New phenomena in the containment problem for simplicial arrangements

TL;DR

This paper investigates the containment problem for ideals of intersection points in simplicial line arrangements, revealing new cases where the containment $I^{(3)} ot\subseteq I^2$ holds or fails, involving irreducible curves of higher degree.

Contribution

It presents novel examples of simplicial arrangements where the containment relation behaves unexpectedly, especially involving irreducible higher-degree curves, expanding understanding of the containment problem.

Findings

Containment $I^{(3)} ot\\subseteq I^2$ can occur in simplicial arrangements.

All points are defined over \\Q$, involving all intersection points.

Examples involve irreducible higher-degree curves, unlike previous cases.

Abstract

In this note we consider two simplicial arrangements of lines and ideals $I$ of intersection points of these lines. There are $127$ intersection points in both cases and the numbers $t_i$ of points lying on exactly $i$ configuration lines (points of multiplicity $i$) coincide. We show that in one of these examples the containment $I^{(3)} \subseteq I^2$ holds, whereas it fails in the other. We also show that the containment fails for a subarrrangement of $21$ lines. The interest in the containment relation between $I^{(3)}$ and $I^2$ for ideals of points in $\P^2$ is motivated by a question posted by Huneke around $2000$. Configurations of points with $I^{(3)} \not\subseteq I^2$ are quite rare. Our example reveals two particular features: All points are defined over $\Q$ and all intersection points of lines are involved. In examples studied by now only points with multiplicity $i\geq 3$ were considered. The novelty of our arrangements lies in the geometry of the element in $I^{(3)}$ which witness the noncontainment in $I^2$. In all previous examples such an element was a product of linear forms. Now, in both cases there is an irreducible curve of higher degree involved.

Tables

Table 1. Table 1:   The equations of lines of 𝒜 ​ ( 31 , 3 ) 𝒜 31 3 \mathcal{A}(31,3) .

$𝒜(31,3)$
$x+y+iz=0$,	for $i\in \{0,2,3,4,5,6,8\}$,
$2x-y+jz=0$,	for $j\in \{4,6,7,8,9,10,12\}$,
$3x+kz=0$,	for $k\in \{8,10,11,12,13,14,16\}$,
$x-2y+lz=0$,	for $l\in \{2,4,6\}$,
$4x+y+mz=0$,	for $m\in \{14,16,18\}$,
$5x-y+nz=0$,	for $n\in \{18,20,22\}$,
$z=0$

Table 2. Table 2:   The coordinates of points of 𝒜 ​ ( 31 , 3 ) 𝒜 31 3 \mathcal{A}(31,3) .

