The module of Valabrega-Valla of the Jacobian ideal of points in projective plane

TL;DR

This paper investigates the Valabrega-Valla module of the Jacobian ideal for points in the projective plane, providing classifications for configurations of 5 and 6 points and identifying cases where the module is nonzero.

Contribution

It offers a complete classification of the Valabrega-Valla module for 5 and 6 points in the projective plane, highlighting cases with non-vanishing modules for specific point configurations.

Findings

The module is nonzero for certain special configurations.

Complete classification for 5 and 6 points.

Identification of configurations with vanishing modules.

Abstract

The module of Valabrega-Valla of the Jacobian ideal of a reduced projective variety $V$ is the torsion of the Aluffi algebra. One considers the problem of its vanishing in the case of where $V$ is a reduced set of points in the projective plane. It is shown that the module is nonzero for several cases of a special configuration class therein -- called $(s-r)$-{fold collinear configuration}. A complete classification of types is given for $5$ and $6$ points in regard to this problem.

Equations

$$\mathcal{V}{\mathcal{V}}_{J\subset I}:=\underset{t\ge 2}{⨁}\frac{J\cap I^t}{J{I}^{t-1}}.$$

$${\mathcal{A}}_{R/J}(I/J)\simeq \underset{t\ge 0}{⨁}{I}^t/J{I}^{t-1}\twoheadrightarrow {\mathcal{R}}_{R/J}(I/J)\simeq ⨁I^t/J\cap I^t.$$

$$I(\mathbb{X})=\bigcap _{i=1}^{s}I(p_i)=(z,f(x,y)),$$

$$I(\mathbb{X})$$

$$\Theta =\left[\begin{array}{ccc}z& -a_{1}z& x-a_{1}y\\ z& -a_{2}z& x-a_{2}y\\ f_x& f_y& 0\end{array}\right].$$

$$\begin{array}[]{cc}-y&0\\ x&0\\ 0&-y\\ 0&x\end{array}$$

$$0\to S^2(-s)⨁S(-(2s-4))\to S^4(-(s-1))\to S\to S/\mathfrak{a}\to 0.$$

$$0\to S^2(-(2s-1))⨁S(-(3s-5))\to S^4(-2(s-1))\to S\to S/(f)\mathfrak{a}\to 0.$$

$$0\to S^6(-(2s-1))⨁S^2(-(3(s-2)))\to S^9(-(2(s-1)))\to S\to S/{\mathfrak{a}}^2\to 0$$

$$0$$

$$0\to \frac{S}{(f)\cap {\mathfrak{a}}^2}\to \frac{S}{(f)}⨁\frac{S}{{\mathfrak{a}}^2}\to \frac{S}{(f,{\mathfrak{a}}^2)}\to 0.$$

$$\mathrm{HS}\left(\frac{R}{(f)\cap {\mathfrak{a}}^2},t\right)=\frac{1+t+t^2+\cdots +t^{2s-3}-3t^{2s-2}-t^{2s-1}-t^{2s}-\cdots -2t^{3s-7}}{(1-t)}.$$

$$\mathrm{HS}\left(\frac{R}{(f)\mathfrak{a}},t\right)=\frac{1+t+t^2+\cdots +t^{2s-3}-3t^{2s-2}-t^{2s-1}-t^{2s}-\cdots -t^{3s-7}-t^{3s-6}}{(1-t)}.$$

$$I({\mathbb{X}}_{\mathrm{in}})$$

$$J:=(zl,z{l}_{1,(s-1)}{l}_{(s-2),s},l_{(s-2),s}\prod _{i=1}^{s-3}{l}_{i,(s-1)}).$$

$$\left[\begin{array}{cc}{l}_{1,(s-1)}{l}_{(s-2),s}& 0\\ l& {\prod }_{i=2}^{s-3}{l}_{i,(s-1)}\\ 0& z\end{array}\right].$$

$$0\to R(-4)\oplus R(-(s-1))\to R(-2)\oplus R(-3)\oplus R(-(s-2))\to R\to R/J\to 0.$$

$$\mathrm{HS}(R/J,t)=\frac{1-t^2-t^3+t^4-t^{s-2}+t^{s-1}}{(1-t{)}^3}=\frac{1+2t+2t^2+t^3+\dots +t^{s-3}}{1-t}.$$

$$0\to R/I({\mathbb{X}}_{\mathrm{off}})\to R/I({\mathbb{X}}_1)⨁R/I({\mathbb{X}}_2)\to R/(I({\mathbb{X}}_1),I({\mathbb{X}}_2))\to 0.$$

$$R/(I({\mathbb{X}}_1),I({\mathbb{X}}_2))\simeq k[z]/(z^2),R/I({\mathbb{X}}_2)\simeq k[z].$$

$$\mathrm{HS}(R/I({\mathbb{X}}_{\mathrm{off}}))$$

$$q:=xz-yz,c:=z{y}^2-y{z}^2,f:=\sum _{i=0}^{s-2}{g}_i{x}^{s-2-i}{y}^i+y{z}^{s-3},$$

$$\Theta =\left[\begin{array}{ccc}z& -z& x-y\\ 0& 2yz-z^2& y^2-2yz\\ f_x& f_y& f_z\end{array}\right],$$

$$f_x=\sum _{i=1}^{s-2}(s-2-i)g_i{x}^{s-3-i}{y}^i,f_y=\sum _{i=1}^{s-2}(i)g_i{x}^{s-2-i}{y}^{i-1}+z^{s-3},f_z=(s-3)y{z}^{s-4}.$$

$$I=(xz-yz,y^{2}z,y{z}^2,z^3,G,(y-x)f_x,(y-x)H,y^2{f}_x),$$

$$\mathfrak{b}:=(G,(y-x)f_x,(y-x)H,y^2{f}_x),$$

$$y^{2s-7}={\lambda }_{1}G+{\lambda }_2(y-x)H+{\lambda }_3{y}^2{f}_x\in \mathfrak{b},$$

$${\lambda }_1$$

$${\lambda }_2$$

$${\lambda }_3$$

Full text

Introduction

Let $V\subset{\mathbb{P}}^{d}={\mathbb{P}}^{d}_{k}$ denote a reduced projective variety over a perfect field $k$, with homogeneous coordinate ring $A=R/J$, where $R=k[x_{0},\ldots,x_{d}]$ and $J$ is the homogeneous defining ideal of $V$ in its embedding. By definition, the Jacobian ideal of the $k$-algebra $A$ is the Fitting ideal of order $d+1-(d-\dim V)=\dim V+1$ of the module of Kähler $k$-differentials $\Omega_{A/k}$ of $A$. Since $\Omega_{A/k}$ depends only on $A$, and not on any particular presentation of $A$, and so do the Fitting ideals of $\Omega_{A/k}$, then this notion is dependent only upon the given projective embedding of $V$. This is as much invariance one can dispose of.

