS(7)$.

######  Remark 4.7.

We refer the reader to Section 3.5 in [31] to see examples of different types
of colourings that give different chromatic numbers for the same $H$. In
particular, in Example 8 of [31], the strong vertex colouring of $H$ as in
(4.1) gives $\chi(H)=7$ (recall that a mapping $c$ is a strong colouring of
vertices of $H$ if for all $e\in E$ it is $|c(e)|=|e|$).

For a non empty hypergraph $H$, i.e. $H\subset 2^{V}$, we define the ideal

$J_{H}:=\bigcap_{\sigma\in H}\mathfrak{p}_{\sigma}\subseteq k[V]$

called the cover ideal of $H,$ where for a subset of $V$,
$\sigma:=\\{{i_{1}},{i_{2}},\ldots,{i_{n}}\\}\subseteq V,$ the ideal

$\mathfrak{p}_{\sigma}:=(x_{i_{1}},x_{i_{2}},\ldots,x_{i_{n}})\subseteq k[V]$

denotes the prime ideal generated by the variables indexed by $\sigma$.

For a hypergraph $H=(V,B)$, we denote by $\tau(H):=\min_{b\in B}\\{|b|\\}.$

We study some properties of the cover ideals of $B$. The following results
show a relation between the coverability of a hypergraph $H$ and that the
Containment problem can fail.

######  Theorem 4.8.

Let $H=(V,B)$ be a hypergraph. If $H$ is not $d$-coverable then
$J_{H}^{(\tau(H))}\not\subseteq J_{H}^{d}.$

###### Proof.

We put $\tau:=\tau(H)$ and $w:=x_{1}\cdot x_{2}\cdots x_{n}$. In order to
prove the statement it is enough to show that $w\in J_{H}^{(\tau)}$ but
$w\notin J_{H}^{d}.$ For each $b\in B$ the ideal $\mathfrak{p}_{b}$ has height
$|b|\leq\tau$. Therefore $x_{b}\in(\mathfrak{p}_{b})^{\tau}$. Thus
$w\in\mathfrak{p}_{b}^{\tau}$ for each $b\in B$. This implies $w\in
J_{H}^{(\tau)}.$ By contradiction, assume $w\in J_{H}^{d}.$ Thus, there exist
$w_{1},\ldots,w_{d}\in J_{H}$ such that $w=w_{1}\cdots w_{d}.$ We
$U_{j}:=\\{x_{u}\in V\ |\ x_{u}\ \text{divides}\ w_{j}\\}$, then
$U_{1},\ldots,U_{d}$ is a partition of $V$. Thus, for each $b\in B$ we have
$w_{i}\in\mathfrak{p}_{b}$, therefore $U_{i}\cap e\neq 0$ with $i=1,\ldots,d.$
This contradicts that $H$ is not $d$-coverable. ∎

Recall that an $m$-colouring of $(V,B)$ is called an $m$-bicolouring if the
vertices of each $b\in B$ are coloured with exactly two colours. A Steiner
Triple Systems $(V,B)$ admitting an $m$-bicolouring is $m$-bicolourable. Thus,
in a bicolouring of a Steiner Triple System $(V,B)$, every triple has two
elements in one colour class and one in another class, so there are no
monochromatic triples nor polychromatic triples (i.e. triples receiving three
colours). For instance, for a deep investigation of colouring properties of
Steiner Triple Systems the reader can see [15].

As a consequence, we get a failure of the containment for the cover ideals
associated to Steiner Triple Systems $(V,B)$ of type $S(t,n,v).$

######  Proposition 4.9.

If $v>3$ and $S(2,3,v)=(V,B)$ is a Steiner Triple System, then
$J_{B}^{(3)}\not\subseteq J_{B}^{2}$.

###### Proof.

It is enough to show that $B$ in not 2-coverable. Assume by contradiction
$V=U_{1}\cup U_{2}.$ By definition, for each $\\{i,j,k\\}\in B$ we have
$\\{i,j,k\\}\cap U_{1}\neq\emptyset$ and $\\{i,j,k\\}\cap U_{2}\neq\emptyset$.
This implies that $S(2,3,v)$ is 2-bicolourable contradicting a well known fact
about Steiner Triple Systems, see [50]. ∎

We end the paper showing the failure of the containment for the cover ideals
associated to Steiner Systems.

######  Proposition 4.10.

Let $\mathcal{S}=(V,B)$ be a Steiner System with parameters $S(t,t+a,v)$ where
$1\leq a\leq t-2$ and $v>(a+1)t$. Then, $J_{B}^{(t+a)}\not\subseteq
J_{B}^{t}.$

###### Proof.

Note that, from Theorem 4.8, it is enough to show that $B$ is not
$t$-coverable. Assume by contradiction there is a partition of $V$ in $t$
colour classes, $V=C_{1}\cup\cdots\cup C_{t}$ such that $B\cap
C_{j}\neq\emptyset$ for each $j=1,\ldots,t$ and $B$ a block of $\mathcal{S}.$
We denote by $c_{j}$ the number of elements in $C_{j}$ for $j=1,\ldots,t$.
Note that $c_{j}\leq a+1$. Indeed if $i_{1},i_{2},\ldots,i_{a+2}\in C_{j}$ are
different elements, then a block containing $i_{1},i_{2},\ldots,i_{a+2}$
cannot intersect $t$ colour classes. This implies $v\leq(a+1)t$. ∎

The next example shows that Theorem 4.8 does not characterize the failure of
the containment.

