Contents - CSDL Khoa học

K ¨ AHLER DIFFERENTIAL MODULES AND CONFIGURATIONS OF POINTS IN P

²

Tran Nguyen Khanh Linh^a∗, Le Ngoc Long^a

aDepartment of Mathematics, University of Education, Hue University, Vietnam

∗Corresponding author: Email: tnkhanhlinh@hueuni.edu.vn

Article history Received: 07/05/2021

Received in revised form: 24/05/2021|Accepted: 26/07/2021

Abstract

Given a finite set of points in the projective plane, we use the module of K¨ahler dif- ferentials to investigate the configurations of these points. More precisely, depending on the values of the Hilbert function of the module of K¨ahler differential 3-forms we determine whether the set of points lies on a non-singular conic or on two different lines or on a line.

Keywords: K¨ahler differential module; finite set of points; configuration of points;

Hilbert function.

Contents

1 Introduction 64

2 K¨ahler differential modules for sets of points 65

3 Several configurations of points in the projective plane 68 Acknowledgements. The authors were supported by Hue University under grant number DHH2021-03-159.

Article identifier: http://tckh.dlu.edu.vn/index.php/tckhdhdl/article/view/887 Article type: (peer-reviewed) Full-length research article/review article Copy right ©2022 The author(s).

Licensing: This article is licensed under a CC BY-NC-ND 4.0

1 Introduction

The theory of K¨ahler differentials has been used very extensively in the literature as a tool to study finitely generated algebras. Many structural properties of the algebra can be reflected in terms of the modules of K¨ahler differentials, for instance, the smoothness criterion, the regular criterion, the ramification criterion of the algebra (see Kunz, 1986; Scheja and Storch, 1988). In de Dominicis and Kreuzer, 1999, the authors introduced methods using this theory into the study of finite sets of points in the projective n-spacePⁿ over a field K of charac- teristic zero. Explicitly, given a set of points X⊆Pⁿ with homogeneous coordinate ring R=K[X₀, ...,X_n]/I_X, the module of K¨ahler differential 1-forms ofR/K isΩ¹_R/K =J/J², whereJ is the kernel of the multiplication mapµ :R⊗_KR→R,f⊗g7→ f g. One of main results of de Dominicis and Kreuzer, 1999 is the formula

HF_Ω1

R/K(i) = (n+1)HF_X(i−1)− ∑ⁿ

j=1

HF_X(i−d_j)

whenXis a complete intersection of hypersurfaces of degreesd₁, ...,d_n. Also, they showed that the module Ω¹_R/K contains more information about X. Later, in Kreuzer et al., 2015, 2019, these differential algebra techniques have been developed for arbitrary 0-dimensional schemes inPⁿ.

In this paper we are interested in applying the theory of K¨ahler differentials to investigate the geometrical structure of finite sets of points in the projective spaces. More precisely, we first look at the Hilbert functions of the modules of K¨ahler differential m-forms for a finite set of points in the projectiven-spacePⁿover a fieldKof characteristic zero and use these to determine the smallest projective space which contains the set of points. Then we restrict our attention to finite sets of points inP²and show whether a set of points lies on a non-singular conic, on two different lines or on a line by detecting the the Hilbert functions of the modules of K¨ahler differential 3-forms.

Now let us describe the content of the paper in details. In Section 2, we first introduce the modules of K¨ahler differentialm-forms and their basic properties for a finite setXof points in Pⁿ. In particular, we define the matrix of coordinates of points of Xand show that the rank of the matrix is exactly the smallest positive integer msuch that the module of K¨ahler differentialm-forms is not zero (see Theorem 2.6). Also, we show that the Hilbert function of the module of K¨ahler differentialm-forms for a subset ofXis bounded by that one forX. Next, in Section 3, we use the modules of K¨ahler differentialm-forms forXto investigate the configurations of Xin caseX⊆P². Basically, we apply the Hilbert function of the module of K¨ahler differential 3-forms to characterize the geometrical properties of X. Several nice properties about the shapes of X are provided. For example, Proposition 3.2 shows thatX lies on a conic, not a double line if and only if HF_Ω3

R/K(3) =1 and HF_Ω3

R/K(i)≤1 for alli≥0.

Specially, Theorem 3.8 shows us how to determine the setXto be on a line, on two different lines, or on a non-singular conic by checking the certain values of the Hilbert functions of Ω³_R/K.

All examples in this paper were calculated by using the computer algebraic system ApCo- CoA (see The ApCoCoA Team, 2021). Some of the results in this paper were improved and developed from part of Linh, 2015.