Now, since $\Omega_{A/k}$ admits a module presentation as the cokernel of the transposed Jacobian matrix $\Theta$ of a set of generators of $J$, modulo $J$, then the Jacobian ideal of $A$ is the ideal of $A$ generated by the $(d-\dim V)$-minors of this Jacobian matrix. By abuse, one often takes the ideal of $(d-\dim V)$-minors back in the polynomial ring $R$ as the Jacobian ideal of the ideal $J$. If this malpractice is followed then, in order to keep the invariant properties of the Jacobian ideal of $A$, one should always take those minors summed to $J$ – that is, the ideal $I:=(J,I_{d-\dim V}(\Theta))\subset R$.

In this paper we focus on the problem as to when the so-called Valabrega–Valla module vanishes for the pair $J\subset I$, where $J$ is a homogeneous ideal in a polynomial ring over a field and $I$, denotes its Jacobian ideal. For that a basic principle was established in [8, Example 2.19] and successfully applied to a collection of classical geometric situations, such as the rational normal curve (codimension $d-1$) and certain Segre and Veronese embeddings. A little later, this problem was applied in [10] for special classes of linear determinantal ideals (rational normal scrolls and alike).

In [9] the problem was circumscribed to the case of a set of reduced points when the latter admits a minimal set of generators in degree $2$. In [7, Theorem 1.2] a characterization of the vanishing of the module of Valabrega-Valla was given in terms of the first syzygy module of the form ideal $J^{*}$ in the associated graded ring of $I$. In [13] a vast extension of these matters was taken all the way to the environment of modules, bringing up some hard features of Cohen–Macaulay and Gorenstein algebras naturally arising from Rees algebras of modules.

Given an inclusion of ideals $J\subset I$, the module of Valabrega–Valla is defined as the graded module

[TABLE]

As such, it has appeared elsewhere in a different context (see [14], also [15, 5.1]). As it turns out, provided $I$ has a regular element module $J$, the Valabrega–Valla module is the $A$-torsion of the so-called embedded Aluffi algebra of $I/J$, hence its interest for the geometric purpose [6]. Dealing directly with the Valabrega–Valla module makes the structure of the Aluffi algebra itself sort of invisible. On the bright side, in the case where $\mathfrak{a}=I/J$ is the Jacobian ideal of $A=R/J$, the heavy work is transferred to understanding its nature. Besides, for some mysterious reason, the existence of non-trivial torsion is often delivered at the level of degree $2$ of ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}_{J\subset I}$.

The Jacobian ideal $I/J\subset A:=R/J$ will be said to be ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion free if ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}_{J\subset I}=\{0\}$, i.e., if $J\cap I^{t}=JI^{t-1}$ for all $t\geq 2$.

The main goal of this work is to understand the nature of the ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion freeness of the Jacobian ideal $I$ of an ideal $J\subset R:=k[x_{0},\ldots,x_{n}]$ of a finite set of distinct points in projective space. Note that, since $R/J$ is Cohen–Macaulay, $R/J$ is reduced if and only if ${\rm ht}\,(I)\geq{\rm ht}\,(J)+1$, which is equivalent to saying that $J$ is an ideal of reduced points if and only if $I$ contains a regular element modulo $J$ or, still, to assert that $I$ is an ${\mathfrak{m}}$-primary ideal, where ${\mathfrak{m}}=(x_{0},\ldots,x_{n})$. As such, the only algebraic subtlety away from the geometric data is the number and the degrees of a set of minimal generators of the ${\mathfrak{m}}$-primary ideal $I$. Indeed, this emphasis will prevail throughout the paper.

The results and examples in [9] show that the answer to this problem may not have a general shape, even when one assumes that the defining ideal is generated by quadrics – a condition that is only guaranteed when the number of points in ${\mathbb{P}}^{n}$ is at most $2n$. In ${\mathbb{P}}^{2}$ this forces the number of points to be at most $4$, not a very bright situation. The idea of this work is to go beyond, by rather focusing on special configuration of points in the projective plane ${\mathbb{P}}^{2}$, without imposing constraints on the number of points. However, we do consider the ‘next’ cases: $5$ and $6$ points are completely classified.

The reason to look at reduced plane points is that, along with the results of [8] on hypersurfaces, it would give a reasonable picture of the Jacobian torsion for the subvarietis of ${\mathbb{P}}^{2}$. The next difficult step would be the case of curves in ${\mathbb{P}}^{3}$, a problem that we may tackle in the near future. It is the authors expectation that new suitable techniques be brought in to get a bird view of the problem in all cases.

The outline of the paper is as follows.

In section 1, we give a quick view of the main characters for arbitrary ideals in a Noetherian ring (for further details we refer to [8]). Then proceed to the case of the defining ideal of a reduced set of points in projective space. In this regard, Proposition 1.3 gives a criterion to check ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion freeness of the Jacobian ideal of a finite set of points in a projective space.

In section 2, we consider the following configuration of points in ${\mathbb{P}}^{2}$: given integers $s\geq 4$ and $0\leq r\leq s-3$, will say that a finite set of points ${{\mathbb{X}}}$ is an $(s-r)$-fold collinear configuration when $(s-r)$ among them lie on a straight line. We prove that if $r=0,1$, then the Jacobian ideal of the defining ideal $I({{\mathbb{X}}})$ of ${{\mathbb{X}}}$ is not ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion free (Theorem 2.2). In the case that $r=2,\,s\geq 8$ or $r=3,\,s\geq 9$, we show that the Jacobian ideal of ${{\mathbb{X}}}$ is not ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion free. Clearly, all theses cases have a simple geometry, which however does not seem to yield automatically the ‘asymptotic’ behavior of the problem in consideration.

In section 3 we consider the defining ideal $I({{\mathbb{X}}})$ of a set ${{\mathbb{X}}}$ of $5$ and $6$ points in ${\mathbb{P}}^{2}$. In the case of $5$ distinct points it is known that there are $5$ mutually distinct configurations. We show that the Jacobian ideal of $I({{\mathbb{X}}})$ is ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion-free exactly in the following configurations:

• •

${{\mathbb{X}}}$ is in general linear position.

• •

${{\mathbb{X}}}$ is a $3$-fold collinear configuration.

• •

${{\mathbb{X}}}$ is a $3$-fold collinear configuration such that the straight line through the remaining two points intersect in a point of ${{\mathbb{X}}}$.

Finally, let ${{\mathbb{X}}}\subset{\mathbb{P}}^{2}$ be a set of six points. For $6$ points, there are eleven distinct configurations (see Figure  1). We show that the Jacobian ideal of $I({{\mathbb{X}}})$ is ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion-free if and only if ${{\mathbb{X}}}$ is in one of the following configurations:

• •

${{\mathbb{X}}}$ is in general linear position.