######  Example 4.11.

Let $B$ denote the blocks of a Steiner quadruple system $SQS({8})=S(3,4,8)$ on
the vertex set $V=\\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8}\\}$,

$\begin{array}[]{rrl}B:=\\{&\\{x_{1},x_{2},x_{3},x_{4}\\},\\{x_{1},x_{2},x_{5},x_{6}\\},\\{x_{1},x_{2},x_{7},x_{8}\\},\\{x_{1},x_{3},x_{5},x_{7}\\},\\{x_{1},x_{3},x_{6},x_{8}\\},&\\\
&\\{x_{1},x_{4},x_{5},x_{8}\\},\\{x_{1},x_{4},x_{6},x_{7}\\},\\{x_{2},x_{3},x_{5},x_{8}\\},\\{x_{2},x_{3},x_{6},x_{7}\\},\\{x_{2},x_{4},x_{5},x_{7}\\},&\\\
&\\{x_{2},x_{4},x_{6},x_{8}\\},\\{x_{3},x_{4},x_{5},x_{6}\\},\\{x_{3},x_{4},x_{7},x_{8}\\},\\{x_{5},x_{6},x_{7},x_{8}\\}&\\}.\end{array}$

From Proposition 4.10, $B$ is not 3-coverable. Therefore Theorem 4.8 ensures
$J_{B}^{(4)}\not\subseteq J_{B}^{3}.$ However, one can check that, for
instance, $x_{1}x_{2}\cdots x_{7}\in J_{B}^{(3)}\setminus J_{B}^{2},$ so the
failure of the containment $J_{B}^{(3)}\subseteq J_{B}^{2}$ cannot be
motivated from Theorem 4.8.

##  5\. Conclusions

Several conjectures have been posed on the Containment problem, creating an
active area of current interests and ongoing investigations. In this paper, we
show that the Stable Harbourne Conjecture and the Stable Harbourne – Huneke
Conjecture hold for the defining ideal of a Complement of a Steiner
configuration of points in $\mathbb{P}^{n}_{k}$. Moreover, given a hypergraph
$H$, we also study the relation between its colourability and that the
Containment problem can fail for the cover ideal associated to $H$. We wish to
continue the study of Steiner configurations of points and their Complements,
since they are special subsets of star configurations whose Hilbert Function
is the same as sets of generic points while geometrically they are far of
being generic.

We end this section recalling some open questions that are still under
investigations and posing new ones.

We recall from [3], that from a combinatorial point of view, two Steiner
systems having the same parameters could have very different properties and
such differences effect the homological invariants. Using experiments with [1]
and [32] we ask:

######  Question 5.1.

Let $(V,B)$ be a Steiner system of type $S(t,n,v)$, and $X_{\mathcal{H},B}$
the associated Steiner configuration of points. Assume that the hyperplanes in
$\mathcal{H}$ are chosen generically. Do the Hilbert function and the graded
Betti numbers of $X_{\mathcal{H},B}$ only depend on $t,n,v$?

######  Question 5.2.

Let $(V,B)$ be a Steiner system of type $S(t,n,v)$, and $X_{\mathcal{H},B}$
the associated Steiner configuration of points. Assume that the hyperplanes in
$\mathcal{H}$ are chosen generically. Are the Hilbert function and the graded
Betti numbers of $X_{\mathcal{H},B}$ generic with respect to the Hilbert
function? (i.e. the same as a set of $|B|$ generic points in
$\mathbb{P}^{n}$?)

Given a hypergraph ${H}$, we also study the relation between its colourability
and the failure of the containment problem for the cover ideal associated to
$H$. We suggest the following

######  Question 5.3.

Can different types of colourings of a hypergraph give different answers to
the Containment problem and related conjectures?

We also thank one of the referees to point out [42, 43], where the author
studies graph partitioning (fragmentation criteria) that has many fields of
applications in engineering, especially in applied sciences as applied
chemistry and physics, computer science and automation, electronics and
telecommunication. See [2, 21, 49] just to cite some of them.

######  Question 5.4.

Can different types of colourings of hypergraphs give also different answers
to fragmentation criteria?

######  Acknowledgments 5.5.

This paper represents the second part of a project started in April 2018 when
E. Guardo and L. Milazzo met together with the idea of making an
interdisciplinary project between their own fields of research, such as
Combinatorics and Commmutative Algebra/Algebraic Geometry. L. Milazzo died in
2019 but all the authors recognize his contribution in Section 4. We thank M.
Gionfriddo and S. Milici for encouraging us to continue this project. We also
thank Zs. Tuza for his useful comments in the revised version of the paper.
Favacchio and Guardo’s work has been supported by the Università degli Studi
di Catania, ”Piano della Ricerca 2020/2022 Linea di intervento 2”. The first
three authors are members of GNSAGA of INdAM (Italy). The computer softwares
CoCoA [1] and Macaulay2 [32] were indispensable for all the computations we
made.

The authors thank all the referees for the useful suggestions/comments that
improved the first version of the paper.

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#  On the molecular nature of the $\Omega_{c}(3120)$ and its analogy with the
$\Omega(2012)$

Natsumi Ikeno<EMAIL_ADDRESS>Department of Agricultural, Life and
Environmental Sciences, Tottori University, Tottori 680-8551, Japan  Wei-Hong
Liang<EMAIL_ADDRESS>Department of Physics, Guangxi Normal University,
Guilin 541004, China  Guangxi Key Laboratory of Nuclear Physics and
Technology, Guangxi Normal University, Guilin 541004, China  Eulogio Oset
<EMAIL_ADDRESS>Department of Physics, Guangxi Normal University, Guilin
541004, China  Departamento de Física Teórica and IFIC, Centro Mixto
Universidad de Valencia - CSIC, Institutos de Investigación de Paterna, Aptdo.
22085, 46071 Valencia, Spain

###### Abstract

We make a study of the $\Omega_{c}(3120)$, one of the five $\Omega_{c}$ states
observed by the LHCb collaboration, which is well reproduced as a molecular
state from the $\Xi^{*}_{c}\bar{K}$ and $\Omega^{*}_{c}\eta$ channels mostly.
The state with $J^{P}=3/2^{-}$ decays to $\Xi_{c}\bar{K}$ in $D$-wave and we
include this decay channel in our approach, as well as the effect of the
$\Xi^{*}_{c}$ width. With all these ingredients, we determine the fraction of
the $\Omega_{c}(3120)$ width that goes into $\Xi_{c}\pi\bar{K}$, which could
be a measure of the $\Xi^{*}_{c}\bar{K}$ molecular component, but due to a
relatively big binding, compared to its analogous $\Omega(2012)$ state, we
find only a small fraction of about 3%, which makes this measurement difficult
with present statistics. As an alternative, we evaluate the scattering length
and effective range of the $\Xi^{*}_{c}\bar{K}$ and $\Omega^{*}_{c}\eta$
channels which together with the binding and width of the $\Omega_{c}(3120)$
state, could give us an answer to the issue of the compositeness of this state
when these magnitudes are determined experimentally, something feasible
nowadays, for instance, measuring correlation functions.

