Show Summary Details
More options …

# Open Mathematics

### formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

IMPACT FACTOR 2018: 0.726
5-year IMPACT FACTOR: 0.869

CiteScore 2018: 0.90

SCImago Journal Rank (SJR) 2018: 0.323
Source Normalized Impact per Paper (SNIP) 2018: 0.821

Mathematical Citation Quotient (MCQ) 2018: 0.34

ICV 2017: 161.82

Open Access
Online
ISSN
2391-5455
See all formats and pricing
More options …
Volume 16, Issue 1

# On the freeness of hypersurface arrangements consisting of hyperplanes and spheres

Ruimei Gao
/ Qun Dai
• Corresponding author
• Department of Science, Changchun University of Science and Technology, Changchun, 130022, China
• Email
• Other articles by this author:
/ Zhe Li
Published Online: 2018-04-23 | DOI: https://doi.org/10.1515/math-2018-0041

## Abstract

Let V be a smooth variety. A hypersurface arrangement 𝓜 in V is a union of smooth hypersurfaces, which locally looks like a union of hyperplanes. We say 𝓜 is free if all these local models can be chosen to be free hyperplane arrangements. In this paper, we use Saito’s criterion to study the freeness of hypersurface arrangements consisting of hyperplanes and spheres, and construct the bases for the derivation modules explicitly.

Keywords: Hypersurface arrangement; Freeness; Hyperplane; Sphere

MSC 2010: 52C35; 32S22

## 1 Introduction

A hypersurface arrangement 𝓜 in a smooth variety V is a reduced divisor D consisting of a union of smooth hypersurfaces, such that at each point D is locally analytically isomorphic to a hyperplane arrangement. For hypersurface arrangements, many researchers made focus on the study of Milnor fibers, higher homotopy groups and Alexander invariants of the hypersurface complements, such as [1, 2, 3, 4]. Besides the topological properties, the freeness of a hypersurface arrangement could also be considered. We say a hypersurface arrangement is free if D is itself a free divisor on V. The study of free hyperplane arrangements was initiated by H. Terao in [5], and has been playing the central role in this area. Recently, there have been several studies to determine when a hyperplane arrangement is free, e.g., [6, 7, 8, 9] and so on. However, it is still very difficult to determine the freeness. Freeness of hyperplane arrangements implies several interesting geometric and combinatorial properties of the arrangements, for example see [6, 10, 11]. Therefore, there were many works on the freeness of hyperplane arrangements, especially on Coxeter arrangements and the cones over Catalan and Shi arrangements[12, 13, 14, 15, 16].

In [17], H. Schenck and S. Tohǎneanu studied the freeness of Conic-Line arrangements in P2 and their results are the first to give an inductive criterion for freeness of nonlinear arrangements. Until now, the papers about the freeness of hypersurface arrangements are few. In this paper, we will consider the freeness of hypersurface arrangements consisting of hyperplanes and spheres, and will construct bases for the derivation modules of hypersurface arrangements.

The paper is organized as follows: in Section 2, we recall the basic definitions and generalize Saito’s criterion to hypersurface arrangements consisting of hyperplanes and spheres. In Section 3, for the hypersurface arrangement consisting of n spheres, the hypersurface arrangement containing a free hyperplane arrangement and n spheres, we present the constructions of bases for the derivation modules respectively.

## 2 Preliminaries and Notations

We begin with some basic concepts and notations of arrangements, for more information see P. Orlik and H. Terao [18].

Let V be an -dimensional vector space on 𝕂 with a coordinate system {x1,…,x} ⊂ V*. Let S = S(V*) be the symmetric algebra of V* and Der𝕂(S) be the module of derivations

$DerK(S)={θ:S→S| θ(fg)=fθ(g)+gθ(f), f,g∈S}.$

Define Di = /∂xi, 1 ≤ i, then D1,…,D\ℓ is a basis for Der𝕂(S) over S.

#### Definition 2.1

A nonzero element θ ∈ Der𝕂(S) is of polynomial degree p if θ = $\begin{array}{}\sum _{k=1}^{\ell }{f}_{k}{D}_{k}\end{array}$ and the maximum of the degrees of coefficient polynomials f1,…,f (get rid of 0) is p. In this case we write pdegθ = p.