• •

${{\mathbb{X}}}$ is a $4$-fold collinear configuration.

• •

$\mathbb{X}$ is a $4$-fold collinear configuration such that the straight line through the remaining two points intersect in a point of ${{\mathbb{X}}}$.

• •

${{\mathbb{X}}}$ is a $3$-fold collinear configuration such that the remaining three points are in general linear position.

• •

${{\mathbb{X}}}$ is a $3$-fold collinear configuration such that the straight line through two of the remaining points intersect in a point of ${{\mathbb{X}}}$.

• •

${{\mathbb{X}}}$ is a $3$-fold collinear configuration such that two straight lines through of the remaining points intersect in points of ${{\mathbb{X}}}$.

• •

${{\mathbb{X}}}$ is a $3$-fold collinear configuration such that the three straight lines through of the remaining points intersect in points of ${{\mathbb{X}}}$.

Acknowledgment

The second author thanks the Instituto de Ciências Matemáticas e da Computação (ICMC, São Carlos, Brazil) and the Department of Mathematics of the Federal University of Sergipe (UFS, Brazil) for providing a suitable atmosphere for her stay in the frame of a sabbatical leave. Both authors thank Zaqueu Ramos and Aron Simis for insightful discussions on the preliminary versions of this paper. Simis, in particular, has been helpful in suggesting a couple of improvements in the style of some proofs

1. The module of Valabrega-Valla of points

Given a Noetherian ring $R$ and ideals $J\subset I$ of $R$, the module of Valabrega-Valla is defined as graded module

[TABLE]

There is a natural surjective $R/J$-algebra homomorphism from the Aluffi algebra to the Rees algebra of $I/J$

[TABLE]

The Aluffi algebra is an algebraic version of characteristic cycles in intersection theory [6]. The connection of module of Valabrega-Valla with the Aluffi algebra is as follows:

Proposition 1.1**.**

([8, Proposition 2.5])* If $I$ has a regular element module $J$, then the module of Valabrega–Valla is the $R/J$-torsion of the Aluffi algebra of $I/J$.*

The vanishing of ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}_{J\subset I}$ has close relation with the theory of $I$-standard base ( in the sense of Hironaka), Artin Rees number and relation type number (See [11] and [10]).

Let ${{\mathbb{X}}}:=\{p_{1},\ldots,p_{s}\}$ be a set of $s$ distinct points in ${\mathbb{P}}^{n}:={\mathbb{P}}_{k}^{n}$, where $k$ is an algebraically closed field of characteristic zero and $n\geq 2$. The defining ideal of ${{\mathbb{X}}}$ is the ideal $I({{\mathbb{X}}})=\bigcap_{i=1}^{s}I(p_{i})\subseteq R=k[x_{0},\ldots,x_{n}]$ where $I(p_{i})$ is the prime ideal generated by $n$ linear forms. Note that the coordinate ring $R/I({{\mathbb{X}}})$ of ${{\mathbb{X}}}$ is a reduced ring of dimension one and hence is a Cohen-Macaulay ring. The multiplicity of $R/I({{\mathbb{X}}})$ is the number of points in ${{\mathbb{X}}}$ [2, Corollary 3.10]. Since $R/I({{\mathbb{X}}})$ is locally regular on the punctured spectrum, by the Jacobian criterion it translates into the property that the Jacobian ideal $I=(I({{\mathbb{X}}}),I_{n}(\Theta))$ is ${\mathfrak{m}}$-primary, where $I_{n}(\Theta)$ is the ideal of $n$-minors of the Jacobian matrix $\Theta$ of $I({{\mathbb{X}}})$ and ${\mathfrak{m}}$ denotes the maximal irrelevant ideal of the polynomial ring $R$. In other words, there is a suitable power ${\mathfrak{m}}^{l}$ that lands into $I$. Therefore, if the defining ideal of ${{\mathbb{X}}}$ is minimally generated in single degree $m\geq 2$ and ${\mathfrak{m}}^{n(m-1)}\subset I$, then the Valabrega-Valla module vanishes [9].

Let us introduce the following terminology:

Definition 1.2**.**

The Jacobian ideal $I/J\subset A:=R/J$ will be said to be ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion free if ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}_{J\subset I}=\{0\}$, i.e., if $J\cap I^{t}=JI^{t-1}$ for all $t\geq 2$.

Given a ring $A$ and an ideal $\mathfrak{a}=(a_{1},\ldots,a_{n})\subseteq A$, one lets $\varphi:A[T_{1},\ldots,T_{n}]\twoheadrightarrow\mathcal{R}_{A}(\mathfrak{a})=A[\mathfrak{a}t]$ be the graded map sending $T_{i}$ to $a_{i}t$. The relation type number of $\mathfrak{a}$ is the largest degree of any minimal system of homogeneous generators of the kernel $\varphi$. Since the isomorphism $A[T_{1},\ldots,T_{n}]/\mathrm{Ker}\varphi\simeq\mathcal{R}_{A}(\mathfrak{a})$ is graded, an application of the Schanuel lemma to the graded pieces shows that the notion is independent of the set of generators of $\mathfrak{a}$.

A key result in the case of points reads like this:

Proposition 1.3**.**

Let ${{\mathbb{X}}}$ be a finite set of $s$ points in ${\mathbb{P}}^{n}$ and $I$ stands for the Jacobian ideal of $I({{\mathbb{X}}})$. Assume that $I({{\mathbb{X}}})\cap I^{t}=I({{\mathbb{X}}})I^{t-1}$ for every $2\leq t\leq s$. Then the Jacobian ideal of $I({{\mathbb{X}}})$ is ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion-free.

Proof.

By [8, Corollary 2.17] it suffices to prove that the relation type number of $I/I({{\mathbb{X}}})$ is at most $s$. But as $R/I({{\mathbb{X}}})$ is a $1$-dimensional Cohen Macaulay graded ring, then [12, Lemma 6.3] implies that the relation type number of $I/I({{\mathbb{X}}})$ is bounded by the multiplicity of $R/I({{\mathbb{X}}})$ which is the number of points in ${{\mathbb{X}}}$. ∎

Remark 1**.**

(a) By Proposition  1.3, to check the ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion-freeness it is enough to check for $t\leq s$, where $s$ is the number of points. This bound is not sharp in general – see, e.g., Proposition 3.1.