2 K¨ahler differential modules for sets of points

In the following let K be a field of characteristic zero, let S=K[X₀, . . . ,X_n], and let X= {p₁, . . . ,p_s} be a set of s distinctK-rational points in the projective n-space Pⁿ. One may suppose that no point of X lies on the hyperplane at infinity Z⁺(X₀). This enables us to write p_j= (1 : p_j1:· · ·: p_jn)with p_j1, . . . ,p_jn∈K for j=1, . . . ,s. Each point p_j∈Xhas its associated homogeneous prime ideal inSgiven by

p_j:=hp_j1X₀−X₁, . . . ,p_jnX₀−X_ni ⊆S.

Then the homogeneous vanishing ideal ofXisI_X=^Ts_j=1p_jand the homogeneous coordinate ofXisR=S/I_X. The ringRis a Cohen-Macaulay ring of dimension 1 and is also a graded K-algebra (see e.g. Kreuzer and Robbiano, 2005, Section 6.3 or Linh, 2015, Section 2.4).

The ringR^e:=R⊗_KRis known as theenveloping algebraof the algebraR/K. We have the canonical multiplication mapµ:R^e→Rwithµ(f⊗g) = f gfor all f,g∈R. It is clearly true that this map is homogeneous of degree zero and its kernelJ :=Ker(µ)is a homogeneous ideal ofR^e. For a homogeneous system of generators ofJ see Kunz, 2005, Theorem G.7.

These notions lead us to the following definition of K¨ahler differential modules of the algebra R/K.

Definition 2.1. Letmbe a positive integer.

(a) The gradedR-moduleΩ¹_R/K :=J/J²is called themodule of K¨ahler differentialsof R/K. The homogeneousK-linear mapd:R→Ω¹_R/Kgiven by f 7→ f⊗1−1⊗f+J² is called theuniversal derivationofR/K.

(b) The exterior power Ω^m_R/K :=^Vm_RΩ¹_R/K is called the module of K¨ahler differential m- formsofR/K.

Byx_iwe denote the image of X_iinRfori=0, . . . ,n. Then we have deg(dx_i) =deg(x_i) =1 andΩ¹_R/K =Rdx₀+· · ·+Rdx_n. Hence Ω^m_R/K =0 if m≥n+2, and for m≥1, Ω^m_R/K is a finitely generated gradedR-module and its Hilbert function is defined by

HF_Ω^m

R/K(i):=dim_K(Ω^m_R/K)_i, for all i∈Z.

Moreover, from Kunz, 1986, Proposition 4.12 we get the following presentation forΩ^m_R/K. Proposition 2.2. For1≤m≤n+1, the graded R-moduleΩ^m_R/K has a presentation

Ω^m_R/K∼=Ω^m_S/K/(I_XΩ^m_S/K+dI_XΩ^m−1_S/K) where dI_X=hdF|F ∈I_Xi.

In view of Kreuzer et al., 2019, Proposition 4.4, the Hilbert polynomial of the module of K¨ahler differentialm-formsΩ^m_R/K can be described as follows.

Proposition 2.3. For1≤m≤n+1, the Hilbert polynomial ofΩ^m_R/K satisfies HP_Ω^m

R/K(z) =

(s if m=1, 0 if m≥2.

In addition, the regularity index ofΩ^m_R/K is bounded byri(Ω^m_R/K)≤2 ri(R) +m.

The next result found in Scheja and Storch, 1988, Proposition 85.12 is useful for the proof of Theorem 2.6.

Lemma 2.4. For two R-modules M and N, there is a canonical isomorphism V_n

R(M⊕N)∼=∑ⁿ_i=0^Vn−i_R (M)⊗_R^Vi_R(N).

Lemma 2.5. Let1≤m≤n+1and letK be the graded R-module K =h(^∂F

∂x₀, . . . ,_∂x^∂F

n)∈Rⁿ⁺¹|F∈I_Xi.

There is an exact sequence of graded R-modules

0−→K ∧_R^Vm_R(Rⁿ⁺¹)−→^Vm+1_R (Rⁿ⁺¹)−→Ω^m+1_R/K(m+1)−→0. (1) Proof. By de Dominicis and Kreuzer, 1999, Proposition 1.3, we have an epimorphism of gradedR-modulesϕ:Rⁿ⁺¹→Ω¹_R/K(1)withϕ((F₀, ...,F_n)) =F₀dx₀+· · ·+F_ndx_nforF₀, ...,F_n∈ R and the kernel ofϕ is K . Thus the exact sequence (1) follows from Scheja and Storch, 1988, p. X. 83.