##  I Introduction

The $\Omega(2012)$ was reported by the Belle collaboration in Ref. Yelton  _et
al._ (2018) and estimulated much work from the theoretical side, some of it
from the quark model perspective, assuming it to be a low-lying $p$-wave
exited $3/2^{-}$state Xiao and Zhong (2018); Aliev  _et al._ (2018a, b);
Polyakov  _et al._ (2019); Liu  _et al._ (2020); Arifi  _et al._ (2022); Wang
_et al._ (2023), or from the molecular picture generated by the interaction of
the $\bar{K}\Xi^{*}(1530)$ and $\eta\Omega$ channels, decaying to $\bar{K}\Xi$
Valderrama (2018); Lin and Zou (2018); Pavao and Oset (2018); Huang  _et al._
(2018a); Lu  _et al._ (2020); Ikeno  _et al._ (2020); Liu  _et al._ (2021).
The molecular picture is reinforced by the fact that the state was predicted
before its experimental observation in Refs. Hofmann and Lutz (2006); Sarkar
_et al._ (2005). It also gets extra support since even using quark models a
molecular structure was claimed in Refs. Wang  _et al._ (2007, 2008). The use
of the Weinberg compositeness condition also led the authors of Ref. Gutsche
and Lyubovitskij (2020) to advocate the molecular character of the state.

In order to test the molecular nature of the $\Omega(2012)$ the Belle
collaboration conducted some tests, particularly looking at the decay into
$\bar{K}\pi\Xi$, a signal of the $\bar{K}\Xi^{*}(1530)$ component of the
state. A first experiment Jia  _et al._ (2019) reported a ratio smaller than
11.9% for the decay rate into $\bar{K}\pi\Xi$ versus $\bar{K}\Xi$, which might
challenge the molecular picture, although not necessarily, as explained in
Refs. Lu  _et al._ (2020); Ikeno  _et al._ (2020). However, a posterior
experiment Belle (2022) corrected this ratio and provided

$\mathcal{R}^{\Xi\pi\bar{K}}_{\Xi\bar{K}}=0.97\pm 0.24\pm 0.07,$ (1)

and it was concluded that this ratio is consistent with the molecular
interpretation of the $\Omega(2012)$ given in Refs. Valderrama (2018); Pavao
and Oset (2018); Huang  _et al._ (2018a); Gutsche and Lyubovitskij (2020).

The basic idea on the molecular picture is that the $\Omega(2012)$ is a
particular case of the interaction of the octet of pseudoscalar mesons with
the baryons of the decuplet of the $\Delta(1232)$ Hofmann and Lutz (2006);
Sarkar  _et al._ (2005).

Now we give a jump to the $\Omega_{c}$ states discovered by the LHCb
collaboration Aaij  _et al._ (2017). In this work five states were reported,
$\Omega_{c}(3000)$, $\Omega_{c}(3050)$, $\Omega_{c}(3066)$,
$\Omega_{c}(3090)$, $\Omega_{c}(3119)$. More recently two additional states
have been found, $\Omega_{c}(3185)$ and $\Omega_{c}(3327)$ Aaij  _et al._
(2023). The $\Omega_{c}$ states have also raised a wave of interest in the
theoretical community, and, actually, many predictions about them and related
states had been done, some from the quark model point of view Ebert  _et al._
(2008); Roberts and Pervin (2008); Garcilazo  _et al._ (2007); Migura  _et
al._ (2006); Ebert  _et al._ (2011); Valcarce  _et al._ (2008); Shah  _et al._
(2016); Vijande  _et al._ (2013); Yoshida  _et al._ (2015); Chen  _et al._
(2015, 2016); Chiladze and Falk (1997); Manohar and Georgi (1984); Agaev  _et
al._ (2017a, b); Wang (2017), and others from the molecular perspective
Hofmann and Lutz (2005); Jimenez-Tejero  _et al._ (2009); Romanets  _et al._
(2012); Xin  _et al._ (2023). After the experimental discovery, work followed
with several works trying to explain the states from the quark model
perspective Karliner and Rosner (2017); Wang  _et al._ (2017); Wang and Zhu
(2017); Chen and Liu (2017), pentaquark structures Yang and Ping (2018); Huang
_et al._ (2018b); Kim  _et al._ (2017); An and Chen (2017); Ali  _et al._
(2017); Anisovich  _et al._ (2017), lattice QCD Padmanath and Mathur (2017).