(b) Let ${{\mathbb{X}}}$ be a set of $s$ collinear points with $s\geq 4$ in ${\mathbb{P}}^{2}$. Then the defining ideal of ${{\mathbb{X}}}$ is $I({{\mathbb{X}}})=(z,f(x,y))$, where $f(x,y)$ is a reduced product of $s$ linear forms in $x,y$. Then the relation type of $I/I({{\mathbb{X}}})$ on $R/I({{\mathbb{X}}})$ is the degree of the dual form of $f(x,y)$, which is itself (for the details see Proposition 2.2). Thus, the relation type is $s$ (maximal possible), but $I({{\mathbb{X}}})\cap I^{2}\neq I({{\mathbb{X}}})I$ as can be readily obtained by considering the element $y^{2s-4}f$.

2. $(s-r)$-fold collinear configurations in ${\mathbb{P}}^{2}$

We introduce the following configuration of points in ${\mathbb{P}}^{2}$.

Definition 2.1**.**

Let $s\geq 4$ and $0\leq r\leq s-3$ be integers. An $(s-r)$-fold collinear configuration is a finite set of $s\geq 4$ plane points such that exactly $(s-r)$ of its points lie on a straight line.

Theorem 2.2**.**

Let $\mathbb{X}\subset{\mathbb{P}}^{2}$ be an $(s-r)$-fold collinear configuration. If $r=0,1$, then the Jacobian ideal of $I(\mathbb{X})$ is not ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion free.

Proof.

Suppose that $r=0$, i.e., all points are collinear. Say, the points lie on the line $\{z=0\}$. Then the ideal of a point $p_{i}(1\leq i\leq s)$ has the form $I(p_{i}):=(z,\ell_{i}(x,y))$, for some $k$-linear form $\ell_{i}(x,y)\in k[x,y]$. Moreover, these linear forms are independent. Then, it is an easy exercise to get

[TABLE]

where $f=\prod_{i=1}^{s}\ell_{i}(x,y)$.

By using Euler formula, the Jacobian ideal of $I(\mathbb{X})$ is $I=(z,\partial f/\partial x,\partial f/\partial y)$. Now, since $R/I({{\mathbb{X}}})\simeq k[x,y]/(f)$ and $I/I({{\mathbb{X}}})\simeq(\partial f/\partial x,\partial f/\partial y)/(f)$, one can argue with these simplified structures. Since the ideal $(\partial f/\partial x,\partial f/\partial y)$ is a complete intersection, hence of linear type, it follows that the Aluffi algebra of $(\partial f/\partial x,\partial f/\partial y)/(f)$ is isomorphic to its symmetric algebra (see [8]), whose ideal of relations contains no form of bidegree $(0,d)$, with $d\geq 1$. On the other hand, the defining relations of its Rees algebra contains the equation of the dual curve to $f$, which is $f$ itself read in the relational variables. This gives a relation of bidegree $(0,s)$, $s\geq 4$. By Proposition 1.1, this relation gives a non-trivial torsion.

Now let $\mathbb{X}$ be an $(s-1)$-fold collinear configuration. As before, assume that the points $p_{1},\ldots,p_{s-1}$ lie on $\{z=0\}$, while $p_{s}\notin\{z=0\}$. Let $l_{i,s}\in k[x,y]$ denote the linear form defining the unique straight line through $p_{i}$ and $p_{s}$, for $i=1,\ldots,s-1$. Then a simple calculation yields

[TABLE]

since $\{z,l_{1,s},l_{2,s}\}$ is a regular sequence both linear forms $l_{1,s},l_{2,s}$ are factors of $\prod_{i=1}^{s-1}l_{i,s}$.

Next, writing $p_{i}=[a_{i}:1:0]$ for $i=1,\ldots,s-1$ and $p_{s}:=[0,0,1]$, hence $l_{i,s}=x-a_{i}y$. Setting $f:=\prod_{i=1}^{s-1}l_{i,s}$, the Jacobian matrix of $I(\mathbb{X})$ is

[TABLE]

The $2\times 2$-minors fixing the last row yield the following subset $\{xf_{x},yf_{x},xf_{y},yf_{y}\}\subset k[x,y]$ of a set of minimal generators of the Jacobian ideal $I$. Set $\mathfrak{a}:=(xf_{x},yf_{x},xf_{y},yf_{y})$, as an ideal of the ring $k[x,y]$.

Claim. $(f)\cap\mathfrak{a}^{2}\neq(f)\mathfrak{a}$.

To see this, note that, as an ideal of codimension $2$ in $k[x,y]$, $\mathfrak{a}$ is $(x,y)$-primary, hence perfect. Thus, its matrix of syzygies is a $4\times 3$ matrix containing the obvious columns

[TABLE]

Counting degrees and letting $S=k[x,y]$, the minimal free resolution of $S/\mathfrak{a}$ over $S$ has the form

[TABLE]

Therefore, the minimal free resolution of $S/(f)\mathfrak{a}$ over $S$ has the form

[TABLE]

Now consider the ideal $\mathfrak{a}^{2}$. By the same token, it is a perfect ideal of codimension $2$. Moreover, since $f_{x}$ and $f_{y}$ have no common factors, it will turn out that $\mathfrak{a}^{2}$ is minimally generated by $9$ forms of degree $2(s-1)$. It follows that its syzygy matrix is $9\times 8$. Since the linear syzygies in (2) generate $6$ independent linear syzygies of $\mathfrak{a}^{2}$, it follows that the remaining two syzygies have the same degree $3(s-2)$. It follows that the minimal free $S$-resolution of $S/\mathfrak{a}^{2}$ has the form

[TABLE]

Now, consider the ideal $(f,\mathfrak{a}^{2})$. Drawing upon the Euler relation, it is easy to see that this ideal is minimally generated by $f$ and $5$ more among the original generators of $\mathfrak{a}^{2}$. Moreover, it inherits $4$ linear syzygies among those of $\mathfrak{a}^{2}$. Therefore, its minimal free $S$-resolution has the shape

[TABLE]

Finally, take the exact squence

[TABLE]

Applying to this exact sequence the information gathered in (5),(2), we find the Hilbert series of the left-most term:

[TABLE]

Using (4), one has

[TABLE]

As the two Hilbert series are different, it follows that $(f)\cap\mathfrak{a}^{2}\neq(f)\mathfrak{a}$. This completes the proof of the claim.

The Jacobian ideal $I$ is minimally generated by the monomials $xz,yz,z^{2}$ and $\mathfrak{a}$. By the above claim, there exists a polynomial $g\in(f)\cap\mathfrak{a}^{2}\subset S$ such that $g\neq(f)\mathfrak{a}$. Since the ideal $I({{\mathbb{X}}})I$ is generated by polynomials which contain the variable $z$ and $f\mathfrak{a}$, it follows that the polynomial $g$ does not belong to $I({{\mathbb{X}}})I$. This proves that the Jacobian ideal of $I({{\mathbb{X}}})$ is not ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion free. ∎

The configuration studied so far requires that $r\leq s-3$, hence $r\leq 1$ for $s=4$. However, for $s\geq 5$, $r=2$ is a kosher value. We now digress on a slight degeneration of such a configuration.