$𝒜(31,3)$
double	(2,-2,-3),	(6,-6,-1),	(7,-7,-3),	(13,-13,-3),	(3,-1,-1),	(13,-7,-3),
	(5,1,-1),	(11,7,-3),	(22,2,-3),	(2,6,-1),	(17,7,-3),	(11,13,-3),
	(7,2,-3),	(3,0,-1),	(14,-5,-3),	(16,-7,-3),	(7,-1,-2),	(23,-5,-6),
	(25,-7,-6),	(5,0,-1),	(17,-2,-3),	(8,7,-3),	(10,5,-3),	(23,7,-6),
	(25,5,-6),	(9,1,-2),	(-22,-8,3),	(-22,-5,6),	(-26,-7,6),	(-22,1,6),
	(-26,-1,6),	(-2,8,3),	(-22,7,6),	(-26,5,6),	(-6,-8,1),	(-13,-14,3),
	(-13,-8,3),	(-11,8,3),	(-2,8,1),	(-11,14,3),	(-8,-22,3),	(8,-11,-3),
	(8,-26,-3),	(10,-7,-3),	(14,7,-3),	(-16,22,3),	(16,11,-3),	(16,26,-3),
	(7,0,-2),	(23,-8,-6),	(23,4,-6),	(25,-4,-6),	(25,8,-6),	(9,0,-2)
triple	(4,-4,-3),	(2,-2,-1),	(8,-8,-3),	(4,-4,-1),	(14,-14,-3),	(16,-16,-3),
	(3,-3,-1),	(11,-11,-3),	(2,0,-1),	(16,-10,-3),	(8,4,-3),	(16,-4,-3),
	(6,0,-1),	(8,10,-3),	(6,2,-1),	(20,4,-3),	(4,4,-1),	(16,8,-3),
	(8,16,-3),	(10,14,-3),	(5,3,-1),	(13,11,-3),	(8,1,-3),	(5,-2,-1),
	(16,-1,-3),	(3,2,-1),	(-16,-5,3),	(-34,-8,9),	(-38,-10,9),	(-32,2,9),
	(-40,-2,9),	(-8,5,3),	(-34,10,9),	(-38,8,9),	(-14,-16,3),	(-16,-20,3),
	(-11,-10,3),	(-16,-14,3),	(-8,14,3),	(-8,20,3),	(-10,16,3),	(-13,10,3)
quadruple	(2,1,0),	(-1,4,0),	(1,5,0),	(10,-10,-3),	(8,-2,-3),	(14,-8,-3),
	(11,-5,-3),	(11,1,-3),	(13,-1,-3),	(16,2,-3),	(10,8,-3),	(13,5,-3),
	(14,10,-3),	(10,-1,-3),	(4,-1,-1),	(11,-2,-3),	(13,-4,-3),	(4,1,-1),
	(14,1,-3),	(11,4,-3),	(13,2,-3)
quintuple	(10,-4,-3),	(4,-2,-1),	(10,2,-3),	(14,-2,-3),	(14,4,-3),	(4,2,-1)
sextuple	(4,0,-1)
octuple	(-1,1,0),	(1,2,0),	(0,1,0)

Equations

$$I^{(m)}\subset I^r$$

$$I^{(m)}\subseteq I^r$$

$$I^{(3)}\subseteq I^2$$

$$f=l_1\cdot ...\cdot l_s\in I^{(3)}$$

$$I^{(3)}⊈I^2.$$

$$I^{(m)}=\mathbb{K}[{\mathbb{P}}^N]\cap \left(\bigcap _{\mathfrak{p}\in \mathrm{Ass}(I)}(I^m{)}_{\mathfrak{p}}\right),$$

$$I^{(m)}=\{f\in \mathbb{K}[x_0,\dots ,x_N]:\text{f vanishes to order }\ge \text{m in all points }P\in V(I)\}.$$

$$I^{(m)}=\bigcap _{i=1}^{t}I(P_i{)}^m.$$

$$t(\mathcal{A})=(t_2(\mathcal{A}),t_3(\mathcal{A}),\dots ,t_s(\mathcal{A})).$$

$$t_2=54,t_3=42,t_4=21,t_5=6,t_6=1,t_8=3$$

$$x\pm auz=0,y\pm bz=0,$$

$$F_{10}=x^7{y}^3-x^{7}y{z}^2-\frac{63}{4}{x}^5{y}^3{z}^2+\frac{63}{4}{x}^{5}y{z}^4+\frac{189}{4}{x}^3{y}^3{z}^4-\frac{189}{4}{x}^{3}y{z}^6-27x{y}^3{z}^6+27xy{z}^8$$

$$x\pm auz=0,y\pm bz=0,$$

$$f_1=z,f_2=x^2+y^2,f_3=11x^6+15x^4{y}^2+45x^2{y}^4+9y^6.$$

$$12-3-4-4=1>0$$

$$\frac{2093688}{17}{f}_1^{12}-\frac{9398511}{34}{f}_1^{10}{f}_2+\frac{2995218}{17}{f}_1^8{f}_2^2-\frac{64485153}{1088}{f}_1^6{f}_2^3$$