We follow here the molecular line and recall two independent works on the
issue, an update of Ref. Jimenez-Tejero  _et al._ (2009) done in Ref. Montaña
_et al._ (2018) to the light of the experimental results Aaij  _et al._ (2023)
and the work of Ref. Debastiani  _et al._ (2018). Both of them use as input
for the interaction the exchange of vector mesons based on the local hidden
gauge approach Bando  _et al._ (1985, 1988); Meissner (1988); Nagahiro  _et
al._ (2009) between several coupled channels. There is only one free
parameter, a cut off to regularize the loop functions, which is adjusted to
get the mass of one state. The mass of the other states and the widths are
then genuine prediction of the models. There is one difference between these
works. In Ref. Montaña  _et al._ (2018) SU(4) symmetry is used to obtain the
baryon wave functions, while in Ref. Debastiani  _et al._ (2018) the wave
functions involving $c$ quark are taken, as in Refs. Capstick and Isgur
(1986); Roberts and Pervin (2008), isolating the heavy quarks and imposing the
symmetry of the wave function on the light quarks. In Ref. Montaña  _et al._
(2018) two $\Omega_{c}$ states were reproduced using as coupled channels
pseudoscalar meson-baryon $1/2^{+}$ states, the $\Omega_{c}(3050)$ and
$\Omega_{c}(3090)$ with $J^{P}=1/2^{-}$. In Ref. Debastiani  _et al._ (2018)
the same states were obtained, with practically the same properties, but in
addition an extra state was obtained, the $\Omega(3120)$, with
$J^{P}=3/2^{-}$, coming from the interaction of pseudoscalar mesons with
baryons of $3/2^{+}$, concretely the $\Xi^{*}_{c}\bar{K}$,
$\Omega^{*}_{c}\eta$, and $\Xi^{*}D$, which were not considered in Ref.
Montaña  _et al._ (2018). The agreement of the results of Refs. Montaña  _et
al._ (2018) and Debastiani  _et al._ (2018) for the two states
$\Omega_{c}(3050)$ and $\Omega_{c}(3090)$ is not accidental. Even if SU(4)
symmetry is used in Ref. Montaña  _et al._ (2018), the important part of the
interaction comes from the exchange of light vector mesons in which case the
heavy quarks act as spectators and one is in practice projecting over SU(3),
and the two pictures coincide. The two works of Refs. Montaña  _et al._
(2018); Debastiani  _et al._ (2018) conclude that the $\Omega_{c}(3120)$ state
is not obtained as a $1/2^{-}$ state, but in Ref. Debastiani  _et al._ (2018)
the state is obtained as a $3/2^{-}$ state. As mentioned above, the coupled
channels in this case are $\Xi^{*}_{c}\bar{K}$, $\Omega_{c}^{*}\eta$, and
$\Xi^{*}D$, with threshold masses $3142\textrm{ MeV}$, $3314\textrm{ MeV}$ and
$3327\textrm{ MeV}$, respectively. The analogy with the $\Omega(2012)$ is
clear. In this latter case, the coupled channels are $\Xi^{*}\bar{K}$,
$\Omega\eta$. There is one extra channel, $\Xi^{*}D$, in the case of
$\Omega_{c}(3120)$, but this channel plays a minor role in Ref. Debastiani
_et al._ (2018). Indeed, the transition potential from $\Xi^{*}D$ to
$\Xi_{c}^{*}\bar{K}$, $\Omega_{c}^{*}\eta$ is suppressed compared to that of
$\Omega_{c}^{*}\eta$ to $\Xi^{*}_{c}\bar{K}$, the mass of the channel is more
than $200\textrm{ MeV}$ above the mass of the $\Omega_{c}(3120)$ and the
$\Omega_{c}(3120)$ wave function at the origin in coordinate space is almost
$20$ times smaller than that of the $\Xi^{*}_{c}\bar{K}$ channel. We shall
neglect this channel in our study, knowing that its small effect can be
incorporated by small changes in the cut off parameter, which is fitted to the
data. Thus, the analogy of the $\Omega(2012)$ and $\Omega_{c}(3120)$ is more
apparent. There is also another common feature: in both cases, the state is
not observed in any of the building blocks, instead the $\Omega(2012)$ is
observed in the $\Xi\bar{K}$ channel and the $\Omega_{c}(3120)$ in the
$\Xi_{c}\bar{K}$ one. Both channels appear in $D$-wave, they have not much
relevance in the structure of the $\Omega(2012)$ and $\Omega_{c}(3120)$
states, but they provide the largest source of the width of the states, which
is quite small, mostly due to the $D$-wave character of the decay.

In the present work, we retake the case of the $\Omega_{c}(3120)$ state and,
by analogy to what was done in Ref. Pavao and Oset (2018), we introduce the
$\Xi_{c}\bar{K}$ channel in $D$-wave in addition to the $\Xi^{*}_{c}\bar{K}$
and $\Omega^{*}_{c}\eta$ channels in $s$-wave, conduct a fit to the mass and
width of the $\Omega_{c}(3120)$ state and evaluate the partial decay widths
into $\Xi_{c}\bar{K}$ and $\pi\Xi_{c}\bar{K}$. The width was zero in Ref.
Debastiani  _et al._ (2018) since the $\Xi_{c}\bar{K}$ decay channel was not
included and the width of the $\Xi^{*}_{c}$ was also omitted. At the same
time, we evaluate the molecular probabilities of $\Xi_{c}^{*}\bar{K}$ and
$\Omega^{*}_{c}\eta$ and find about 63% and 10% respectively, similar to those
of the $\Xi^{*}\bar{K}$ and $\Omega\eta$ channels in the $\Omega(2012)$,
indicating a large molecular component of the $\Omega_{c}(3120)$ wave
function. The partial decay width into $\pi\Xi_{c}\bar{K}$ is found small, of
the order of 3%, much smaller than the corresponding $\pi\Xi\bar{K}$ one in
the $\Omega(2012)$ case, the reason being that the $\Xi^{*}_{c}\bar{K}$ is now
more bound than the $\Xi^{*}\bar{K}$ in the case of the $\Omega(2012)$ and the
width of the $\Xi^{*}_{c}$ is much smaller than the binding energy of the
$\Xi_{c}^{*}\bar{K}$ component. In order to find in experimental confirmation
for the molecular structure of the $\Omega_{c}(3120)$ state, we evaluate the
scattering length and effective range of the $\Xi^{*}_{c}\bar{K}$ and
$\Omega_{c}^{*}\eta$ channels, which can be accessible in the future measuring
correlation functions, and recall the works of Refs. Song  _et al._ (2022);
Dai  _et al._ (2023); Song  _et al._ (2023); Li  _et al._ (2023) where it is
found that the knowledge of the binding, scattering length, and effective
range of the coupled channels that build up a molecular state can determine
with a fair accuracy the molecular probability of the state.