Theorem 2.3**.**

Let $\mathbb{X}\subset{\mathbb{P}}^{2}$ be an $(s-2)$-fold collinear configuration of $s\geq 8$ distinct points. Then the Jacobian ideal of $I({{\mathbb{X}}})$ is not ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion-free.

Proof.

There are two sub-cases of this configuration, according to which the collinearity line and the straight line through the remaining two points intersect in ${{\mathbb{X}}}$ or off ${{\mathbb{X}}}$. Let ${{\mathbb{X}}}_{\mathrm{in}}$ and ${{\mathbb{X}}}_{\mathrm{off}}$ stand for these configurations of points, respectively.

We compute a set of minimal generators of $I({{\mathbb{X}}}_{\mathrm{in}})$ and $I({{\mathbb{X}}}_{\mathrm{off}})$ and this will work fine for any $s\geq 5$. Take $p_{1},\ldots,p_{s-2}$ to lie on the straight line $\{z=0\}$. Let $p_{s-1}$ and $p_{s}$ the remaining points off the collinear line. Let $l_{i,s-1}$ and $l_{i,s}$ denote, respectively, the linear forms defining the straight lines through $p_{i}$ and $p_{s-1}$ and through $p_{i}$ and $p_{s}$, for $i=1,\ldots,s-2$. Finally, let $l$ denote the linear form defining the straight line through $p_{s-1}$ and $p_{s}$.

A simple calculation, based on the same elementary principles as for (1), yields

[TABLE]

Now setting

[TABLE]

Clearly $J\subseteq I({{\mathbb{X}}}_{\mathrm{off}})$. By construction, $J$ is generated by $2\times 2$ minors of the $2\times 3$ Hilbert-Burch matrix

[TABLE]

Hence the minimal free $R$-resolution of $R/J$ has the form

[TABLE]

It follows that the Hilbert series of $R/J$ is

[TABLE]

Write ${{\mathbb{X}}}_{\mathrm{off}}:={{\mathbb{X}}}_{1}\cup{{\mathbb{X}}}_{2}$, where ${{\mathbb{X}}}_{1}$ is the set of $s^{\prime}=s-1$ points in $(s^{\prime}-1)$-fold collinear configuration and ${{\mathbb{X}}}_{2}=\{p_{s}\}$. One has the short exact sequence

[TABLE]

Direct inspection gives

[TABLE]

Using the above exact sequence and Theorem 2.2, we get

[TABLE]

which proves that $J=I({{\mathbb{X}}}_{\mathrm{off}})$.

Now we prove ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}_{I({{\mathbb{X}}}_{\mathrm{off}})\subset I}\neq 0$. Similar argument apply to the configuration ${{\mathbb{X}}}_{\mathrm{in}}$. Writing $p_{i}=[1:a_{i},0]$ with $a_{i}\neq 0,1$ for $i=1,\ldots,s-2$, $p_{s-1}=[0:0:1]$ and $p_{s}=[1:1:1]$. It follows that the ideal $I({{\mathbb{X}}}_{\mathrm{off}})$ is generated by

[TABLE]

where each $g_{i}$ is a certain polynomial expression of the $a_{i}$’s. The Jacobian matrix of $I({{\mathbb{X}}}_{\mathrm{off}})$ is

[TABLE]

where

[TABLE]

The ideal generated by the $2\times 2$ minors fixing the first two rows and the generators $q$ and $c$ yield the subset $\{xz-yz,yz^{2},zy^{2},z^{3}\}$ of a set of minimal generators of the Jacobian ideal $I$. Setting $\mathfrak{a}:=(xz-yz,yz^{2},zy^{2},z^{3})$. One can verify that the polynomials $zf_{x},\,zf_{y},\,zf_{z}$ belong to the ideal $\mathfrak{a}$ and the Jacobian ideal is minimally generated by

[TABLE]

where $G=\sum_{i=0}^{s-2}g_{i}x^{s-2-i}y^{i}$ and $H=\sum_{i=1}^{s-2}(i)g_{i}x^{s-2-i}y^{i-1}$. Set

[TABLE]

a zero dimensional homogeneous ideal. Then $\mathfrak{b}\cap k[y]\neq 0$. The following relation for suitable $\alpha_{i},\beta_{i},\gamma_{i}$ – certain polynomials like expression in the $a_{i}$’s – shows that $y^{2s-7}\in\mathfrak{b}$ and $2s-7$ is minimum with this property.

[TABLE]

where

[TABLE]

Now consider the polynomial $E:=y^{2s-8}G-yz^{2s-3}$. Clearly, $E$ dose not belong to $I({{\mathbb{X}}}_{\mathrm{off}})I$. We show that $E\in I({{\mathbb{X}}}_{\mathrm{off}})\cap I^{2}$, which proves that ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}\neq 0$ in degree $2$. We have

[TABLE]

and

[TABLE]

where

[TABLE]

with $\beta_{ij},\gamma_{ij}\in k$. Finally, $yz^{2s-3}=(yz^{5})z^{2s-8}\in I^{2}$, which proves that $E\in I({{\mathbb{X}}}_{\mathrm{off}})\cap I^{2}$. ∎

Recall that a finite set of $s\geq 3$ distinct points in ${\mathbb{P}}^{2}$ are in *general linear position *if no subset of three points lie on a line.

Theorem 2.4**.**

Let ${{\mathbb{X}}}\subseteq{\mathbb{P}}^{2}$ be an $(s-3)$-fold collinear configuration of $s\geq 9$ distinct points. Suppose, moreover, that the remaining three points are in general linear position. Then the Jacobian ideal of $I({{\mathbb{X}}})$ is not ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion-free.

Proof.