#### Definition 2.2

For a hypersurface arrangement 𝓜 in V, the derivation module D(𝓜) is defined by

$D(M)={θ∈DerK(S)| θ(αX)∈αXS for all X∈M},$

where X = ker(αX), 𝓜 is called free if D(𝓜) is free.

#### Definition 2.3

Let 𝓜 be a free hypersurface arrangement and let {θ1,…,θ} be a basis for D(𝓜). We call pdegθ1,…, pdegθ the exponents of 𝓜 and write

$expM={pdegθ1,…,pdegθℓ}.$

#### Definition 2.4

Given derivations θ1,…,θD(𝓜), define the coefficient matrix M(θ1,…,θ) by Mi,j = θj(xi), thus

$M(θ1,…,θℓ)=θ1(x1)...θℓ(x1)...............θ1(xℓ)...θℓ(xℓ),$

and θj = $\begin{array}{}\sum _{i=1}^{\ell }{\mathsf{M}}_{i,j}{D}_{i}.\end{array}$

#### Definition 2.5

Let 𝓜 be a hypersurface arrangement, the product

$Q(M)=∏X∈MαX$

is called a defining polynomial of 𝓜, where X = ker(αX).

For hyperplane arrangements, Saito’s criterion provides a wonderful method to prove the freeness. Next, we will prove it also holds for 𝓜, where 𝓜 is a hypersurface arrangement in ℝ consisting of linear hyperplanes and spheres.

#### Lemma 2.6

If θ1,…,θD(𝓜), then det M(θ1,…,θ) ∈ Q(𝓜)S.

#### Proof

Let X ∈ 𝓜, and let X = ker(αX), then

$detM(θ1,…,θℓ)=fdetθ1(x1)...θℓ(x1)..........θ1(αX)...θℓ(αX)..........θ1(xℓ)...θℓ(xℓ),$

If $\begin{array}{}{\alpha }_{X}=\sum _{k=1}^{\ell }{c}_{k}{x}_{k},\text{then}\phantom{\rule{thinmathspace}{0ex}}f={c}_{k}\in \mathbb{R};\text{If}\phantom{\rule{thinmathspace}{0ex}}{\alpha }_{X}=\sum _{k=1}^{\ell }\left({x}_{k}-{a}_{k}{\right)}^{2}-r,\end{array}$ then f = 2(xkak). For any case, det M(θ1,…,θ) is divisible by αX. Since X is arbitrary, det M(θ1,…,θ) ∈ Q(𝓜)S. □

#### Lemma 2.7

Let Mn be an n × n matrix with the (p, q) entry as follows:

$Mpq=xpxqif p≠q,xp2−rif p=q,$

where 1 ≤ p, qn, r ∈ ℝ. Therefore,

$detMn=(−r)n−1(∑k=1nxk2−r).$

#### Proof

We will prove the lemma by induction on n.

1. For the case n = 1, M1 = $\begin{array}{}{x}_{1}^{2}\end{array}$r, then det M1 = $\begin{array}{}{x}_{1}^{2}\end{array}$r.

2. We assume that for the case n the result holds, that is det $\begin{array}{}{M}_{n}=\left(-r{\right)}^{n-1}\left(\sum _{k=1}^{n}{x}_{k}^{2}-r\right).\end{array}$

For the case n + 1 we have

$Mn+1=MnNNTxn+12−r,$

where N = (x1xn+1,…,xnxn+1)T. Therefore,

$detMn+1=detMnNNTxn+12+detMnOn×1NT−r=det−rEnNO1×nxn+12+(−r)detMn=(−r)n(xn+12)+(−r)(−r)n−1(∑i=1nxi2−r)=(−r)n(∑k=1n+1xk2−r),$

where On×1, On are the n × 1 and 1 × n null matrices respectively, and En is the n × n identity matrix. □

#### Lemma 2.8

Let

$S={(x1,…,xℓ)∣∑k=1ℓ(xk−ak)2=r∈R+}$

be the ( – 1)-dimensional sphere in with center (a1, a2, …, a) and radius $\begin{array}{}\sqrt{r},\end{array}$ define

$θq=∑p=1ℓfpqDp,1≤q≤ℓ,$

where

$fpq=(xp−ap)(xq−aq)if p≠q,(xp−ap)2−rif p=q.$

Then θ1,…,θD({𝓢}) and det M(θ1,…,θ) ≐ Q({𝓢}).