##  II Formalism

We take the results from Ref. Debastiani  _et al._ (2018) for the transition
potential between the $\Xi_{c}^{*}\bar{K}$ and $\Omega^{*}_{c}\eta$ channels
and introduce the $\Xi_{c}\bar{K}$ $D$-wave channel phenomenologically, as
done in Ref. Pavao and Oset (2018). The potential is given by

$V=\begin{matrix}\begin{matrix}~{}\Xi_{c}^{*}\bar{K}&~{}~{}~{}\Omega^{*}_{c}\eta~{}~{}&~{}~{}~{}\Xi_{c}\bar{K}~{}~{}\end{matrix}&\\\\[7.11317pt]
\begin{pmatrix}F&\frac{4}{\sqrt{3}}\,F&~{}\alpha\,q^{\,2}_{\rm
on}~{}~{}\\\\[5.69054pt] \frac{4}{\sqrt{3}}\,F&0&\beta\,q^{\,2}_{\rm
on}\\\\[5.69054pt] ~{}~{}\alpha\,q^{\,2}_{\rm on}~{}&~{}\beta\,q^{\,2}_{\rm
on}~{}&0\\\\[7.11317pt]
\end{pmatrix}&\begin{matrix}~{}\Xi_{c}^{*}\bar{K}\\\\[5.69054pt]
~{}\Omega^{*}_{c}\eta\\\\[5.69054pt] ~{}\Xi_{c}\bar{K}\\\\[5.69054pt]
\end{matrix}\end{matrix}$ (2)

with

$F=-\dfrac{1}{4f^{2}}\;(k^{0}+k^{\prime\,0});~{}~{}~{}f=93\;{\rm MeV},$ (3)
$q_{\rm
on}=\dfrac{\lambda^{1/2}(s,m^{2}_{\bar{K}},m^{2}_{\Xi_{c}})}{2\,\sqrt{s}},$
(4)

and $k^{0},k^{\prime\,0}$ the energies of the initial, final meson. In Eq. (2)
$\alpha,\beta$ are unknown parameters, to be fitted to the width of the
$\Omega_{c}(3120)$ state.

The scattering matrix between these three channels is given by

$T=[1-VG]^{-1}\,V,$ (5)

where $G$ is the diagonal matrix of loop function for the meson-baryon states,
${\rm
diag}(G_{\Xi^{*}_{c}\bar{K}},G_{\Omega^{*}_{c}\eta},G_{\Xi_{c}\bar{K}})$, with

$\displaystyle G_{i}(\sqrt{s})$ $\displaystyle=$
$\displaystyle\int_{|\vec{q}\,|<q_{\rm max}}\dfrac{{\rm
d}^{3}q}{(2\pi)^{3}}\;\dfrac{1}{2\,\omega_{i}(q)}\;$ (6)
$\displaystyle\times\dfrac{M_{i}}{E_{i}(q)}\;\dfrac{1}{\sqrt{s}-\omega_{i}(q)-E_{i}(q)+i\varepsilon},$

for the $S$-wave $\Xi_{c}^{*}\bar{K}$ and $\Omega^{*}_{c}\eta$ channel, and

$\displaystyle G_{\Xi_{c}\bar{K}}(\sqrt{s})$ $\displaystyle=$
$\displaystyle\int_{|\vec{q}\,|<q_{\rm max}}\dfrac{{\rm
d}^{3}q}{(2\pi)^{3}}\;\left(\dfrac{q}{q_{\rm
on}}\right)^{4}\,\dfrac{1}{2\,\omega_{\bar{K}}(q)}$ (7)
$\displaystyle\times\dfrac{M_{\Xi_{c}}}{E_{\Xi_{c}}(q)}\;\dfrac{1}{\sqrt{s}-\omega_{\bar{K}}(q)-E_{\Xi_{c}}(q)+i\varepsilon},$

for the $D$-wave $\Xi_{c}\bar{K}$ channel, where
$\omega_{i}(q)=\sqrt{m_{i}^{2}+\vec{q}^{\;2}}$,
$E_{i}(q)=\sqrt{M_{i}^{2}+\vec{q}^{\;2}}$, with $m_{i},M_{i}$ the masses of
meson and baryon in channel $i$. For the parameter $q_{\rm max}$, we shall
take a value around $650\,\textrm{ MeV}$, as in Ref. Debastiani  _et al._
(2018), fine tuned to get the right energy of the state. The parameters
$\alpha,\beta$ will be chosen to get the width of the $\Omega_{c}(3120)$
state. We have from Ref. Aaij  _et al._ (2017) 111In Ref. Aaij  _et al._
(2023) the $\Omega_{c}(3120)$ width is changed to $0.60\pm 0.63\textrm{ MeV}$,
which overlaps with the results of Eq. (8). We carry our analysis using the
data of Eq. (8). The conclusions of the paper do not change from using one or
the other data.,

$\begin{split}&M_{\Omega_{c}(3120)}=3119.1\pm 0.3\pm
0.9^{+0.3}_{-0.5}\;\textrm{ MeV},\\\\[5.69054pt]
&\Gamma_{\Omega_{c}(3120)}=1.1\pm 0.8\pm 0.4\;\textrm{ MeV}.\end{split}$ (8)

In order to see the relevance of the $\Xi_{c}^{*}$ decay width in the width of
the $\Omega_{c}(3120)$ state, we calculate the $T$ matrix including the
$\Xi_{c}^{*}$ selfenergy in the loop. For this we follow the prescription
given in Ref. Dai  _et al._ (2022), recommended when the width of the particle
is small, instead of the popular convolution method used in Ref. Pavao and
Oset (2018), substituting the $G$ function for the $\Xi_{c}^{*}\bar{K}$
channel by