We compute a set of minimal generators of $I({{\mathbb{X}}})$ and this will work for any $s\geq 6$. Assume that $p_{1},\ldots,p_{s-3}$ lie on the straight line $\{z=0\}$. Let $p_{s-2},\,p_{s-1},\,p_{s}$ the remaining points off the collinear line which are in general linear position. Let $l_{i,j}$ denote the linear form defining the straight line through points $p_{i}$ and $p_{j}$. The defining ideal of $3$ points in general linear position is generated by three conics

[TABLE]

Consider the polynomial $G:=l_{1,(s-2)}l_{2,(s-1)}\prod_{i=3}^{s-3}l_{is}$. Setting

[TABLE]

By construction, the generators of $J$ vanishes on ${{\mathbb{X}}}$ and hence $J\subseteq I({{\mathbb{X}}})$. We claim that $J=I({{\mathbb{X}}})$. Setting $\mathfrak{a}:=(zq_{1},\,zq_{2},\,zq_{3})$. Consider the short exact sequence

[TABLE]

Direct inspection gives that $(\mathfrak{a}:_{R}G)=(z)$. One has

[TABLE]

Next write ${{\mathbb{X}}}={{\mathbb{X}}}_{1}\cup{{\mathbb{X}}}_{2}$, where ${{\mathbb{X}}}_{1}$ is the set of $(s-3)$ collinear points and ${{\mathbb{X}}}_{2}$ is the set of $3$ points in general linear position. By Theorem 2.2, we get $I({{\mathbb{X}}}_{1})=(z,f(x,y))$, where $f(x,y)$ is a reduced polynomial of degree $s-3$. We have

[TABLE]

Also $R/I({{\mathbb{X}}}_{2})$ has the Hilbert series

[TABLE]

Since $\mathfrak{b}:=(I({{\mathbb{X}}}_{1}),I({{\mathbb{X}}}_{2}))=(x^{2},xy,y^{2},z)$, it follows that $R_{2}=\mathfrak{b}_{2}$, hence

[TABLE]

Using the exact sequence (2) in this setup, we get

[TABLE]

which proves the claim.

We may assume that $p_{i}=[a_{i}:1:0]$ for $i=1,\ldots,s-3$ with $a_{i}\neq 0,1,-1$ and, by a projective transformation, $p_{s-2}=[1:0:1],\,p_{s-1}=[0:1:1],\,p_{s}=[0:0:1]$. In this setting, one gets $\mathfrak{a}=(zxy,\,x^{2}z-xz^{2},\,y^{2}z-yz^{2})$ and

[TABLE]

where $f(x,y)=\prod_{i=1}^{s-3}(x-a_{i}y)$ and $e=\prod_{i=1}^{s-3}a_{i}$. The Jacobian matrix of $I({{\mathbb{X}}})$ is

[TABLE]

Let $I_{1}:=(\mathfrak{a},I_{2}(\Theta^{\prime}))$, where $\Theta^{\prime}$ is a submatrix of $\Theta$ that we delete the last row. Thus the ideal $I_{1}$ is minimally generated by $\mathfrak{a}$ and the monomials $xz^{3},yz^{3},z^{4}$. We have the following relations

[TABLE]

Note that by symmetry, the second relation holds for $xyf_{y}$. Using the ideal $I_{1}$ and the above relation, we conclude that

[TABLE]

Setting $\mathfrak{b}:=(f,\,x^{2}f_{x},\,x^{2}f_{y},\,y^{2}f_{x},\,y^{2}f_{y})$. The ideal $\mathfrak{b}$ is zero dimensional homogeneous ideal. By the same argument as in the proof of Theorem 2.3, we conclude that $y^{2s-9}$ belongs to $\mathfrak{b}$ and the power is minimum. Also the polynomial $y^{2s-10}f-yz^{3s-14}\in I({{\mathbb{X}}})\cap I^{2}$, but not in $I({{\mathbb{X}}})I$ which prove that ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}\neq 0$ in degree $2$. ∎

3. Five and six points in ${\mathbb{P}}^{2}$

Let ${{\mathbb{X}}}$ be a set of $4$ points in ${\mathbb{P}}^{2}$. There exists only three configurations on their geometry. More precisely, points are in general linear position, ${{\mathbb{X}}}$ is $3$-fold collinear configuration and $X$ is $4$-fold collinear configuration. The Jacobian ideal of $I({{\mathbb{X}}})$ is not ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion-free. In fact, the general linear position follows by [9, Section 3.1] and the other configurations follow by Theorem 2.2.

Let $\mathbb{X}=\{p_{1},p_{2},p_{3},p_{4},p_{5}\}$ be a set of $5$ points in $\mathbb{P}^{2}$. There exist only five configurations on their geometry:

• (1)

${{\mathbb{X}}}$ is in general linear position.

• (2)

${{\mathbb{X}}}$ is a $5$-fold collinear configuration.

• (3)

${{\mathbb{X}}}$ is a $4$-fold collinear configuration.

• (4)

${{\mathbb{X}}}$ is a $3$-fold collinear configuration.

• (5)

${{\mathbb{X}}}$ is a $3$-fold collinear configuration such that the straight line through the remaining two points intersect in a point of ${{\mathbb{X}}}$.

The following result characterizes ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion freeness of five points in their geometry.

Proposition 3.1**.**

Let $\mathbb{X}$ be a set of $5$ distinct points in $\mathbb{P}_{k}^{2}$. The Jacobian ideal of $I({{\mathbb{X}}})$ is ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$- torsion-free if and only if ${{\mathbb{X}}}$ is the configurations (1),(4),(5).

Proof.

By Theorem 2.2, it is enough to show that for the configurations (1),(4),(5), ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}_{J\subset I}=0$.

Configuration (1). Note that the ideal of $4$ points in general linear position is generated by two conics. Then the ideal of ${{\mathbb{X}}}^{\prime}:={{\mathbb{X}}}\setminus\{p_{5}\}$ is generated by conics say $q_{1},q_{2}$ in $R=k[x,y,z]$. It is well known that there exists an unique conic $q$ passes through $5$ points. Since $I({{\mathbb{X}}})\subseteq I({{\mathbb{X}}}^{\prime})$, it follows that $q=aq_{1}+bq_{2}$ for certain uniquely determined nonzero scalars $a,b\in k$. Since $I({{\mathbb{X}}})\subseteq I(p_{5})=(\ell_{1},\ell_{2})$, we obtain that $q=L_{1}\ell_{1}+L_{2}\ell_{2}$, where $L_{1},L_{2}$ are linear forms in $R$. We claim that

[TABLE]

Setting $J:=(q,q_{1}\ell_{1},q_{1}\ell_{2})$. One has $J\subseteq I({{\mathbb{X}}}^{\prime})\cap I(p_{5})=I({{\mathbb{X}}})$. On the other hand, the ideal $J$ is generated by $2\times 2$ minors of the Hilbert-Burch matrix

[TABLE]

Therefore, the minimal free $R$-resolution of $R/J$ has the form

[TABLE]

Thus the Hilbert series of $R/J$ is

[TABLE]

Since $I({{\mathbb{X}}})=I({{\mathbb{X}}}^{\prime})\cap I(p_{5})$, one has a short exact sequence

[TABLE]

Direct inspection gives

[TABLE]

We obtain that

[TABLE]

Since $J\subseteq I({{\mathbb{X}}})$, it follows that $J=I({{\mathbb{X}}})$.