From the coordinates of points p₁, ...,p_sofX, we form the matrix A_X= (p_jk)_i=1,...,s

j=0,...,n

∈Mat_s×(n+1)(K),

and we letρ_X denote the rank of the matrixA_X. The next theorem shows how the rankρ_X can be detected via the Hilbert functions of the K¨ahler differential modules forX.

Theorem 2.6. The setXhasρ_X=m if and only ifHF_Ω^m

R/K(i)6=0for some i andHF

Ω^m+1_R/K(i) = 0for all i∈Z.

In order to prove Theorem 2.6, we need only prove the following two lemmas.

Lemma 2.7. If m>ρ_X, thenHF_Ω^m

R/K(i) =0for all i∈Z.

Proof. SetM=K ∧_R^Vρ_R^X(Rⁿ⁺¹)andN=^Vρ_R^X+1(Rⁿ⁺¹). According to Lemma 2.5, it suf- fices to show that HF_M(i) =HF_N(i)for alli≥0. Let{e₁, . . . ,e_n+1}be a standard basis of the graded-freeR-moduleRⁿ⁺¹, and letV be the space of solutions of the homogeneous system of linear equations with coefficient matrixA_X. Then we havet:=dim_K(V) =n+1−ρ_X. Let {v₁, ...,v_t}be a K-basis ofV. W.l.o.g. we may assume thatv_j=e_j+∑ⁿ⁺¹_k=t+1a_jke_k for j=1, ...,t. For each j∈ {1, ...,t}we see thatX_j−1+∑ⁿ⁺¹_k=t+1a_jkX_k−1∈I_X, and hencev_j∈K . This implieshv₁, . . . ,v_ti_R⊆K ⊆Rⁿ⁺¹.

Now we need only prove thatM⁰:=hv₁, . . . ,v_ti_R∧_R^Vρ_R^X(Rⁿ⁺¹) =N. InM⁰, we have v_j∧e_t+1∧ · · · ∧e_n+1=e_j∧e_t+1∧ · · · ∧e_n+1.

If we can show thate_i1∧ · · · ∧e_i

ρX+1 ∈M⁰for all{i₁, . . . ,i_k} ⊆ {1, . . . ,t},{i_k+1, . . . ,i_ρ

X+1} ⊆ {t+1, . . . ,n+1} and 1≤k≤ ρ_X implies e_j1∧ · · · ∧e_j

ρX+1 ∈M⁰ for all {j₁, . . . ,j_k+1} ⊆ {1, . . . ,t}and{j_k+2, . . . ,j_ρ

X+1} ⊆ {t+1, . . . ,n+1}. Then the set {e_i1∧ · · · ∧e_i

ρX+1 | {i₁, . . . ,i_ρ

X+1} ⊆ {1, . . . ,n+1},∃i_k∈ {1, . . . ,t} }

is contained inM⁰, and henceM⁰=N, since this set generatesNas anR-module.

To seee_j1∧ · · · ∧e_j

ρX+1 ∈M⁰, we represent e_j1∧ · · · ∧e_jρ

X+1= (e_j1+

n+1

∑

`=t+1

a_{j1`</sub>e<sub>`})∧e_j2∧ · · · ∧e_jρ

X+1

− ⁿ⁺¹∑

`=t+1

(−1)^ρXa_j1`(e_j2∧ · · · ∧e_j

ρX+1∧e_`)

=v_j1∧e_j2∧ · · · ∧e_j

ρX+1

− ⁿ⁺¹∑

`=t+1

(−1)^ρXa_j1`(e_j2∧ · · · ∧e_jk+1)∧(e_jk+2∧ · · · ∧e_j

ρX+1∧e_`).

It follows from the hypothesis thate_j1∧ · · · ∧e_j

ρX+1 ∈M⁰, as desired.

Lemma 2.8. We haveHF

Ω^ρ_R/K^X (i)6=0for some i∈Z.

Proof. Using the notation given in the proof of Lemma 2.7, letU=he_t+1, . . . ,e_n+1i_R, V = hv₁, . . . ,v_ti_R, andN =^Vρ_R^X(Rⁿ⁺¹). ThenU,V are graded-free R-modules of rankρ_X andt, respectively. We want to show thatV∧_R^Vρ_R^X−1(Rⁿ⁺¹)(N. We haveV⊕U =Rⁿ⁺¹and

V∧_R^Vρ_R^X−1(Rⁿ⁺¹) =V∧_R^Vρ_R^X−1(V⊕U)

=V∧_{R ρ}

X−1

∑

i=0

Vρ_X−1−i

R (V)⊗_R^Vi_R(U)

=

ρ_X−1

∑

i=0

Vρ_X−i

R (V)⊗_R^Vi_R(U),

where the second equality follows from by Lemma 2.4. SinceU∩V =h0i, the graded-free R-moduleV∧_R^Vρ_R^X−1(Rⁿ⁺¹)has rank

rank(V∧_R^Vρ_R^X−1(Rⁿ⁺¹)) =

ρ_X−1

∑

i=0

n+1−ρ_X ρ_X−i

· ^ρ_i^X .

