The Jacobian ideal of a hyperplane arrangement
Max Wakefield, Masahiko Yoshinaga
超平面排列的雅可比理想是多项式环中的理想，其生成元是定义多项式的排列的偏导数。在这篇文章中，我们证明了一个安排可以重建从它的雅可比理想。
The Jacobian ideal of a hyperplane arrangement is an ideal in the polynomial ring whose generators are the partial derivatives of the arrangements defining polynomial. In this article, we prove that an arrangement can be reconstructed from its Jacobian ideal.
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Multiplier ideals of hyperplane arrangements
我们计算乘子理想的超平面安排通过解释这些理想的空间的弧，由于艾因，Lazarsfeld和作者。
In this article, we study the $k$-Lefschetz properties for non-Artinian algebras, proving that several known results in the Artinian case can be generalized in this setting. Moreover, we describe how to characterize the graded algebras having the $k$-Lefschetz properties using sectional matrices. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements, with particular attention to the class of free arrangements.
Closing two recent conjectures related to the Jacobian ideal of hyperplane arrangements
本文解决了Burity-Simis-Tohăneanu关于中心超平面排列的雅可比理想的两个猜想：一个得到证实，另一个在秩至少为4时被反例否定，并揭示了Pfaffian障碍与排列的电路和余维二平坦的关系。
Lefschetz properties and the Jacobian algebra of 3-dimensional hyperplane arrangements
本文研究高维商空间和1维几乎完全交的弱与强莱夫谢茨性质，并将结果应用于超平面构型的雅可比代数分析。
Logarithmic sheaves attached to arrangements of hyperplanes
非奇簇上的简化除子定义对数1-形式的层。我们引入一个特定的凝聚层，它的双对偶与这个层重合。它有一些很好的性质，例如，即使除数是奇异的，它的剩余正合序列仍然成立，并且它有一个简单的局部自由分解。我们专门的情况下，当除数是一个安排超平面在射影空间，并与稳定性的层的组合学的安排。我们还扩展了一个Torelli型定理，非通用的安排，它允许一个重建的安排，从附加到它的层。
In this article we describe two new characterizations of freeness for hyperplane arrangements via the study of the generic initial ideal and of the sectional matrix of the Jacobian ideal of arrangements.
In this paper, we study the class of free hyperplane arrangements. Specifically, we investigate the relations between freeness over a field of finite characteristic and freeness over $\mathbb{Q}$.
On homological properties of some Cynk-Szemberg octic hyperplane arrangements
本文研究了Cynk-Szemberg八次超平面排列的同调性质，定义了其类型并提出了本质超平面排列的非自由准则。
Freeness and multirestriction of hyperplane arrangements
推广了Yoshinaga在3维空间中的一个结果，证明了四维空间中的一个中心超平面排列是自由的，如果它对一个固定超平面的重数限制是自由的，并且它的约化特征多项式等于这个限制的特征多项式.我们表明，同样的陈述在任何维度都成立时，施加一定的驯服假设。
In this article, we study the weak and strong Lefschetz properties, and the related notion of almost revlex ideal, in the non-Artinian case, proving that several results known in the Artinian case hold also in this more general setting. We then apply the obtained results to the study of the Jacobian algebra of hyperplane arrangements.
We introduce a new class of arrangements of hyperplanes, called (strictly) plus-one generated arrangements, from algebraic point of view. Plus-one generatedness is close to freeness, i.e., plus-one generated arrangements have their logarithmic derivation modules generated by dimension plus one elements, with relations containing one linear form coefficient. We show that strictly plus-one generated arrangements can be obtained if we delete a hyperplane from free arrangements. We show a relative freeness criterion in terms of plus-one generatedness. In particular, for plane arrangements, we show…
We study the Hadamard product of the linear forms defining a hyperplane arrangement with those of its dual, which we view as generating an ideal in a certain polynomial ring. We use this ideal, which we call the ideal of pairs, to study logarithmic derivations and critical set varieties of arrangements in a way which is symmetric with respect to matroid duality. Our main result exhibits the variety of the ideal of pairs as a subspace arrangement whose components correspond to cyclic flats of the arrangement. As a corollary, we are able to give geometric explanations of some freeness and projec…
We establish a general theory for projective dimensions of the logarithmic derivation modules of hyperplane arrangements. That includes the addition-deletion and restriction theorem, Yoshinaga-type result, and the division theorem for projective dimensions of hyperplane arrangements. They are generalizations of the free arrangement cases, that can be regarded as the special case of our result when the projective dimension is zero. The keys to prove them are several new methods to determine the surjectivity of the Euler and the Ziegler restriction maps, that is combinatorial when the projective…
Addition theorems for Ziegler pairs of hyperplane arrangements
受Terao自由性猜想启发，本文通过加法定理从复射影平面上的例子构造出任意维数和大小的Ziegler对，即具有相同底层拟阵但不同对数导子模的超平面配置对。
The{N/D}-Conjecture for Nonresonant Hyperplane Arrangements
本文通过Walther的结果验证了满足非共振条件的加权超平面排列的n/d猜想，并推导出强拓扑单叶猜想。
We show that the deletion theorem of a free arrangement is combinatorial, i.e., whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice. In fact, we give an explicit sufficient and necessary condition for the deletion theorem in terms of characteristic polynomials. This gives a lot of corollaries including the existence of free filtrations. The proof is based on the result about the form of minimal generators of a logarithmic derivation module of a multiarrangement which satisfies the $b_2$-equality.