$$+\frac{18708003}{4352}{f}_1^4{f}_2^4+\frac{1258659}{4352}{f}_1^2{f}_2^5-\frac{493695}{4352}{f}_2^6+\frac{2121309}{1088}{f}_1^6{f}_3$$

$$-\frac{402561}{4352}{f}_1^4{f}_2{f}_3-\frac{158697}{4352}{f}_1^2{f}_2^2{f}_3+\frac{2619}{128}{f}_2^3{f}_3-\frac{3979}{4352}{f}_3^2,$$

$$g=\left(\genfrac{}{}{0ex}{}{12-1}{2}\right)-12\left(\genfrac{}{}{0ex}{}{2}{2}\right)=43.$$

$$x\pm auz=0,y\pm bz=0,$$

$$t_2=54,t_3=42,t_4=21,t_5=6,t_6=1,t_8=3.$$

$$I_{\mathcal{A}(31,2)}^{(3)}\subseteq I_{\mathcal{A}(31,2)}^2.$$

$$9-4-4=1>0.$$

$$-\frac{38320128}{107}{f}_1^{10}+\frac{80453952}{107}{f}_1^8{f}_2-\frac{42393996}{4107}{f}_1^6{f}_2^2+\frac{50759217}{214}{f}_1^4{f}_2^3$$

$$-\frac{20519091}{856}{f}_1^2{f}_2^4+\frac{67086}{107}{f}_2^5-\frac{3811059}{214}{f}_1^4{f}_3+\frac{1778227}{856}{f}_1^2{f}_2{f}_3-\frac{6089}{107}{f}_2^2{f}_3.$$

$$\chi ({\mathcal{B}}_{21},t)=1-20t+141t^2$$

$$I^{(3)}⊈I^2$$

Full text

Keywords simplicial arrangements, arrangements of lines, containment problem, symbolic powers

Mathematics Subject Classification (2010) 14N20, 13A15, 13F20, 52C35

1 Introduction

The following problem has attracted a lot of attention in the last two decades.

**Containment Problem. **

Determine all pairs of positive integers $(m,r)$ such that the containment

[TABLE]

holds for all homogeneous ideals $I\subseteq\mathbb{K}[x_{0},\ldots,x_{N}]$ in the ring of polynomials over a field $\mathbb{K}$.

In $2000$ Ein, Lazarsfeld and Smith [8] in characteristic zero and Hochster and Huneke [14] in positive characteristic discovered that the containment (1) holds provided $m\geq Nr$.

Theorem 1.1** (Ein-Lazarsfeld-Smith, Hochster-Huneke).**

Let $I\subseteq\mathbb{K}[x_{0},\dots,x_{N}]$ be a homogeneous ideal. Then there is

[TABLE]

for all $m\geq Nr$.

This ground-breaking result prompted a natural question about the optimality of the bound $m\geq Nr$. A number of examples suggested the following conjectural improvement (see [1, Conjecture $8.4.2$], or [12, Conjecture $4.1.1$], or [2, Conjecture $1.1$])

Conjecture 1.2**.**

Let $I$ be a homogeneous ideal. Ten

[TABLE]

for all $m\geq Nr-(N-1)$.

The first non-trivial case is $N=2$ and $r=2$. Then there is always $I^{(4)}\subset I^{2}$ and it is very easy to give examples with $I^{(2)}\nsubseteq I^{2}$. Huneke asked around $2000$ if

[TABLE]

holds for all ideals defining points in $\mathbb{P}^{2}$.

This is not the case. The first non-containment example was announced in [7] and soon after additional non-containment examples were discovered and described in [4], [13], [19], [15], [16], [10].

Such examples are quite rare and they all follow the same pattern, in particular they are related to line arrangements. More precisely, let $\mathcal{L}=\{L_{1},\ldots,L_{s}\}$ be an arrangement of lines in $\mathbb{P}^{2}$ and let $\mathcal{P}=\{P_{1},\ldots,P_{t}\}$ be the set of all points contained in at least $2$ lines from $\mathcal{L}$. Let $I$ be the ideal of those points which are contained in at least $3$ lines. By the Zariski-Nagata Theorem [9, Theorem 3.14] the product

[TABLE]

and sometimes it happens that $f\not\in I^{2}$ (here $l_{i}$ is the equation of $L_{i}$).