Moreover,N is a graded-freeR-module of rank rank(N) =∑^ρ_i=0^{X n+1−ρ}_ρ ^X

X−i

· ^ρ_i^X

, and conse- quently rank(N)−rank(V∧_R^Vρ_R^X−1(Rⁿ⁺¹)) =1 orV∧_R^Vρ_R^X−1(Rⁿ⁺¹)(N. An application of the the exact sequence (1) withV ⊆K yields HF

Ω^ρ_R/K^X (i)6=0 for somei∈Z. Due to Lemmas 2.7 and 2.8, HF_Ω^m

R/K(i)6=0 for someiand HF

Ω^m+1_R/K(i) =0 for alli∈Zif and only ifρ_X=m, and Proposition 2.6 is completely proved.

Now let us look at an explicit example.

Example 2.9. Let X={p₁, ...,p₆} ⊆P⁴ be the set of six Q-rational points, where p₁ = (1 : 0 : 0 : 0 : 1), p₂= (1 :−1 : 1 :−1 : 2), p₃= (1 : 1 : 1 : 1 : 2), p₄= (1 : 2 : 4 : 8 :−1), p₅= (1 :−2 : 4 :−8 : 11), and p₆= (1 : 3 : 9 : 27 :−14)are on the intersection of a twisted cubic curve defined by X₀X₂−X₁²,X₁X₃−X₂²,X₀X₃−X₁X₂ and an hyperplane defined by F₁=X₀+X₁+X₂−X₃−X₄.

By formingAXand calculating its rank, we haveρ_X=4. The homogeneous vanishing ideal I_XofXis generated by{F₁,F₂,F₃,F₄,F₅}with

F₂=X₁²−2X2X₃+X₃²−X₂X₄+X₃X₄,

F₃=X₁X₂+²³₂₂X₂X₃−₁₁⁸X₃²−₁₁³X₁X₄+₁₁⁴X₂X₄−³₄X₃X₄, F₄=X₂²−₂₂¹X₂X₃−₁₁³X₃²+₁₁³X₁X₄−₁₁⁴X₂X₄−¹₄X₃X₄, F₅=X₁X₃−₂₂¹X₂X₃−₁₁³X₃²+₁₁³X₁X₄−₁₁⁴X₂X₄−¹₄X₃X₄ and HF_X=HF_R: 1 4 6 6· · ·. In this case we also have HF_Ω4

R/K(4) =16=0 and HF_Ω5 R/K

(i) =0 for alli∈Z.

Remark 2.10. WhenX={p₁, . . . ,p_s} ⊂Pⁿis a set ofsdistinctK-rational points that lie on a line, the Hilbert functions ofXand ofΩ^m_R

X/K satisfy HF_X: 1 2 3 4 · · · s−1s s · · ·

HF_Ω1

RX/K : 0 2 4 6· · · (2s−2) (2s−1) (2s−2) (2s−3)· · · (s+1)s s· · · HF_Ω2

RX/K : 0 0 1 2 · · · (s−2) (s−1) (s−2) · · ·2 1 0 0· · · and HF_Ω^m

RX/K(i) =0 form≥3 and for alli∈Z.

We end this section with a relation between the Hilbert functions of the K¨ahler differential modules forXand its subsets. This property is also true whenXis a 0-dimensional scheme inPⁿ.

Lemma 2.11. LetYbe a subset ofXand let R_Ybe the homogeneous coordinate ofY. Then 1≤m≤n+1and for all i∈Zwe have

HF_Ω^m

RY/K(i)≤HF_Ω^m

R/K(i).

Proof. We know thatI_X⊆I_Y, and so there is a surjective homomorphism of ringsπ :R→ R_Y=S/I_Y. This map induces a surjective homomorphism of gradedR-modulesφ :Ω¹_R/K→ Ω¹_R

Y/K. For 1≤m≤n+1, the map φ induces a surjective homomorphism of graded R- modulesφ^(m):Ω^m_R/K →Ω^m_R

Y/K. Therefore we get the inequality HF_Ω^m

RY/K(i)≤HF_Ω^m

R/K(i)for alli∈Z.