#### Proof

We can see

$θq=∑p=1ℓfpqDp=(xq−aq)∑p=1ℓ(xp−ap)Dp−rDq,$

and

$θq[∑k=1ℓ(xk−ak)2−r]=2(xq−aq)[∑k=1ℓ(xk−ak)2−r]∈[∑k=1ℓ(xk−ak)2−r]S,$

Thus θqD({𝓢}) for 1 ≤ q.

By Lemma 2.7, we obtain

$detM(θ1,…,θℓ)=(−r)ℓ−1[∑k=1ℓ(xk−ak)2−r]≐Q({S}).$ □

Next, we will show Saito’s criterion for hypersurface arrangements.

#### Theorem 2.9

Given θ1,…,θD(𝓜), the following two conditions are equivalent:

1. det M(θ1,…,θ) ≐ Q(𝓜).

2. θ1,…,θ form a basis for D(𝓜) over S.

#### Proof

(1)⇒(2) The proof is exactly the same with that of Saito’s criterion in [18].

(2)⇒(1) By Lemma 2.6, we can write det M(θ1,…,θ) = fQ(𝓜) for some fS. Fix X ∈ 𝓜, if X is a hyperplane, then {X} is a free hyperplane arrangement; if X is a sphere, by Lemma 2.8 and (1)⇒(2), {X} is a free hypersurface arrangement. Assume η1,…,η is the basis of X, then QXη1,…,QXηD(𝓜), where QX = Q(𝓜)/αX. Since each QXηi is an S-linear combination of η1,…,η, then there exists an × matrix N with entries in S, such that

$M(QXη1,…,QXηℓ)=M(θ1,…,θℓ)N.$

Thus we have

$Q(M)QXℓ−1≐detM(QXη1,…,QXηℓ)∈detM(θ1,…,θℓ)S=fQ(M)S.$

Therefore f divides $\begin{array}{}{Q}_{X}^{\ell -1}\end{array}$ for all X ∈ 𝓜. Since the polynomials $\begin{array}{}\left\{{Q}_{X}^{\ell -1}{\right\}}_{X\in \mathcal{M}}\end{array}$ have no common factor, f ∈ ℝ*. □

#### Corollary 2.10

If 𝓢 is an ( − 1)-dimensional sphere in, then {𝓢} is a free hypersurface arrangement with

$exp{S}={2,…,2},$

where 2 appears ℓ times.

#### Proof

The result is obtained directly from Lemma 2.8 and Theorem 2.9. □

## 3 Main results

In this section, we will consider the freeness for hypersurface arrangements containing hyperplanes and spheres, and give the explicit bases for the derivation modules of the free ones. First, we show that the hypersurface arrangement having n spheres is free.

#### Theorem 3.1

Let 𝓜n be the hypersurface arrangement consisting of n spheres 𝓢1,…,𝓢n, where

$Si={(x1,…,xℓ)∣∑k=1ℓ(xk−ak)2=ri, (a1,a2,…,aℓ)∈Rℓ, ri∈R+}.$

Define derivations $\begin{array}{}{\phi }_{1}^{n},\dots ,{\phi }_{\ell }^{n}\end{array}$ by

$M(φ1n,…,φℓn)=AnAn−1⋯A1,$

where Ai is an × matrix and the (p, q) entry of Ai is

$(Ai)pq=(xp−ap)(xq−aq) if p≠q,(xp−ap)2−ri if p=q.$

Then $\begin{array}{}{\phi }_{1}^{n},\dots ,{\phi }_{\ell }^{n}\end{array}$ form a basis for D(𝓜n) and exp 𝓜n = {2n,…,2n}, where 2n appears ℓ times.