The novelty of our non-containment example is that whereas the ideal of points is determined by lines, it is not their product which sits in $I^{(3)}\setminus I^{2}.$ More precisely, our main results are the following

Theorem A**.**

There exists an arrangement of $31$ lines which intersect in the total of $127$ points such that for the ideal $I$ of these $127$ points there is

[TABLE]

Moreover, there is an element $f$ of degree $33$ in $I^{(3)}$, which is not contained in $I^{2}$ and which is a product of

• $\bullet$

$21$ of arrangement lines and

• $\bullet$

an irreducible curve of degree $12$.

Theorem B**.**

There exists an arrangement of $21$ lines which intersect in the total of $115$ points such that for the ideal $I$ of these $115$ points there is

[TABLE]

Moreover, there is an element $f$ of degree $31$ in $I^{(3)}$, which is not contained in $I^{2}$ and which is a product of

• $\bullet$

all arrangement lines and

• $\bullet$

an irreducible curve of degree $10$.

A number of elementary but tedious calculations is omitted. Instead we provide a Singular script [17] which provides easy verification of our claims.

2 Preliminaries

In this section we define the basic object we are interested in and state the central conjecture in the field, which motivated our research here.

Let $I\subseteq\mathbb{K}[x_{0},\ldots,x_{N}]$ be a homogeneous ideal in the ring of polynomials over a field $\mathbb{K}$.

Definition 2.1**.**

(Symbolic power) For $m\geq 1$, the $m$-th symbolic power of $I$ is the ideal

[TABLE]

where the intersection is taken over all associated primes of $I$.

Symbolic powers of ideals are of geometric interest due to Zariski-Nagata Theorem [9, Theorem 3.14].

Theorem 2.2**.**

(Zariski-Nagata) Let $I$ be a radical homogeneous ideal, and let char$(\mathbb{K})=0$. For $m\geq 1$

[TABLE]

In the situation when $V(I)$ is a finite set of points $P_{1},\ldots,P_{t}\in\mathbb{P}^{N}$, the symbolic power is particularly easy to compute:

[TABLE]

An arrangements of lines $\mathcal{A}$ is a finite set of mutually distinct lines $L_{1},\ldots,L_{s}$. An arrangement of lines determines a finite set of points $\mathcal{P}=\{P_{1},\ldots,P_{t}\}$ in $\mathbb{P}^{2}$, where at least $2$ of arrangement lines intersect. For $i\geq 2$, we denote by $t_{i}(\mathcal{A})$ the number of points in $\mathcal{P}$ where exactly $i$ lines from $\mathcal{A}$ intersect. These numbers define the $t$-vector $t(\mathcal{A})$ of $\mathcal{A}$

[TABLE]

It is a basic combinatorial invariant of $\mathcal{A}$.

For line arrangements defined over $\mathbb{R}$, the following property has been distinguished.

Definition 2.3** (Simplicial arrangement).**

We say that an arrangement $\mathcal{A}$ of real lines is simplicial if every connected component of its complement $\mathbb{P}^{2}(\mathbb{R})\setminus\mathcal{A}$ is a triangle.

It is expected, but not known, if (apart of $3$ obvious infinite families described in [11]) there are only finitely many sporadic examples. A list of such examples was constructed by Grünbaum in [11] and extended recently by Cuntz in [3].

3 Simplicial arrangements $\mathcal{A}(31,2)$ and $\mathcal{A}(31,3)$

The arrangements we study here come from [11], where they are called $\mathcal{A}(31,2)$ and $\mathcal{A}(31,3)$.