3 Several configurations of points in the projective plane

In this section, we restrict our attention to the sets of K-rational pointsX={p₁, . . . ,p_s}in the projective planeP². More precisely, we will discuss some geometrical configurations of pointsXinP² which are reflected in terms of the Hilbert function of the module of K¨ahler differential 3-forms. Firstly, we look at a representation of this module. We denote

∂I_X:=h^∂F

∂X_i |F∈I_X,0≤i≤ni ⊆S.

The ideal∂I_Xis also known as then-th Jacobian ideal(or then-th K¨ahler different) ofR/K (see e.g. Kunz, 1986, Section 10). We can use this notation to give an explicit description for the module of K¨ahler differential 3-forms ofR/K as follows.

Lemma 3.1. There is an isomorphism of graded R-modules Ω³_R/K∼= (S/∂I_X)(−n−1).

In particular,HF_Ω3

R/K(i) =HF_S/∂I

X(i−n−1)for all i∈Z. Proof. See Kreuzer et al., 2019, Corollary 2.3.

Now we may give a criterion for the setXto lie on a conicC =Z⁺(C), where

C=

∑

0≤i,j≤2

a_{i j}X_iX_j (a_{i j}∈K). (2)

Proposition 3.2. A set of s distinct K-rational points X⊆P² lies on a conic C =Z⁺(C) which is not a double line if and only if we haveHF_Ω3

R/K(i)≤1for all i∈ZandHF_Ω3

R/K(3) = 1.

Proof. We may assume thatC6=a`<sup>2</sup> for any linear form`∈S anda∈K and thata₀₀ 6=0 (after a change of coordinates). We have _∂X^∂C

i =∑²_j=0a_{i j}X_j∈∂I_X fori=0,1,2, asC∈I_X. PutA := (a_{i j})_i,j=0,1,2∈Mat₃(K). Then rank(A) =1 if and only ifA is of the form

A =









a₀₀ a₀₁ a₀₂ a₀₁ ^a

2 01

a₀₀

a01a02

a₀₀

a₀₂ ^a01_a^a02

00

a²₀₂ a00







 .

In this case we get C = _a¹

00`<sup>2</sup> with ` = ^∂C

∂X0 =∑²_j=0a_0jX_j, which is impossible. Hence rank(A)≥2 and suppose that two first rows of A are linearly independent. Then _∂X^∂C

0,

∂C

∂X1 form a regular sequence forS. Notice that J:=h^∂C

∂X0,_∂X^∂C

1i_S⊆ h_∂X^∂C

i |0≤i≤2i ⊆∂I_X. Hence, according to Lemma 3.1, for alli∈Zwe have

HF_Ω3 R/K

(i) =HF_S/∂I

X(i−3)≤HF_S/J(i−3)≤1.

On the other hand, we also have∂I_X⊆ hX₀,X₁,X₂i_S, becauseXdoes not lie on a line. This implies HF_Ω3

R/K(3) =1.

Conversely, assume that HF_Ω3

R/K(i)≤1 for alli∈Zand HF_Ω3

R/K(3) =1. Theorem 2.6 guar- antees that X lies neither on a line nor on a double line. If X does not lie on any conic, then the homogeneous vanishing idealI_X is generated in degrees greater than 2, and hence HF_∂I

X(1) =0. By Lemma 3.1, we get HF_Ω3

R/K(4) =3>1, contradiction. ThereforeXmust lie on a conic.

Due to the proof of Proposition 3.2,Xlies on a line if and only if HF_Ω3

R/K(i) =0 for alli∈N. Lemma 3.3. If the setX={p₁, ...,p_s} ⊆P²lies on a non-singular conicC =Z⁺(C), then HF_Ω3

R/K(i) =0for i6=3andHF_Ω3

R/K(3) =1.

Proof. LetC be given in (2) and letA = (a_{i j})_i,j=0,1,2∈Mat₃(K) be the coefficient matrix ofC. It is well-known that C is a non-singular conic if and only if rank(A) =3. The last condition yieldshX₀,X₁,X₂i_S=h_∂X^∂C

i |0≤i≤2i_S⊆∂I_X. HencehX₀,X₁,X₂i_S=∂I_X, and the claim follows by Lemma 3.1.

Next we consider the case that the setXlies on two different lines.

Proposition 3.4. Let X⊆P² be a set of s distinct K-rational points which lie on on two different lines.

(a) If s=5and none of four points ofXlie on a line, then HF_Ω3

R/K : 0 0 0 1 1 0 · · ·.