$\displaystyle\tilde{G}_{\Xi^{*}_{c}\bar{K}}(\sqrt{s})$ $\displaystyle=$
$\displaystyle\int_{|\vec{q}\,|<q_{\rm max}}\dfrac{{\rm
d}^{3}q}{(2\pi)^{3}}\;\dfrac{1}{2\,\omega_{\bar{K}}(q)}\;\dfrac{M_{\Xi_{c}^{*}}}{E_{\Xi_{c}^{*}}(q)}$
$\displaystyle\times$
$\displaystyle\dfrac{1}{\sqrt{s}-\omega_{\bar{K}}(q)-E_{\Xi_{c}^{*}}(q)+i\frac{\sqrt{s^{\prime}}}{2E_{\Xi_{c}^{*}}(q)}\Gamma_{\Xi_{c}^{*}}(\sqrt{s^{\prime}})},$

where

$s^{\prime}=[\sqrt{s}-\omega_{\bar{K}}(q)]^{2}-\vec{q}^{\,2},$ (10)

and

$\Gamma_{\Xi_{c}^{*}}(M_{\rm inv})=\dfrac{M_{\Xi_{c}^{*}}}{M_{\rm
inv}}\,\left(\dfrac{q^{\prime}}{q^{\prime}_{\rm on}}\right)^{3}\,\Gamma_{\rm
on}\;\theta(M_{\rm inv}-m_{\pi}-M_{\Xi_{c}}),$ (11)

with $M_{\rm inv}=\sqrt{s^{\prime}}$ the invariant mass of $\pi\Xi_{c}$,
$\Gamma_{\rm on}=2.25\textrm{ MeV}$ the average width of $\Xi_{c}^{*+}$ and
$\Xi_{c}^{*0}$ from PDG, and

$\begin{split}q^{\prime}_{\rm
on}&=\dfrac{\lambda^{1/2}(M^{2}_{\Xi_{c}^{*}},m^{2}_{\pi},M^{2}_{\Xi_{c}})}{2\,M_{\Xi_{c}^{*}}},\\\\[5.69054pt]
q^{\prime}&=\dfrac{\lambda^{1/2}(M^{2}_{\rm
inv},m^{2}_{\pi},M^{2}_{\Xi_{c}})}{2\,M_{\rm inv}},\end{split}$ (12)

We also need to evaluate the couplings at the pole. For that we must use $G$
in the second Riemann sheet

$G^{(II)}_{i}(\sqrt{s})=G_{i}(\sqrt{s})+\left\\{\begin{array}[]{ll}0,&~{}~{}{\rm
for}\;{\rm Re}(\sqrt{s})<\sqrt{s}_{{\rm th},i}\\\\[5.69054pt]
i\dfrac{M_{i}\,k}{2\pi\sqrt{s}},&~{}~{}{\rm for}\;{\rm
Re}(\sqrt{s})\geqslant\sqrt{s}_{{\rm th},i},\end{array}\right.$ (13)

with $\sqrt{s}_{{\rm th},i}$ the threshold mass of channel $i$, and

$k=\dfrac{\lambda^{1/2}(s,m_{i}^{2},M_{i}^{2})}{2\sqrt{s}},~{}~{}~{}~{}{\rm
Im}(k)>0.$ (14)

Only the $\Xi_{c}\bar{K}$ channel goes to the second Riemann sheet at the
pole.

The couplings are defined at the pole as

$g_{i}g_{j}=\lim_{\sqrt{s}\to\sqrt{s_{p}}}(\sqrt{s}-\sqrt{s_{p}})\;T_{ij},$
(15)

with $\sqrt{s_{\rm p}}$ the energy of the pole, which allows to have the
relative phase of one coupling to another, with one of them chosen with
arbitrary phase. We choose $g_{\Xi_{c}^{*}\bar{K}}$ positive. We evaluate the
couplings neglecting the width of the $\Xi_{c}^{*}$.

Once the couplings are evaluated we can calculate the molecular probabilities
of the $S$-wave channels as

$\mathcal{P}_{i}=-g_{i}^{2}\;\dfrac{\partial G_{i}}{\partial\sqrt{s}}.$ (16)

Finally, in order to evaluate the partial decay width into the
$\Xi_{c}\bar{K}\pi$ channel, in analogy to Ref. Ikeno  _et al._ (2020), we
evaluate the amplitude for the diagram of Fig. 1 as

Figure 1: Mechanism for $\Omega_{c}(3120)$ to decay into $\Xi_{c}\bar{K}\pi$
via primary decay into $\Xi_{c}^{*}\bar{K}$.

$t_{\Omega_{c}\to\pi\bar{K}\Xi_{c}}=g_{\Omega_{c},\bar{K}\Xi^{*}_{c}}\;\dfrac{1}{M_{\rm
inv}(\pi\Xi_{c})-M_{\Xi_{c}^{*}}-i\Gamma_{\Xi_{c}^{*}}/2}\;g_{\Xi_{c}^{*},\pi\Xi_{c}}\;\tilde{p}_{\pi},$
(17)

where

$\tilde{p}_{\pi}=\dfrac{\lambda^{1/2}(M^{2}_{\rm
inv}(\pi\Xi_{c}),m_{\pi}^{2},M^{2}_{\Xi_{c}})}{2\,M_{\rm inv}(\pi\Xi_{c})},$
(18)

with $g_{\Xi_{c}^{*},\pi\Xi_{c}}$ given using $\Gamma_{\Xi_{c}^{*}}$ as

$\Gamma_{\Xi_{c}^{*}}=\dfrac{1}{2\,\pi}\;\dfrac{M_{\Xi_{c}}}{M_{\Xi_{c}^{*}}}\;g^{2}_{\Xi_{c}^{*},\pi\Xi_{c}}\,\tilde{p}^{3}_{\pi},$
(19)

with $\Gamma_{\Xi_{c}^{*}}$ given by Eq. (11).

The mass distribution for the $3$ body decay of the mechanism of Fig. 1 is
given by

$\dfrac{{\rm d}\Gamma_{\Omega_{c}}}{{\rm d}M_{\rm
inv}(\pi\Xi_{c})}=\dfrac{1}{(2\pi)^{3}}\;\dfrac{M_{\Xi_{c}}}{M_{\Omega_{c}}}\;p_{\bar{K}}\tilde{p}_{\pi}\;|t_{\Omega_{c}\to\pi\bar{K}\Xi_{c}}|^{2},$
(20)

with

$p_{\bar{K}}=\dfrac{\lambda^{1/2}(M_{\Omega_{c}}^{2},m^{2}_{\bar{K}},M^{2}_{\rm
inv}(\pi\Xi_{c}))}{2\,M_{\Omega_{c}}},$ (21)

and the integration over $M_{\rm inv}(\pi\Xi_{c})$ produces the width for
$\Omega_{c}(3120)$ decay to $\pi\bar{K}\Xi_{c}$.