Configurations $\mathcal{A}(31,2)$ and $\mathcal{A}(31,3)$ are non isomorphic simplicial arrangements of $31$ lines with the total number of $127$ intersection points. Moreover we have

[TABLE]

and all other $t_{i}=0.$

3.1 Configuration $\mathcal{A}(31,3)$

This configuration can be realized in the following way. We begin with ten lines:

[TABLE]

where $u=\frac{\sqrt{3}}{2}$, $a=0,1,2,4$ and $b=0,1$. These ten lines are visualized in Figure 1.

Then we rotate these lines by $60^{\circ}$ and $120^{\circ}$ around the point $(0:0:1)$. In this way, we obtain $30$ lines. The last line is the line at infinity $z=0$. As a result we obtain a configuration of lines indicated in Figure 2. Taking the product of linear forms defining the ten initial lines we obtain the following polynomial

[TABLE]

and taking the product of all $31$ lines we get a polynomial $F_{31}$ of degree $31$. We are interested in the Jacobian ideal $Jac(F_{31})$ defined by this polynomial. The radical $I_{\mathcal{A}(31,3)}$ of this ideal describes all $127$ intersection points among arrangement lines.

By construction the arrangement is invariant under the group $G=\mathbb{Z}_{3}\times\mathbb{Z}_{2}$ generated by the rotation matrix $A=\begin{bmatrix}\frac{1}{2}&-u&0\\ u&\frac{1}{2}&0\\ 0&0&1\end{bmatrix}$ and the reflection $B=\begin{bmatrix}-1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}$, which together generate the dihedral group $D_{6}$. This group acts on the set of $127$ intersection points so that, there are the following orbits

[TABLE]

It is helpful to consider the sub-arrangement $\mathcal{B}_{21}$ consisting of $7$ lines:

[TABLE]

where $a=1,4$, $b=0,1$ and images of these lines under $A$ and $A^{2}$. This $21$ lines intersect altogether in $115$ points, with multiplicities $t_{2}=72$, $t_{3}=40$ and $t_{4}=3$. The difference between the $127$ and $115$ points is one full orbit represented by point $(9:2u:4u)$.

The points in this orbit are now contained each in only one of the $21$ lines. In order to get an element in $I^{(3)}_{\mathcal{A}(31,3)}$ we need to complete the $21$ lines by a divisor $\Gamma$ vanishing in these $12$ points to order $2$ and passing through the remaining $72$ points, which are double points for $\mathcal{B}_{21}$.

To this end we consider $X=\mathbb{P}^{2}/G$. The ring of invariant polynomials $\mathbb{K}[x,y,z]^{G}$ is generated by

[TABLE]

Using Moliens’s Theorem (see [18], Theorem $2.2.1$), we see that the space of invariant polynomials of degree $12$ has dimension $12$.

Since vanishing to order $2$ at a smooth point of $X$ imposes $3$ conditions and the $72$ points split into $4$ orbits of order $6$ and $4$ orbits of order $12$, counting conditions

[TABLE]

we conclude that the desired divisor $\Gamma$ exists (it is invariant under $G$, so it pulls back from $X$). Computing with Singular, we are able to identify the equation of $\Gamma$:

[TABLE]

in terms of the invariant generators $f_{1}$, $f_{2}$, $f_{3}$.

Considering the equation of $\Gamma$ in the ring $\mathbb{K}[f_{1},f_{2},f_{3}]$, it is easy to check that there is just one singular point, which is locally simple crossing. This implies that $\Gamma$ is irreducible.

For the non-containment $I^{(3)}_{\mathcal{A}(31,3)}\nsubseteq I^{2}_{\mathcal{A}(31,3)}$ we used Singular. We do not have a theoretical proof. Summing up claims in this section, we see that Theorem A is proved.