#### Proof

We will prove this result by Theorem 2.9: Saito’s criterion. By Lemma 2.7, we obtain

$detAi=(−ri)ℓ−1[∑k=1ℓ(xk−ak)2−ri].$

Therefore,

$detM(φ1n,…,φℓn)=∏i=1ndetAi≐∏i=1n[∑k=1ℓ(xk−ak)2−ri]=Q(Mn).$

Next, we will prove $\begin{array}{}{\phi }_{i}^{n}\end{array}$D(𝓜n) and deg $\begin{array}{}{\phi }_{i}^{n}\end{array}$ = 2n for any 1 ≤ i by induction on n.

The case n = 1 is clear according to Lemma 2.8 and Corollary 2.10. For the case n + 1, we notice that

$φin+1=∑p=1ℓφin+1(xp)Dp=∑p=1ℓ[∑q=1ℓ(An+1)pqφin(xq)]Dp=∑p=1ℓ[∑q≠p(xp−ap)(xq−aq)φin(xq)+[(xp−ap)2−rn+1]φin(xp)]Dp=∑p=1ℓ[(xp−ap)∑q=1ℓ(xq−aq)φin(xq)]Dp−rn+1∑p=1ℓφin(xp)Dp=∑q=1ℓ(xq−aq)φin(xq)∑p=1ℓ(xp−ap)Dp−rn+1φin=12φin[∑q=1ℓ(xq−aq)2]∑p=1ℓ(xp−ap)Dp−rn+1φin.$

Therefore, for 1 ≤ i and 1 ≤ jn,

$φin+1[∑k=1ℓ(xk−ak)2−rj]=φin[∑q=1ℓ(xq−aq)2]∑p=1ℓ(xp−ap)2−rn+1φin[∑k=1ℓ(xk−ak)2−rj]=φin[∑q=1ℓ(xq−aq)2][∑p=1ℓ(xp−ap)2−rn+1],$

by induction hypothesis,

$φin[∑q=1ℓ(xq−aq)2]∈∏j=1n[∑k=1ℓ(xk−ak)2−rj]S,$

hence,

$φin+1[∑k=1ℓ(xk−ak)2−rj]∈∏j=1n+1[∑k=1ℓ(xk−ak)2−rj]S.$

This means $\begin{array}{}{\phi }_{i}^{n+1}\in D\left({\mathcal{M}}_{n+1}\right)\end{array}$ for any 1 ≤ i, in addition,

$pdegφin+1=pdegφin+2=2n+2=2(n+1).$

We complete the induction, so by Saito’s criterion $\begin{array}{}{\phi }_{1}^{n},\dots ,{\phi }_{\ell }^{n}\end{array}$ form a basis for 𝓜n, and exp 𝓜n = {2n,…,2n}. □

Next, we will study the freeness for the hypersurface arrangements consisting of hyperplanes and spheres, where all the spheres are centered at origin.

#### Theorem 3.2

Assume 𝓐 is a free hyperplane arrangement with a homogeneous basis θ1,…,θ, and exp 𝓐 = {d1,…,d}, $\begin{array}{}{\mathcal{S}}_{i}^{0}\end{array}$ is the sphere centered at origin:

$Si0={(x1,…,xℓ)∣∑k=1ℓxk2=ri∈R+},1≤i≤n,$

and

$Mn=A∪{S10,…,Sn0},$

Define derivations $\begin{array}{}{\phi }_{1}^{n},\dots ,{\phi }_{\ell }^{n}\end{array}$ by

$M(φ1n,…,φℓn)=(AnAn−1⋯A1)M(θ1,…,θℓ),$

where Ai is an × matrix and the (p, q) entry of Ai is

$(Ai)pq=xpxq if p≠q,xp2−ri if p=q,$

then $\begin{array}{}{\phi }_{1}^{n},\dots ,{\phi }_{\ell }^{n}\end{array}$ form a basis for D(𝓜n) and exp 𝓜n = {d1 + 2n,…,d + 2n}.

#### Proof

By Lemma 2.7, we obtain

$detAi=(−ri)ℓ−1(∑k=1ℓxk2−ri).$

Since 𝓐 is a free arrangement with a homogeneous basis θ1,…,θ, by Saito’s criterion,

$detM(θ1,…,θℓ)≐Q(A).$

Therefore,

$detM(φ1n,…,φℓn)=∏i=1n(detAi)detM(θ1,…,θℓ)≐∏i=1n(∑k=1ℓxk2−ri)Q(A)=Q(Mn).$

Next, we will prove $\begin{array}{}{\phi }_{i}^{n}\end{array}$D(𝓜n) and deg $\begin{array}{}{\phi }_{i}^{n}\end{array}$ = di + 2n for any 1 ≤ i by induction on n.