(b) If s≥5and there exist five points such that no four of them lie on a line, then HF_Ω3

R/K(3) =HF_Ω3

R/K(4) =1.

Proof. Suppose thatXlies on two lines defined by`<sub>1</sub>=∑<sup>2</sup><sub>j=0</sub>a<sub>j</sub>X<sub>j</sub>and`₂=∑²_j=0b_jX_jwith a_i,b_j∈K. Note that gcd(`<sub>1</sub>, `₂) =1 and w.l.o.g. assume thata₀b₁−a₁b₀6=0. We first treat the case of five points.

(a) Assume that X1 ={p₁,p₂,p₃} ⊆ Z⁺(`<sub>1</sub>) and X2 ={p<sub>4</sub>,p<sub>5</sub>} ⊆Z<sup>+</sup>(`₂). Then we have I_X1 =h`<sup>0</sup><sub>1</sub>`⁰₂`<sup>0</sup><sub>3</sub>, `₁i_S andI_X2 =h`<sup>0</sup><sub>4</sub>`⁰₅, `<sub>2</sub>i<sub>S</sub> for suitable linear forms`⁰_i=∑²_j=0c_{i j}X_j (i= 1, ...,5,c_{i j} ∈K). Clearly, the initial degree of the ideal I_X is 2 and I_X =I_X1∩I_X2. For a non-zero elementF∈(I_X)₂, we may write

F=F₁`<sub>1</sub>=F<sub>2</sub>`₂+c`<sup>0</sup><sub>4</sub>`⁰₅

for somec∈KandF₁,F₂∈S₁. Since`<sub>1</sub>(p<sub>4</sub>)6=0 and`₁(p₅)6=0, we haveF₁(p₄) =F₁(p₅) = 0, and soF₁=c⁰`<sub>2</sub>withc<sup>0</sup>∈K\ {0}. HenceF ∈(I<sub>X</sub>)<sub>2</sub>if and only ifF =h`₁`₂i_K. Observe that

∂`<sub>1</sub>`₂

∂Xi =`<sub>1</sub><sup>∂`</sup>_∂X²

i+`<sub>2</sub><sup>∂`1</sup>

∂Xi =b_i`<sub>1</sub>+a<sub>i</sub>`₂∈∂I_X (i=0,1,2)

andhb₀`<sub>1</sub>+a<sub>0</sub>`₂,b₁`<sub>1</sub>+a<sub>1</sub>`₂i_K=h`<sub>1</sub>, `₂i_K, asa₀b₁−a₁b₀6=0. Sinceb₂`<sub>1</sub>+a<sub>2</sub>`₂∈ h`<sub>1</sub>, `₂i_K, we obtainh`<sub>1</sub>, `₂i_K= (∂I_X)₁. This implies

HF_Ω3

R/K(4) =HF_S/∂I

X(1) =3−dim_K(h`<sub>1</sub>, `₂i_K) =1.

Furthermore, we have`<sub>1</sub>`⁰₄`⁰₅∈I_X, and hence, fora_i6=0, we get

`<sup>0</sup><sub>4</sub>`⁰₅= _a¹

i(^∂`1`04`05

∂Xi −`<sub>1</sub>(<sup>∂`04`05</sup>

∂Xi ))∈∂I_X.

In particular,h`<sub>1</sub>, `₂, `<sup>0</sup><sub>4</sub>`⁰₅i_S⊆∂I_X. If`<sup>0</sup><sub>4</sub>`⁰₅=H₁`<sub>1</sub>+H<sub>2</sub>`₂for someH₁,H₂∈S₁, thenH₁(p₄) = H₁(p₅) =0 orH₁ =c⁰⁰`<sub>2</sub>, and thus `₂|`<sup>0</sup><sub>4</sub>`⁰₅, this is impossible. Hence `<sup>0</sup><sub>4</sub>`⁰₅∈ h`/ <sub>1</sub>, `₂i_S. It follows that

dim_K(h`<sub>1</sub>, `₂, `<sup>0</sup><sub>4</sub>`⁰₅i_S)₂=dim_K(∂I_X)₂=dim_K(S₂) =6, and consequentlyh`<sub>1</sub>, `₂, `<sup>0</sup><sub>4</sub>`⁰₅i_S=∂I_Xand HF_∂I

X(i) =HF_S(i)for alli≥2. Therefore HF_Ω³

R/K(i) = 0 for alli>4.

(b) Because the setXlies on two different lines, Proposition 3.2 implies that HF_Ω3

R/K(3) =1 and HF_Ω3

R/K(4)≤1. LetYbe the set of five points ofXsuch that no four points ofYlie on a line. By (a) and Lemma 2.11, we have 1=HF_Ω3

RY/K(4)≤HF_Ω3

R/K(4)≤1, and hence the last inequality becomes an equality.