By a projective transformation, we may assume that $p_{1},p_{2},p_{3}$ are coordinate points, $p_{4}=[1:1:1]$ and $p_{5}=[a:b:1]$ where $a,b\neq 0,1$ and $a\neq b$. Then the defining ideal $I({{\mathbb{X}}})$ is minimally generated by:

[TABLE]

A calculation yields that the Jacobian ideal $I$ of $I({{\mathbb{X}}})$ minimally generated by $4$ cubics in $R/I({{\mathbb{X}}})$. A computation in [1] yields that the relation type number of the ideal Jacobian ideal $I/I({{\mathbb{X}}})\subseteq R/I({{\mathbb{X}}})$ is two. Therefore, by [8, Corollary 2.17], it is enough to show that $J\cap I^{2}\subseteq JI$, which can be check easily.

Configurations (4),(5). We apply Theorem 2.3 – to find the generators of the defining ideal – and Proposition 1.3. Then $J\cap I^{t}=JI^{t-1}$ for $t\geq 5$. A calculation shows that $J\cap I^{m}=JI^{m-1}$ for $m=2,3,4$, which complete the proof. ∎

Now let $\mathbb{X}$ be a set of $6$ points in $\mathbb{P}^{2}$. There exists only eleven configurations in their geometry [5] (we show these configurations schematically in figure 1):

1. (1)

$\mathbb{X}$ is in general linear position. 2. (2)

$\mathbb{X}$ is a $6$-fold collinear configuration. 3. (3)

$\mathbb{X}$ is a $5$-fold collinear configuration. 4. (4)

$\mathbb{X}$ is a $4$-fold collinear configuration. 5. (5)

$\mathbb{X}$ is a $4$-fold collinear configuration such that the straight line through the remaining two points intersect in a point of ${{\mathbb{X}}}$. 6. (6)

$\mathbb{X}$ is a $3$-fold collinear configuration such that the remaining three points are in general linear position . 7. (7)

${{\mathbb{X}}}$ is a $3$-fold collinear configuration such that the remaining points are collinear and the straight line through them intersect in a point off ${{\mathbb{X}}}$. 8. (8)

${{\mathbb{X}}}$ is a $3$-fold collinear configuration such that the straight line through two of the remaining points intersect in a point of ${{\mathbb{X}}}$. 9. (9)

${{\mathbb{X}}}$ is a $3$-fold collinear configuration such that the two straight line through of the remaining points intersect in points of ${{\mathbb{X}}}$. 10. (10)

${{\mathbb{X}}}$ is a $3$-fold collinear configuration such that the three straight line through of the remaining points intersect in points of ${{\mathbb{X}}}$. 11. (11)

${{\mathbb{X}}}$ is on an irreducible conic.

For $6=\binom{3+1}{2}$ points in general linear position we study the general case of $s=\binom{d+1}{2}$ points with $d\geq 2$.

Let ${{\mathbb{X}}}$ be a finite set of $s=\binom{d+1}{2}$ points in general linear position in ${\mathbb{P}}^{2}$. Then ${{\mathbb{X}}}$ has maximal Hilbert function by [3], that is

[TABLE]

Let $\mathcal{N}$ denote the set of all monomials of degree $d$ in $R=k[x,y,z]$ expect the set of $d+4$ monomials

[TABLE]

Note that $|\mathcal{N}|=\binom{d+2}{d}-(d+4)=s-3$. Let $h_{i}=\sum_{m\in\mathcal{N}}^{s-3}\alpha_{i}^{(j)}m$ denote a $k$-linear combination of all monomials in $\mathcal{N}$ with coefficient $\alpha_{i}^{(j)}\in k$ for $1\leq i\leq d+1$ and $1\leq j\leq s-3$. In [4], it is proved that the defining ideal of $s=\binom{d+1}{2}$ points in general linear position is generated by $d+1$ forms of degree $d$. In the following we find these generators, explicitly.

Proposition 3.2**.**

With assumption and notation as above, the defining ideal of $s=\binom{d+1}{2}$ points in general linear position in ${\mathbb{P}}^{2}$ is minimally generated by form of degree $d$ of the form

[TABLE]

where $1\leq j\leq s-3$ and the coefficient $\alpha_{i}^{(j)}$ are uniquely determined by the coordinate of the points.

Proof.

We may assume that the points are the columns of the matrix

[TABLE]

Since the points are in general linear position, all $3\times 3$ minors of $M$ are non-zero. Consider the following matrices

[TABLE]

and

[TABLE]

The matrix $A$ and $B$ are of size $(s-3)\times\dfrac{d^{2}-d-2}{2}$ and $(s-3)\times(d-2)$, respectively. Thus the concatenation of $A$ and $B$ is a square matrix of size $(s-3)\times(s-3)$. To find $\alpha_{i}^{(j)}$ it is enough to solve the matrix equations

[TABLE]

for $i=1,\ldots,d-1$ and

[TABLE]

Since all $3\times 3$ minors of $M$ are non-zero, the determinant of the matrix \left[\begin{array}[]{l|l}A&B\end{array}\right] dose not vanish. Therefore, the systems (9) and (10) has unique solutions. Furthermore, by Cramer’s rule, $\alpha_{i}^{(j)}\neq 0$. Consider the ideal $J\subset R=k[x,y,z]$ generated by the form of degree $d$ as in the statement, where $\alpha_{i}^{(j)}$ are uniquely determined solution of (9) and (10). Thus the ideal $J$ vanishes on ${{\mathbb{X}}}$ and hence $J\subseteq I({{\mathbb{X}}})$. We show that $J$ and $I({{\mathbb{X}}})$ have the same Hilbert function, hence must be equal.

We claim that the Gröbner basis of $J$ with respect to the deg-revlex term ordering with $x>y>z$ is the set

[TABLE]

where $G$ is the generating set of $J$ and $\beta^{(j)}$ is a certain polynomial expression of the $\alpha$’s. For this, we consider the $S$-polynomials of elements in this set. First, we look at the $S$-polynomial of $f_{1}$ and $f_{2}$

[TABLE]

which upon division by the generators of $J$ is reduces to $f_{d+2}:=y^{d}z+z\sum_{m\in\mathcal{N}}\beta^{(j)}m$, where $\beta^{(j)}$ is a certain polynomial like expression in the $\alpha$’s. Since the initial monomial $y^{d}z$ of $f_{d+2}$ is not divisible by the initial term of any generators of $J$ we add $f_{d+2}$ to the generating set of $J$. By reducing the terms in the $\mathrm{S}$-polynomial $S(f_{i},f_{j})$ with $1\leq i<j\leq d+2$ which are divisible by the initial term of $f_{k}\ (k=1,\ldots,d+2)$, we conclude that $S(f_{i},f_{j})$ reduces to zero which prove the claim.