We introduce a new algebra associated with a hyperplane arrangement $\mathcal{A}$, called the Solomon-Terao algebra $\mbox{ST}(\mathcal{A},η)$, where $η$ is a homogeneous polynomial. It is shown by Solomon and Terao that $\mbox{ST}(\mathcal{A},η)$ is Artinian when $η$ is generic. This algebra can be considered as a generalization of coinvariant algebras in the setting of hyperplane arrangements. The class of Solomon-Terao algebras contains cohomology rings of regular nilpotent Hessenberg varieties. We show that $\mbox{ST}(\mathcal{A},η)$ is a complete intersection if and only if $\mathcal{A}$ …
Residue ideals of hyperplane arrangements
引入留数理想研究超平面配置的对数微分形式模，建立其基本性质并应用于限制自由性，对图配置给出显式生成集，建立与覆盖理想和Stanley-Reisner理论的新联系。
A conjecture on the Poincar{é}-Betti series of modules of differential operators on a generic hyperplane arrangement
霍尔姆研究了一般（中心）超平面排列上的高阶微分算子（广义导子）的模。我们用他的结果来确定这些模的Hilbert级数。我们还提出了一个猜想的庞加莱-贝蒂系列;这些都是已知的模块推导通过工作的Yuzvinsky和玫瑰和寺尾。
Free paths of arrangements of hyperplanes
研究超平面排列的自由路径问题，证明若排列及其去掉两条超平面后的排列自由，则至少其一自由，并在三维情形下给出更强结论。
Totally free arrangements of hyperplanes
在$\ell$维向量空间$V$中的超平面的中心排列$\A$称为{ \it totally free}，如果多重排列$(\A, m)$对于任何重数$ m : \A\to \Z_{> 0}$都是自由的。众所周知，无论何时使用$\ell \le 2$，$\A$都是完全免费的。在这篇文章中，我们将证明，不存在任何完全自由的安排以外的明显的，即一个产品的一维安排和二维的。
On the Orlik--Terao ideal and the relation space of a hyperplane arrangement
超平面排列的关系空间是该排列中的超平面的定义形式之间的所有线性依赖关系的向量空间。本文研究了关系空间与排列的Orlik-Terao理想之间的关系。特别是，我们刻画的关系空间的Orlik-Terao理想的生成集。这一结果推广了Schenck和Tohenneanu \cite [定理2.3]{ST}的2-形式安排的一个特征.我们还研究了与关系空间的子集相关联的Orlik-Terao理想的子理想的极小素理想。最后，我们给出了一些例子，证明了对于2-形式排列，Orlik-Terao理想的余维数不一定等于由二次元生成的次理想的余维数。
Partial fraction decompositions on hyperplane arrangements
本文研究多变量部分分式分解，通过交换代数工具给出有理函数在超平面安排上具有理想部分分式分解的条件，并提出算法用于简化散射振幅计算。
Derivation modules of orthogonal duals of hyperplane arrangements
设A是一个n × d的满秩矩阵。A的正交对偶A^{ \perp }是秩为（d-n）的（d-n）× d矩阵，使得A^{ \perp }的每一行与A的每一行正交（在通常的点积下）。我们定义的正交对偶安排，通过确定一个必要的（中心）安排的d超平面在n维空间中的n × d系数系数矩阵的齐次线性型的超平面的内核。当n ≥ 5时，证明了n维本质排列A的拟阵（或交格）若含有模余点，且其补为跨距，则正交对偶排列\A^{\perp}的导子模至少有投射维数[n（n+2）/4] - 3，（[ ]表示天花板）.
Projective duality of arrangements with quadratic logarithmic vector fields
在这些笔记中，我们研究超平面安排，至少有一个对数导数的程度2，这是不是一个组合的程度1对数导数。众所周知，如果超平面排列具有线性对数导数而不是欧拉导数的常数倍，则该排列分解为较小排列的直积。下一个自然的步骤将是研究安排与非平凡的二次对数导数。在这方面，我们提出了一个计算引理，导致一个完整的分类超平面安排秩3有这样一个二次对数推导。这些结果来作为一个后果，看看在各种各样的点双超平面在这种特殊的安排。
Kamiya, Takemura, and Terao initiated the theory of the characteristic quasi-polynomial of an integral arrangement, which is a function counting the elements in the complement of the arrangement modulo positive integers. They gave a period of the characteristic quasi-polynomial, called the LCM-period, and showed that the first constituent of the characteristic quasi-polynomial coincides with the characteristic polynomial of the corresponding hyperplane arrangement. Recently, Liu, Tran, and Yoshinaga showed that the last constituent of the characteristic quasi-polynomial coincides with the char…
In the study of free arrangements, the most useful result to construct/check free arrangements is the addition-deletion theorem. Recently, the multiple version of the addition theorem is proved, called the multiple addition theorem (MAT) to prove the ideal-free theorem. The aim of this article is to give the deletion version of MAT, the multiple deletion theorem (MDT). Also, we can generalize MAT from the viewpoint of our new proof. Moreover, we introduce their restriction version, a multiple restriction theorem (MRT). Applications of them including the combinatorial freeness of the extended C…
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