The width for $\Omega_{c}(3120)\to\Xi_{c}\bar{K}$ decay can be obtained from

$\Gamma=\dfrac{1}{2\pi}\;\dfrac{M_{\Xi_{c}}}{M_{\Omega_{c}}}\;g^{2}_{\Omega_{c},\Xi_{c}\bar{K}}\;p^{\prime}_{\bar{K}},$
(22)

with $p^{\prime}_{\bar{K}}$ the $\bar{K}$ momentum for
$\Omega_{c}\to\Xi_{c}\bar{K}$ decay in the $\Omega_{c}$ rest frame.

##  III Results

We conduct a fit to the position and width of the $\Omega_{c}(3120)$ state
using Eq. (5) with the potential of Eq. (2), the $G$ functions of Eqs. (6) and
(7) using the second Riemann sheet of Eq. (13). We get a good fit to the data
with the parameters

$\begin{split}q_{\rm max}&=674.6\,\textrm{ MeV},\\\ \alpha&=2.6\times
10^{-8}\,\textrm{ MeV}^{-3},\\\ \beta&=2.0\times 10^{-9}\,\textrm{
MeV}^{-3}.\end{split}$ (23)

The pole position appears at

$(3119.13+i\,0.54)\,\textrm{ MeV}$ (24)

implying a width of $1.08\textrm{ MeV}$ in agreement with the central values
of the experiment.

Now we add the $\Xi_{c}^{*}$ selfenergy in Eq. (II) and look at the
$T_{\Xi_{c}^{*}\bar{K}}$ amplitude in the real energy axis. In Fig. 2

Figure 2: $|T_{\Xi_{c}^{*}\bar{K}}|^{2}$ as a function of $\sqrt{s}$ in the
cases with $\Gamma_{\Xi_{c}^{*}}$ and without $\Gamma_{\Xi_{c}^{*}}$,
respectively.

we show the results for $|T_{\Xi_{c}^{*}\bar{K}}|^{2}$ with and without the
$\Xi_{c}^{*}$ width. As we see, the results are very similar, with the masses
and widths practically the same. This means that, unlike in the case of the
$\Omega(2012)$, where the difference between the widths in the analogous
cases, with $\Xi^{*}$ dressed or without, allowed us to determine the decay
width of $\Omega(2012)\to\Xi^{*}\bar{K}\to\Xi\pi\bar{K}$, in the present case
we cannot determine the
$\Omega_{c}(3120)\to\Xi_{c}^{*}\bar{K}\to\Xi_{c}\pi\bar{K}$ with precision
using this procedure. Hence, we use the more accurate one of evaluating the
width using explicitly the mechanism of Fig. 1. For this we need the coupling
of $\Omega_{c}(3120)\to\Xi_{c}^{*}\bar{K}$, which we address below. The
couplings are evaluated at the pole using the $G$ functions in the second
Riemann sheet through Eq. (13) and we find the results of Table 1.

Table 1: The couplings $g_{i}$ of $\Omega_{c}(3120)$ to different channels, $g_{i}G_{i}$ and the molecular probabilities for the $S$-wave channels. | $\Xi_{c}^{*}\bar{K}$ | $\Omega^{*}_{c}\eta$ | $\Xi_{c}\bar{K}$
---|---|---|---
$g_{i}$ | $2.06-i0.02$ | $2.09-i0.01$ | $-0.138$
$g_{i}G_{i}$ | $-36.77+i0.17$ | $-17.64+i0.06$ |
$-g_{i}^{2}\;\dfrac{\partial G_{i}}{\partial\sqrt{s}}$ | $0.63$ | $0.10$ |

We also show there the values of $g_{i}G_{i}$ for the $S$-wave channels, which
are the wave functions at the origin in coordinate space Gamermann  _et al._
(2010). As we can see, the wave function is dominated by the
$\Xi_{c}^{*}\bar{K}$ component. We ignore now the tiny imaginary part of the
couplings and calculate the probability of the
$\Xi_{c}^{*}\bar{K},\Omega^{*}_{c}\eta$ channels through
$-g_{i}^{2}\;\dfrac{\partial G_{i}}{\partial\sqrt{s}}$ Gamermann  _et al._
(2010); Hyodo (2013). We see again that the $\Xi_{c}^{*}\bar{K}$ has the
largest probability of around $63\%$ and the $\Omega_{c}^{*}\eta$ around
$10\%$, hence we have a largely molecular state.

Once we have calculated the couplings we are in a position to evaluate the
$\Omega_{c}(3120)\to\Xi_{c}^{*}\bar{K}\to\Xi_{c}\pi\bar{K}$ for the mechanism
of Fig. 1 through Eqs. (17), (19) and (20). We get

$\Gamma_{\Omega_{c}\to\Xi_{c}\pi\bar{K}}=0.03\textrm{ MeV},$ (25)

while through Eq. (22) we would get

$\Gamma_{\Omega_{c}\to\Xi_{c}\bar{K}}=0.90\textrm{ MeV},$ (26)