Thus, the following set of monomials is a minimal generating set for the initial ideal of $J$:

[TABLE]

Hence for $r\geq d$,

[TABLE]

Therefore for any $r\geq 0$,

[TABLE]

Therefore, $I({{\mathbb{X}}})=J$ follows by (8). ∎

Proposition 3.3**.**

Let ${{\mathbb{X}}}\subseteq{\mathbb{P}}^{2}$ be a set of six points in general linear position. Then $I_{2}(\Theta)=(x,y,z)^{4}$, where $\Theta$ is the Jacobian matrix of $I({{\mathbb{X}}})$. In particular, the Jacobian ideal of $I({{\mathbb{X}}})$ is ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion-free.

Proof.

We may assume that the points are columns of the matrix

[TABLE]

Since the points are in general linear position, the following relations come out

[TABLE]

By Proposition 3.2, the ideal $I({{\mathbb{X}}})$ is generated by the cubics:

[TABLE]

where $\alpha_{i}^{(j)}$ for $i=1,2,3,4,\ j=1,2,3$ are solutions of the matrix equations

[TABLE]

[TABLE]

By relations (11), all $3\times 3$ minors of the following matrix is non-zero.

[TABLE]

Therefore, by Cramer’s rule, $\alpha_{i}^{(j)}\neq 0$ which is a certain polynomials expression of the coordinate of last two points. The Jacobian matrix of $I({{\mathbb{X}}})$ is

[TABLE]

Consider on the ring $R=k[x,y,z]$ the lex term ordering with $x>y>z$. Denote by $[i\ j\ |\ k\ l]$ a $2\times 2$ minor of the Jacobian matrix $\Theta$, where $1\leq i<j\leq 4,\,1\leq k<l\leq 3$. Setting

[TABLE]

[TABLE]

The initial terms of $g_{1},\ldots,g_{8}$ are $x^{4},\,x^{3}y,\,x^{3}z,\,x^{2}y^{2},\,x^{2}yz,\,x^{2}z^{2},\,xy^{3},\,xy^{2}z$, respectively. We have

[TABLE]

where $\lambda_{1}=2\alpha_{1}^{(1)}\alpha_{3}^{(2)}-2\alpha_{1}^{(2)}\alpha_{3}^{(1)}$ and $\lambda_{2}=\alpha_{1}^{(1)}\alpha_{3}^{(3)}-\alpha_{1}^{(3)}\alpha_{3}^{(1)}$. If $(2\alpha_{3}^{(3)}-4\alpha_{1}^{(2)})\neq 0$, then we reduce the terms $x^{3}z,\,x^{2}z^{2},\,xy^{2}z$ by $g_{1},\ldots,g_{8}$ which gives the form, say $g_{9}$ and the latter has initial term $xyz^{2}$. If $(2\alpha_{3}^{(3)}-4\alpha_{1}^{(2)})=0$, then we choose another minor which contain the term $xyz^{2}$ and apply the same process as above. We make the same process to find the forms $g_{10},\ldots,g_{15}$ which have initial terms

[TABLE]

respectively. Thus, the ideal generated by $g_{1},\ldots,g_{15}$ is contained in the ideal $I_{2}(\Theta)$ and $g_{i}$’s are linearly independents over $k$ since its coefficient matrix is upper triangular matrix which is non-singular by above construction. Since $I_{2}(\Theta)\subseteq(x,y,z)^{4}$, it follows that $I_{2}(\Theta)=(x,y,z)^{4}$. The second assertion follows by [9, Lemma 1.4]. ∎

Remark 2**.**

By a similar argument as in Proposition 3.3, we can show that $I_{2}(\Theta)=(x,y,z)^{2d-2}$ for $d=4,5$, which implies that the Jacobian ideal of the defining ideal of a set of $10$ and $15$ points is ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion free.

we derive the following

Conjecture 3.4**.**

Let ${{\mathbb{X}}}\subseteq{\mathbb{P}}^{2}$ be a set of $s=\binom{d+1}{2}$ points in general linear position. If $d\geq 6$, then $(x,y,z)^{2d-2}\subseteq I$, where $I$ is the Jacobian ideal of $I({{\mathbb{X}}})$.

Finally, we characterize ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion freeness of six points.

Proposition 3.5**.**

Let $\mathbb{X}$ be a set of $6$ points in $\mathbb{P}_{k}^{2}$. Then the Jacobian ideal of $I({{\mathbb{X}}})$ is not ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}$-torsion-free if and only if ${{\mathbb{X}}}$ is one of the configurations (2),(3),(7),(11).

Proof.

The configurations (2) and (3) follows by Theorem 2.2. The defining ideal $I({{\mathbb{X}}})=(f,g)$ of ${{\mathbb{X}}}$ for the configurations (7) and (11) minimally generated by a conic $f$ and a cubic $g$ ([5]). The Jacobian ideal $I$ of $J$ is zero dimensional ideal. Thus $I\cap k[z]\neq 0$. We may assume that $z^{n}\in I$ and $n$ is minimum with this property. Then $z^{n-1}f\in J\cap I^{2}$ and dose not belong to $JI$ by the minimality of $n$. Therefore, ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}_{J\subset I}\neq 0$.

Now we prove that ${{\mathcal{V}}}\kern-5.0pt{{\mathcal{V}}}=0$ for the remaining seven configurations.

Configuration (1). Follows by Proposition 3.3.

Configuration (4),(5). we use Theorem 2.3 – to find the generators of the defining ideal – and Proposition(1.3). Then $J\cap I^{t}=JI^{t-1}$ for $t\geq 6$. A computation in [1] shows that $J\cap I^{m}=JI^{m-1}$ for $m=2,3,4,5$.

Configuration (6). By Theorem 2.4, the ideal $I({{\mathbb{X}}})$ is generated by the $4$ forms of degree $3$. The same argument as in Proposition 3.3 implies that $I_{2}(\Theta)=(x,y,z)^{4}$, where $\Theta$ is the Jacobian matrix of $I({{\mathbb{X}}})$. Therefore, the assertion follows by [9, Lemma 1.4].

Configurations (8),(9),(10). By a projective transformation, we may assume that the points in configurations (8) , (9) and (10) are the columns of the matrices

[TABLE]

[TABLE]

where $a,b\in k$ with $a,b\neq 0,1$ and $a\neq b$. By Theorem 2.4, rather by its proof,mn the defining ideal of these configurations are:

[TABLE]

where $\lambda=-ab+a+b-1$ in $I({{\mathbb{X}}}_{8})$.

As the same argument for points in general linear position (Proposition 3.3), we can find $15$ linearly independent forms of degree $4$ among $18$ non-zero minors of the Jacobian matrix of the defining ideals. Thus $I_{2}(\Theta)=(x,y,z)^{4}$ which proves the assertion. ∎