For the case n = 1,

$φi1=∑p=1ℓφi1(xp)Dp=∑p=1ℓ[∑q=1ℓ(A1)pqθi(xq)]Dp=∑p=1ℓ[∑q≠pxpxqθi(xq)+(xp2−r1)θi(xp)]Dp=∑p=1ℓ[xp∑q=1ℓxqθi(xq)]Dp−r1∑p=1ℓθi(xp)Dp=∑q=1ℓxqθi(xq)∑p=1ℓxpDp−r1θi=∑q=1ℓxqθi(xq)θE−r1θi.$

Since θE, θiD(𝓐), we have $\begin{array}{}{\phi }_{i}^{1}\end{array}$D(𝓐) for any 1 ≤ i. And

$pdegφi1=pdeg[∑q=1ℓxqθi(xq)θE]=pdegθi+2=di+2.$

$φi1(∑k=1ℓxk2−r1)=[∑q=1ℓxqθi(xq)θE−r1θi](∑k=1ℓxk2−r1)=∑q=1ℓxqθi(xq)(2∑k=1ℓxk2)−2r1∑q=1ℓxqθi(xq)=2(∑k=1ℓxk2−r1)∑q=1ℓxqθi(xq)∈(∑k=1ℓxk2−r1)S$

that is, $\begin{array}{}{\phi }_{i}^{1}\in D\left(\left\{{\mathcal{S}}_{1}^{0}\right\}\right).\end{array}$ Therefore, $\begin{array}{}{\phi }_{i}^{1}\in D\left(\mathcal{A}\right)\cap D\left(\left\{{\mathcal{S}}_{1}^{0}\right\}\right)=D\left({\mathcal{M}}_{1}\right)\end{array}$ for any 1 ≤ i.

For the case n + 1, by the similar calculation of $\begin{array}{}{\phi }_{i}^{1}\end{array}$, we get

$φin+1=∑q=1ℓxqφin(xq)θE−rn+1φin=12φin(∑q=1ℓxq2)θE−rn+1φin.$

By induction hypothesis,

$φin∈D(Mn)⊆D(⋃i=1n{Si0}),$

we obtain

$φin(∑q=1ℓxq2)∈∏j=1n(∑k=1ℓxk2−rj)S.$

Therefore,

$φin(∑q=1ℓxq2)θE∈D(⋃i=1n{Si0}).$

Combining θED(𝓐) with

$D(Mn)=D(⋃i=1n{Si0})⋂D(A),$

we conclude

$φin(∑q=1ℓxq2)θE∈D(Mn).$

Hence, $\begin{array}{}{\phi }_{i}^{n+1}\in D\left({\mathcal{M}}_{n}\right)\end{array}$ since $\begin{array}{}{\phi }_{i}^{n}\in D\left({\mathcal{M}}_{n}\right).\end{array}$

$φin+1(∑k=1ℓxk2−rn+1)=φin(∑q=1ℓxq2)∑k=1ℓxk2−rn+1φin(∑k=1ℓxk2)=(∑k=1ℓxk2−rn+1)φin(∑q=1ℓxq2)∈(∑k=1ℓxk2−rn+1)S,$

We obtain $\begin{array}{}{\phi }_{i}^{n+1}\in D\left(\left\{{\mathcal{S}}_{n+1}^{0}\right\}\right)\end{array}$ for any 1 ≤ i, therefore

$φin+1∈D({Sn+10})∩D(Mn)=D(Mn+1), 1≤i≤ℓ.$

Moreover,

$pdegφin+1=pdeg[φin(∑q=1ℓxq2)θE]=pdegφin+2=di+2n+2=di+2(n+1), 1≤i≤ℓ.$

We complete the induction. □

#### Corollary 3.3

Let 𝓜n = $\begin{array}{}\mathcal{A}\cup \left\{{\mathcal{S}}_{1}^{0},\dots ,{\mathcal{S}}_{n}^{0}\right\}\end{array}$ be the hypersurface arrangement defined in Theorem 3.2. Then 𝓐 is free if and only if 𝓜n is free.