the sum of them giving $\Gamma_{\Omega_{c}}\sim 1\textrm{ MeV}$ as the central
value of the experiment, Eq. (8). The fraction of $\Omega_{c}(3120)$ decay to
$\Xi_{c}\pi\bar{K}$ is much smaller than the $\Xi\pi\bar{K}$ in the case of
the $\Omega(2012)$. However, the molecular probabilities of the
$\Xi^{*}_{c}\bar{K}$ and $\Xi^{*}\bar{K}$ in both cases are very similar. The
differences stem from the different bindings. In the case of the
$\Omega(2012)$ the diagonal terms in the $V$ matrix of Eq. (2) were zero,
while here they are finite and negative, indicating extra attraction that
reverts into a much bigger binding of about $20\textrm{ MeV}$. This has as a
consequence that the decay of the $\Xi_{c}^{*}\bar{K}$ bound component into
$\Xi^{*}_{c}\bar{K}\to\Xi_{c}\pi\bar{K}$ is more difficult (technically, the
$\Xi_{c}^{*}$ in the diagram of Fig. 1 is more off shell than the $\Xi^{*}$ in
the analogous diagram for $\Omega(2012)$ decay). While the experimental
determination of a fraction of $0.03\textrm{ MeV}$ is certainly challenging
given the present experimental errors in Eq. (8), we look into other
experimental tests that can lead us to determine the nature of that state. In
Refs. Song  _et al._ (2022); Dai  _et al._ (2023); Song  _et al._ (2023); Li
_et al._ (2023) it was discussed in detail how the knowledge of the binding,
scattering length and effective range of the coupled channels provided
excellent information on the molecular compositeness of the states. The
formalism, improved substantially over the possible application of Weinberg
formulas Weinberg (1965), where that range is ignored and leads to unrealistic
results in most cases. Anticipating that these magnitudes can be determined,
for instance, using correlation functions Tolos and Fabbietti (2020), we
determine here the scattering length and effective range of the
$\Xi_{c}^{*}\bar{K}$ and $\Omega_{c}^{*}\eta$ channels using the formulas of
Ref. Molina  _et al._ (2023),

$-\dfrac{1}{a_{j}}=\left.-\dfrac{8\pi\,\sqrt{s}}{2\,M_{j}}\;(T_{jj})^{-1}\right|_{\sqrt{s_{{\rm
th}}},j},$ (27)
$r_{0,j}=\dfrac{1}{\mu_{j}}\;\dfrac{\partial}{\partial\sqrt{s}}\,\left[-\dfrac{8\pi\,\sqrt{s}}{2\,M_{j}}\;(T_{jj})^{-1}+ik_{j}\right]_{\sqrt{s_{{\rm
th}}},j},$ (28)

with $\mu_{j}$ the reduced mass of channel $j$ and $k_{j}$ the momentum of a
particle of the pair in their rest frame. The results obtained are shown in
Table 2.

Table 2: Scattering length $a_{j}$ and effective range $r_{0,j}$ of the $\Xi_{c}^{*}\bar{K}$ and $\Omega_{c}^{*}\eta$ channels. [in units of fm] | $a_{j}$ | $r_{0,j}$
---|---|---
$\Xi_{c}^{*}\bar{K}$ | $1.45-i0.07$ | $-0.08-i0.01$
$\Omega^{*}_{c}\eta$ | $0.44-i0.09$ | $0.26+i0.01$

We can see that the values of $a_{j}$ and $r_{0,j}$ are mostly real, because
the main source of decay is to $\Xi_{c}\bar{K}$, which we saw leads to a very
small width of the $\Omega_{c}(3120)$ state.

As we can see, we are providing new magnitudes, which are additional to the
binding and width of the $\Omega_{c}(3120)$ state. We believe that these
magnitudes should be sufficient to determine the nature of the
$\Omega_{c}(3120)$ state and the precent work should give an incentive to do
experimental work in this direction.

##  IV Conclusions

We made a thorough study of the $\Omega_{c}(3120)$ state, reported in the LHCb
experiment in Ref. Aaij  _et al._ (2017), which was shown in Ref. Debastiani
_et al._ (2018) to be well reproduced as a molecular state from the
interaction of mostly the $\Xi^{*}_{c}\bar{K}$ and $\Omega_{c}^{*}\eta$
channels. The state has $J^{P}=\frac{3}{2}^{-}$ and is distinct from
$\Omega_{c}(3050),\Omega_{c}(3090)(\frac{1}{2}^{-})$ which are generated from
the pseudoscalar-baryon $(\frac{1}{2}^{+})$ interaction in Refs. Montaña  _et
al._ (2018); Debastiani  _et al._ (2018). In addition to the
$\Xi^{*}_{c}\bar{K},\Omega_{c}^{*}\eta$ channels used in Ref. Debastiani  _et
al._ (2018), we include now the $\Xi_{c}\bar{K}$ decay channel, where the
state was observed, which appears in $D$-wave. With the consideration of the
$\Xi_{c}\bar{K}$ channel, we can now obtain the width of the state, which was
not evaluated in Ref. Debastiani  _et al._ (2018). Then we also evaluate the
$\Xi_{c}\pi\bar{K}$ decay width, which could be a measure of the
$\Xi^{*}_{c}\bar{K}$ component of the $\Omega_{c}(3120)$ state, but unlike in
the analogous $\Omega(2012)$ state where the $\Xi\pi\bar{K}$ decay channel is
sizeable and has been measured, providing support for the $\Xi^{*}\bar{K}$
molecular picture of the $\Omega(2012)$ state, here we obtain a very small
fraction of $\Xi_{c}\pi\bar{K}$, because the $\Omega_{c}(3120)$ state is more
bound than the $\Omega(2012)$. In view of this, we determined the scattering
length and effective range of the $\Xi^{*}_{c}\bar{K}$ and
$\Omega_{c}^{*}\eta$ channels and made a call for the experimental
determination of these magnitudes, accessible for instance measuring
correlation functions, which can be instrumental in the determination of the
nature of the $\Omega_{c}(3120)$ state.

## Acknowledgments

One of us, N. I. wishes to acknowledge the hospitality of Guangxi Normal
University, where part of the work was carried out. This work is partly
supported by the National Natural Science Foundation of China under Grant No.
11975083 and No. 12365019, and by the Central Government Guidance Funds for
Local Scientific and Technological Development, China (No. Guike ZY22096024).
This work is also partly supported by the Spanish Ministerio de Economia y
Competitividad (MINECO) and European FEDER funds under Contracts No.
FIS2017-84038-C2-1-P B, PID2020-112777GB-I00, and by Generalitat Valenciana
under contract PROMETEO/2020/023. This project has received funding from the
European Union Horizon 2020 research and innovation programme under the
program H2020-INFRAIA-2018-1, grant agreement No. 824093 of the STRONG-2020
project.

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