Weighted grassmannians and stable hyperplane arrangements

热带超平面排列与Grassmannians的匹配场多胞体的组合突变

Tropical hyperplane arrangements and combinatorial mutations of the matching field polytopes of Grassmannians

本文利用热带超平面排列提出一种直观检查匹配场多胞体组合突变等价的方法，证明块对角与对角匹配场等价，并推广了已有结果。

On Perverse sheaves of a Coxeter hyperplane arrangement of type $\mathcal{A}_n$

Kapranov and schechtman gave quiver description of perverse sheaves on real hyperplane arrangements. We used this description to relate the perverse sheaves on Coxeter hyperplane arrangements of type $\mathcal A_n$ for different values of $n$. As a consequence we prove that the simple perverse sheaves whose stalk on open cells are zero are induced from the perverse sheaves on lower dimension arrangements.

Jumping numbers of hyperplane arrangements

M. Saito recently proved that the jumping numbers of a hyperplane arrangement depend only on the combinatorics of the arrangement. However, a formula in terms of the combinatorial data was still missing. In this note, we give a formula and a different proof of the fact that the jumping numbers of a hyperplane arrangement depend only on the combinatorics. We also give a combinatorial formula for part of the Hodge spectrum and for the inner jumping multiplicities.

超平面配置的Hirzebruch二次型与平坦对数联络

The Hirzebruch quadratic form of a hyperplane arrangement and flat logarithmic connections

本文证明复超平面配置的Hirzebruch二次型在稳定权集上非正，并识别出零集为特殊类型的平坦对数联络，证明使用了Kempf-Ness和框架势不等式。

格拉斯曼流形中的超平面排列

Hyperplane Arrangements in the Grassmannian

研究格拉斯曼流形去除超平面截面后的欧拉特征，提出组合公式并探讨其计算方法，聚焦通用超平面截面和施布特除子，考虑物理相关的特殊施布特排列。

On contact loci of hyperplane arrangements

We give an explicit expression for the contact loci of hyperplane arrangements and show that their cohomology rings are combinatorial invariants. We also give an expression for the restricted contact loci in terms of Milnor fibers of associated hyperplane arrangements. We prove the degeneracy of a spectral sequence related to the restricted contact loci of a hyperplane arrangement and which conjecturally computes algebraically the Floer cohomology of iterates of the Milnor monodromy. We give formulas for the Betti numbers of contact loci and restricted contact loci in generic cases.

Non-resonance D-modules over arrangements of hyperplanes

The aim of this note is a combinatorial description of a category of $D$-modules over an affine space, smooth along the stratification defined by an arrangement of hyperplanes. These $D$-modules are assumed to satisfy certain non-resonance condition. The main result, see Theorem 4.1, generalizes [S.Khoroshkin, $D$-modules over arrangements of hyperplanes, Comm. in Alg. 23(9) (1995), 3481-3504].

Cayley平面与F型余伴随Grassmann流形的导出范畴

Derived categories of the Cayley plane and the coadjoint Grassmannian of type F

本文证明Faenzi–Manivel的Cayley平面完整Lefschetz例外集限制到其F₄型超平面截面后仍完整，并刻画剩余范畴，给出F型齐性簇的首个完整例外集。

超平面排列的强单值猜想

The strong monodromy conjecture for hyperplane arrangements

通过证明Budur、Mustață和Teitler关于C^n上不可约本质中心超平面排列f的b函数根为-n/d的猜想，证明了复超平面排列的强单值猜想。

K-stability of log Fano hyperplane arrangements

In this article, we completely determine which log Fano hyperplane arrangements are uniformly K-stable, K-stable, K-polystable, K-semistable or not.

Face monoid actions and tropical hyperplane arrangements

We study the combinatorics of tropical hyperplane arrangements, and their relationship to (classical) hyperplane face monoids. We show that the refinement operation on the faces of a tropical hyperplane arrangement, introduced by Ardila and Develin in their definition of a tropical oriented matroid, induces an action of the hyperplane face monoid of the classical braid arrangement on the arrangement, and hence on a number of interesting related structures. Along the way, we introduce a new characterization of the types (in the sense of Develin and Sturmfels) of points with respect to a tropica…

A note on Mustata's computation of multiplier ideals of hyperplane arrangements

M. Mustata used jet schemes to compute the multiplier ideals of reduced hyperplane arrangements. We give an alternate proof using a log resolution, which is simpler and allows us to consider non-reduced arrangements. By applying the idea of wonderful models introduced by De Concini--Procesi, we also simplify the result.

The Jacobian ideal of a hyperplane arrangement

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.