When there is a line passing through all but one point ofX, we have the following property.

Lemma 3.5. Let X=Y∪ {p_s} ⊆P², where Y={p₁, . . . ,p_s−1} is a set of s−1 distinct K-rational points on a lineZ⁺(`)and p<sub>s</sub>∈/Z<sup>+</sup>(`). Then

HF_Ω3

R/K(3) =1 and HF_Ω3

R/K(i) =0, ∀i6=3.

Proof. Since I_Y =^Ts−1_j=1p_j, if p_j =h`, `_ji_S with a linear form `<sub>j</sub> ∈S for j= 1, ...,s−1, thenI<sub>Y</sub> is generated by` and F =∏^s−1_j=1`_j. Let us write p_s=hL₁,L₂i_S with suitable L₁=

∑²_j=0a_iX_i,L₂=∑²_j=0b_iX_i∈S₁ such that p₁∈Z⁺(L₁)and p₂∈Z⁺(L₂). SuchL₁,L₂∈S₁ exist, since p_s∈/Z⁺(`). Then we haveI<sub>X</sub>=<sup>Ts</sup><sub>j=1</sub>p<sub>j</sub>=h`,Fi_S∩ hL₁,L₂i_S.

We claim that dim_K(h`,L<sub>1</sub>,L<sub>2</sub>i<sub>K</sub>) =3. If there are a,b,c∈K such that a`+bL₁+cL₂ = 0, then 0=a`(p<sub>1</sub>) +bL<sub>1</sub>(p<sub>1</sub>) +cL<sub>2</sub>(p<sub>1</sub>) =cL<sub>2</sub>(p<sub>1</sub>) and 0=a`(p₂) +bL₁(p₂) +cL₂(p₂) = bL₁(p₂), and sob=c=0 anda`=0. This means thata=b=c=0 and the claim follows.

Consequently, we geth`,L<sub>1</sub>,L<sub>2</sub>i<sub>K</sub>=hX<sub>0</sub>,X<sub>1</sub>,X<sub>2</sub>i<sub>K</sub>. Let`=∑²_j=0c_iX_iwithc_i∈Kand suppose w.l.o.g. thata₀c₁−a₁c₀6=0 andb₀c₁−b₁c₀6=0 (as{`,L<sub>1</sub>,L<sub>2</sub>}is linearly independent over K). Since`L₁, `L<sub>2</sub>∈I<sub>X</sub>, we havea<sub>i</sub>`+c_iL₁,b_i`+c_iL₂∈∂I_Xfor everyi=0,1,2. We have

ha_i`+c<sub>i</sub>L<sub>1</sub>,b<sub>i</sub>`+c_iL₂|i=0,1,2i_K=h`,L₁,L₂i_K=hX₀,X₁,X₂i_K, and subsequently we get∂I_X=hX₀,X₁,X₂i_S. Therefore, Lemma 3.1 yields

HF_Ω3 R/K

(3) =HF_S/∂I

X(0) =1, HF_Ω3 R/K

(i) =HF_S/∂I

X(i−3) =0 for alli6=3.

Note that theCastelnuovo functionofXis defined by

∆HF_X(i):=HF_X(i)−HF_X(i−1)

for alli∈Z. In the proof of the lemma, we see thatI_X=h`,Fi<sub>S</sub>∩hL<sub>1</sub>,L<sub>2</sub>i<sub>S</sub>⊇ h`L₁, `L₂,FL₁,FL₂i_S. It follows that∆HF_X(2) =1, and therefore∆HF_X(i)≤1 for alli≥2 by Kreuzer and Rob- biano, 2005, Corollary 5.5.28. This proves the following corollary.

Corollary 3.6. In the setting of Lemma 3.5, we have∆HF_X(i)≤1for all i≥2.

Using B´ezout’s Theorem (see e.g. Ueno, 1999, Theorem 1.32), we get the following conse- quence for a set of points on a non-singular conic inP².

Corollary 3.7. If Xis a set of s ≥5 distinct K-rational points in P² which are on a non- singular conicC =Z⁺(C), then∆HF_X(2) =2.

Proof. By B´ezout’s Theorem ands≥5,I_Xhas only one non-zero homogeneous polynomial F of degree 2. In this case we must haveF=aCwith somea∈K\ {0}. Let{C,F₁, . . . ,F_t}, deg(F_i)>2, be a homogeneous system of generators ofI_X. Then the Hilbert function ofX is given by

HF_X: 1 3 5 ∗ ∗ · · ·. In particular, we obtain∆HF_X(2) =2.