#### Proof

If 𝓐 is free we can obtain that 𝓜n is free directly from Theorem 3.2. Assume 𝓜n is free, 𝓐 ⊆ 𝓜n, then D(𝓜n) ⊆ D(𝓐). Let φ1,…,φ be a basis for D(𝓜n), then φiD(𝓐) for 1 ≤ i. Write $\begin{array}{}{\phi }_{i}=\sum _{k\ge 0}{\phi }_{i}^{\left(k\right)},\end{array}$ where $\begin{array}{}{\phi }_{i}^{\left(k\right)}\end{array}$ is zero or homogeneous of degree k ≥ 0. Since Q(𝓐)S is generated by homogeneous polynomial Q(𝓐), each homogeneous component $\begin{array}{}{\phi }_{i}^{\left(k\right)}\left(Q\left(\mathcal{A}\right)\right)\phantom{\rule{thinmathspace}{0ex}}\text{of}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\phi }_{i}\left(Q\left(\mathcal{A}\right)\right)\end{array}$ also lies in Q(𝓐)S. This shows that $\begin{array}{}{\phi }_{i}^{\left(k\right)}\in D\left(\mathcal{A}\right)\end{array}$ for k ≥ 0. Since $\begin{array}{}\left[Q\left(\bigcup _{i=1}^{n}\left\{{\mathcal{S}}_{i}\right\}\right)\right]\left(0\right)\ne 0,\end{array}$ there exist $\begin{array}{}{\phi }_{1}^{\left({d}_{1}\right)},\dots ,{\phi }_{\ell }^{\left({d}_{\ell }\right)}\end{array}$ such that

$detM(φ1(d1),…,φℓ(dℓ))≐Q(A).$

By Saito’s criterion, $\begin{array}{}{\phi }_{1}^{\left({d}_{1}\right)},\dots ,{\phi }_{\ell }^{\left({d}_{\ell }\right)}\end{array}$ form a basis for D(𝓐). □

#### Remark 3.4

In Theorem 3.1 and Theorem 3.2 the preconditions did not impose the restrictions on the size relations of r1, r2, …, rn. Hence, if r1 = r2 = … = rn, Theorem 3.1 and Theorem 3.2 also hold. In this case, (𝓜, m) is a hypersurface arrangement with a multiplicity mi for each hypersurface in 𝓜, we call it hypersurface multiarrangement. As defined by G. Ziegler in [19], the module of derivations consists of θ such that θ(αi) ∈ $\begin{array}{}{\alpha }_{i}^{{m}_{i}}S.\end{array}$

#### Example 3.5

Let 𝓜 be a hypersurface arrangement with the defining polynomial

$Q(M)=(x1−x2)(x1−x3)(x2−x3)(x12+x22+x32−1).$

In this case, the hyperplane arrangement 𝓐 ⊆ 𝓜 is the Coxeter arrangement of type A2, it is a free arrangement with exp 𝓐 = {0, 1, 2}. By Theorem 3.2, 𝓜 is a free hypersurface arrangement and D(𝓜) has the basis φ1, φ2, φ3 as follows:

$(φ1,φ2,φ3)=(D1,D2,D3)x12−1x1x2x1x3x2x1x22−1x2x3x3x1x3x2x32−11x1x121x2x221x3x32.$

That is

$φ1=(x12+x1x2+x1x3−1)D1+(x1x2+x22+x2x3−1)D2+(x1x3+x2x3+x32−1)D3,φ2=x1(x12+x22+x32−1)D1+x2(x12+x22+x32−1)D2+x3(x12+x22+x32−1)D3,φ3=x1(x13+x23+x33−x1)D1+x2(x13+x23+x33−x2)D2+x3(x13+x23+x33−x3)D3.$

And exp 𝓜 = {pdegφ1, pdegφ2, pdegφ3} = {2, 3, 4}.