At the end of this section we want to underline another interesting observation about curve from set $I_{\mathcal{A}(31,3)}^{(3)}\setminus I_{\mathcal{A}(31,3)}^{2}$ indicated on Figure 5. The twelve visible double points are the only singular points for this curve, thus we can easily calculate the arithmetic genus, which is

[TABLE]

This is the first known example of the curve, which form an element from the set $I^{(3)}\setminus I^{2}$ and which is not rational at the same time.

3.2 Configuration $\mathcal{A}(31,2)$

This configuration is very similar to $\mathcal{A}(31,3)$. It can be realized starting with lines

[TABLE]

where $a=0,1,2,3$ and $b=0,1$. These lines are visualized in Figure 3.

Rotating again by $60^{\circ}$ and $120^{\circ}$ and taking the line at infinity we obtain the configuration presented in Figure 4.

The multiplicities vector of this configuration is the same as for $\mathcal{A}(31,3)$, i.e., there is

[TABLE]

In particular there are again $127$ intersection points of pairs of arrangement lines. However, a quick Singular check shows that now we have

[TABLE]

This shows, once again (see [10]), that the (non)containment property is quite subtle and cannot be decided by looking at the basis combinatorial invariants only.

3.3 Arrangement $\mathcal{B}_{21}$

Now we consider more closely the arrangement $\mathcal{B}_{21}$ defined in the previous section. We keep the notation introduced there.

The ideal $I_{\mathcal{B}_{21}}$ defines $115$ points. This is the subset of $127$ points defined by $I_{\mathcal{A}(31,3)}$, the difference being one $G-$orbit, consisting of $12$ double points of $\Gamma$.

In order to exhibit an element in $I^{(3)}_{\mathcal{B}_{21}}$, we need to find a divisor vanishing at the $72$ points, where only $2$ of arrangement lines meet.

Revoking again Molien’s Theorem, we see that the dimension of the space of $G-$invariant polynomials of degree $10$ is $9$, thus the expected dimension of invariant polynomials vanishing at $4$ order $6$ and $4$ order $12$ orbits in which the $72$ points split is

[TABLE]

Hence there is a divisor $\Delta$ of degree $10$ vanishing at these $72$ points. We can express its equation in terms of invariant polynomials:

[TABLE]

Since $\Delta$ is smooth, it is irreducible.

The non-containment $I^{(3)}_{\mathcal{B}_{21}}\nsubseteq I^{2}_{\mathcal{B}_{21}}$ is proved again with the aid of Singular [6].

Summing up the claims of this section, we obtain the proof of Theorem B.

As for the curve in Section 3, we also calculate arithmetic genus for the curve, which is indicated as a solid line on Figure 6. Using any computer algebra program one can check that this curve has only four non-reduced singular points, and that its genus is $g=30.$ This means that this curve is not rational.

We conclude this section by noting that the arrangement $\mathcal{B}_{21}$ is not free. Indeed, its characteristic polynomial is

[TABLE]

and it does not split over the integers (see Main Theorem in [20]).

4 Realizability over rational numbers

The first non-containment

[TABLE]

was the dual Hesse arrangement, see [7]. This arrangement cannot be realized over the reals. The first real non-containment example, the Böröczky arrangement of $12$ lines was discovered in [4]. It was realized in [2] and [15] that the Böröczky arrangement can be defined over the rational numbers. Additional examples were provided in [16] and [10]. Such examples are quite rare, so we find it worth to mention that $\mathcal{A}(31,3)$ and $\mathcal{B}_{21}$ can be both realized over $\mathbb{Q}$. We can be quite explicit here. Table 1 contains equations of all $31$ lines, whereas coordinates of their intersection points are provided in Table 2.

Acknowledgements.

We would like to warmly thank T. Szemberg for all helpful remarks, valuable comments and inspiring discussions which greatly improve the original draft of our paper.

The research of Lampa-Baczyńska was partially supported by National Science Centre, Poland, grant 2016/23/N/ST1/01363, the research of Malara was partially supported by National Science Centre, Poland, grant 2016/21/N/ST1/01491.