Combining the above results we get the following classification of sets of points correspond- ing to values of the Hilbert function of the module of K¨ahler differentials.

Theorem 3.8. LetXbe a set of s≥5distinct K-rational points inP².

(a) Xlies on two different lines and no s−1points ofXlie on a line, whenHF_Ω3

R/K(3) =1 andHF_Ω3

R/K(4) =1.

(b) IfHF_Ω3

R/K(3) =1andHF_Ω3

R/K(i) =0for i6=3, thenXeither contains s−1points on a line or lies on a non-singular conic depending on the value of∆HF_X(2)equals 1 or not.

Proof. We see that X lies on a conic if and only if the Hilbert function of Ω³_R/K satisfies HF_Ω3

R/K(4)≤1 and HF_Ω3

R/K(3) =1 by Theorem 2.6 and Proposition 3.2. We consider the following three possibilities for the shapes of the points ofX.

Case 1: If Xlies on two different lines, such that no s−1 points lie on a line, then Proposi- tion 3.4 shows that HF_Ω3

R/K

(3) =HF_Ω3 R/K

(4) =1.

Case 2: In the case that there is a line passing through all but one point ofX, by Lemma 3.5 we have HF_Ω3

R/K(3) =1 and HF_Ω3

R/K(4) =0. Furthermore, Corollary 3.6 shows that

∆HF_X(2) =1.

Case 3: WhenXlies on a non-singular conic, Lemma 3.3 implies HF_Ω3

R/K(3) =1 and HF_Ω3

R/K(4) = 0. In this case we have∆HF_X(2) =2 by Corollary 3.7.

Altogether, the theorem is completely proved.

The following example shows us how to distinguish two sets of K-rational points with the same cardinality using the modules of K¨ahler differentials.

Example 3.9. LetX⊆P²be a set of sixK-rational points on the conicX₀²+2X₁²−X₂², and let Y⊆P²be a set of sixK-rational points on two lines defined by`<sub>1</sub>=x<sub>1</sub>and`₂=x₂. Then the Hilbert functions ofXandYare the same and given by HF_X=HF_Y: 1 3 5 6 6· · ·. Also, their homogeneous vanishing ideals have the same minimal graded free resolution (see Tohaneanu and van Tuyl, 2013, Example 4.1). However, their modules of K¨ahler differentialm-forms have the following Hilbert functions:

HF_Ω1

R/K : 0 3 8 11 10 7 6 6· · ·, HF_Ω1

RY/K : 0 3 8 11 10 7 6 6· · ·, HF_Ω2

R/K : 0 0 3 6 4 1 0· · ·, HF_Ω2

RY/K : 0 0 3 6 5 1 0 0· · ·, HF_Ω³

R/K : 0 0 0 1 0 0· · ·, HF_Ω³

RY/K : 0 0 0 1 1 0 0· · ·.

Thus, by looking at the Hilbert functions of the modules of K¨ahler differentialm-forms, one can distinguish two setsXandY.

References

de Dominicis, G., & Kreuzer, M. (1999). K¨ahler differentials for points in Pⁿ. Journal of Pure and Applied Algebra,141, 153–173.

Kreuzer, M., Linh, T. N. K., & Long, L. N. (2015). K¨ahler differentials and K¨ahler differents for fat point schemes.Journal of Pure and Applied Algebra,2015, 4479–4509.

Kreuzer, M., Linh, T. N. K., & Long, L. N. (2019). K¨ahler differential algebras for 0- dimensional schemes.Journal of Algebra,501, 255–284.

Kreuzer, M., & Robbiano, L. (2005).Computational commutative algebra 2. Springer-Verlag.

Kunz, E. (1986).K¨ahler differentials. Vieweg Verlag, Braunschweig.

Kunz, E. (2005).Introduction to plane algebraic curves. Birkh¨auser.

Linh, T. N. K. (2015).K¨ahler differential modules for 0-dimensional schemes and applica- tions. PhD Thesis, University of Passau.

Scheja, G., & Storch, U. (1988).Lehrbuch der algebra, teil 2. B.G. Teubner, Stuttgart.

The ApCoCoA Team. (2021). Apcocoa: Applied computations in commutative algebra.

Tohaneanu, S. O., & van Tuyl, A. (2013). Bounding invariants of fat points using a coding theory construction.Journal of Pure and Applied Algebra,217, 269–279.

Ueno, K. (1999). Algebraic geometry 1, from algebraic varieties to schemes. Amer. Math.

Soc., Providence.