#### Example 3.6

Let 𝓜 be a hypersurface arrangement with the defining polynomial

$Q(M)=x1x2x3(x1+x2)(x1+x3)(x2+x3)(x1−x2)(x1−x3)(x2−x3)(x12+x22+x32−1)(x12+x22+x32−2).$

In this case, the hyperplane arrangement 𝓐 ⊆ 𝓜 is the Coxeter arrangement of type B3, it is a free arrangement with exp 𝓐 = {1, 3, 5}. By Theorem 3.2, 𝓜 is a free hypersurface arrangement and D(𝓜) has the basis φ1, φ2, φ3 as follows:

$(φ1,φ2,φ3)=(D1,D2,D3)x12−2x1x2x1x3x2x1x22−2x2x3x3x1x3x2x32−2x12−1x1x2x1x3x2x1x22−1x2x3x3x1x3x2x32−1x1x13x15x2x23x25x3x33x35.$

And exp 𝓜 = {pdegφ1, pdegφ2, pdegφ3} = {5, 7, 9}.

## Acknowledgement

The work was partially supported by NSF of China No. 11501051, No. 11601039 and Science and Technology Development Foundation of Jilin Province (No.20180520025JH and No.20180101345JC).

## References

• [1]

Dimca A., Maxim L., Multivariable Alexander invariants of hypersurface complements, Trans. Amer. Math. Soc., 2007, 359, 3505-3528

• [2]

Dimca A., Papadima S., Hypersurface complements, Milnor fibers and higher homotopy groups of arrangements, Ann. of Math., 2003, 158(2), 473-507

• [3]

Hun J., Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs, J. Amer. Math. Soc., 2012, 25(3), 907-927

• [4]

Liu Y., Maxim L., Characteristic varieties of hypersurface complements, Adv. Math., 2014, 306, 451-493

• [5]

Terao H., Arrangements of hyperplanes and their freeness I, II., J. Fac. Sci. Univ., 1980, 27, 293-320 Google Scholar

• [6]

Abe T., Roots of characteristic polynomials and and intersection points of line arrangements, J. Singularities, 2014, 8, 100-117 Google Scholar

• [7]

Abe T., Yoshinaga M., Free arrangements and coefficients of characteristic polynomials, Math. Z., 2013, 275(3), 911-919

• [8]

Yoshinaga M., Characterization of a free arrangement and conjecture of Edelman and Reiner, Invent. Math., 2004, 157(2), 449-454

• [9]

Yoshinaga M., On the freeness of 3-arrangements, Bull. London Math. Soc., 2005, 37(1), 126-134

• [10]

Abe T., Chambers of 2-affine arrangements and freeness of 3-arrangements, J. Alg. Combin., 2013, 38(1), 65-78 Google Scholar

• [11]

Terao H., Generalized exponents of a free arrangement of hyperplanes and Shephard-Todd-Brieskorn formula, Invent. math., 1981, 63, 159-179

• [12]

Abe T., Terao H., Simple-root bases for Shi arrangements, J. Algebra, 2015, 422, 89-104

• [13]

Gao R., Pei D., Terao H., The Shi arrangement of the type D, Proc. Japan Acad. Ser. A Math. Sci., 2012, 88(3), 41-45

• [14]

Suyama D., A Basis Construction for the Shi Arrangement of the Type B or C, Comm. Algebra, 2015, 552, 1435-1448 Google Scholar

• [15]

Suyama D., Terao H., The Shi arrangements and the Bernoulli polynomials, Bull. London Math. Soc., 2012, 44, 563-570

• [16]

Terao H., Multiderivations of Coxeter arrangements, Invent. Math., 2002, 148, 659-674

• [17]

Schenck H., Tohǎneanu S., Freeness of Conic-Line arrangements in P2, Comment. Math. Helv., 2009, 84, 235-258 Google Scholar

• [18]

Orlik P., Terao H., Arrangements of Hyperplanes, Grundlehren der Mathematischen Wissenschaften, 1992, 300, Berlin: Springer-Verlag Google Scholar

• [19]

Ziegler G., Multiarrangements of hyperplanes and their freeness, In Singularities (Iowa City, IA, 1986), Contemp. Math., Amer. Math. Soc., Providence, RI, 1989, 90, 345-359 Google Scholar

Accepted: 2018-01-15

Published Online: 2018-04-23

Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 437–446, ISSN (Online) 2391-5455,

Export Citation