示性主值积分在超平面排列中的应用

Motivic principal value integrals for hyperplane arrangements

本文针对超平面排列，验证了Denef-Jacobs-Veys猜想，将多值有理顶端形式的示性主值积分与秩一局部系统支撑定位的上同调联系起来。

On the local zeta functions and the b-functions of certain hyperplane arrangements

Conjectures of J. Igusa for p-adic local zeta functions and of J. Denef and F. Loeser for topological local zeta functions assert that (the real part of) the poles of these local zeta functions are roots of the Bernstein-Sato polynomials (i.e. the b-functions). We prove these conjectures for certain hyperplane arrangements, including the case of reduced hyperplane arrangements in three-dimensional affine space.

超平面排列补集的Milnor-Witt动机上同调

Milnor-Witt motivic cohomology of complements of hyperplane arrangements

本文计算仿射空间中超平面排列补集的Milnor-Witt动机上同调，给出其代数结构，并通过实现到经典上同调获得推论。

Motivic zeta functions of hyperplane arrangements

For each central essential hyperplane arrangement $\mathcal{A}$ over an algebraically closed field, let $Z_\mathcal{A}^{\hatμ}(T)$ denote the Denef-Loeser motivic zeta function of $\mathcal{A}$. We prove a formula expressing $Z_\mathcal{A}^{\hatμ}(T)$ in terms of the Milnor fibers of related hyperplane arrangements. We use this formula to show that the map taking each complex arrangement $\mathcal{A}$ to the Hodge-Deligne specialization of $Z_{\mathcal{A}}^{\hatμ}(T)$ is locally constant on the realization space of any loop-free matroid. We also prove a combinatorial formula expressing the mot…

Multiplier ideals of hyperplane arrangements

We compute the multiplier ideals of hyperplane arrangements via the interpretation of these ideals in terms of spaces of arcs, due to Ein, Lazarsfeld and the author.

Bernstein-Sato ideals and hyperplane arrangements

We study the relation between zero loci of Bernstein-Sato ideals and roots of b-functions and obtain a criterion to guarantee that roots of b-functions of a reducible polynomial are determined by the zero locus of the associated Bernstein-Sato ideal. Applying the criterion together with a result of Maisonobe we prove that the set of roots of the b-function of a free hyperplane arrangement is determined by its intersection lattice. We also study the zero loci of Bernstein-Sato ideals and the associated relative characteristic cycles for arbitrary central hyperplane arrangements. We prove the mu…

The algebra of differential operators tangent to a hyperplane arrangement

We prove using an extension of Saito's criterion that the algebra of differential operators tangent to a free hyperplane arrangement is generated by the derivations of the arrangement, recovering in this way a result of F. J. Calderón-Moreno.

Length and decomposition of the cohomology of the complement to a hyperplane arrangement

Let $\mathcal A$ be a hyperplane arrangement in $\mathbb C^n$. We show that the number of decomposition factors as a perverse sheaf of the direct image $Rj_*\mathbb C_U $ of the constant sheaf on the complement $U$ to the arrangement is given by the Poincaré polynomial of the arrangement. Furthermore we describe the composition factors of $Rj_*\mathbb C_U $ as certain local cohomology sheaves and give their multiplicity.

Discriminantal Arrangement, $3\times3$ Minors of Plücker Matrix and hypersurfaces in Grassmannian $Gr(3,n)$

We show that points in specific degree 2 hypersurfaces in the Grassmannian $Gr(3, n)$ correspond to generic arrangements of $n$ hyperplanes in $\mathbb{C}^3$ with associated discriminantal arrangement having intersections of multiplicity three in codimension two.

Hyperplane arrangements in CoCoA

We introduce the package \textbf{arrangements} for the software CoCoA. This package provides a data structure and the necessary methods for working with hyperplane arrangements. In particular, the package implements methods to enumerate many commonly studied classes of arrangements, perform operations on them, and calculate various invariants associated to them.

超平面排列补集上复秩一局部系统的非共振定理

Combinatorial Nonresonance Theorems for Hyperplane Arrangement Complements

本文研究超平面排列补集上复秩一局部系统的非共振现象，提出组合充分条件，并改进Bailet、Dimca和Yoshinaga的定理，同时发展限制与提升技术证明线排列的非共振定理。

The Monodromy Conjecture for hyperplane arrangements

The Monodromy Conjecture asserts that if c is a pole of the local topological zeta function of a hypersurface, then exp(2πi c) is an eigenvalue of the monodromy on the cohomology of the Milnor fiber. A stronger version of the conjecture asserts that every pole is a root of the Bernstein-Sato polynomial of the hypersurface. In this note we prove the weak version of the conjecture for hyperplane arrangements. Furthermore, we reduce the strong version to the following conjecture: -n/d is always a root of the Bernstein-Sato polynomial of an indecomposable essential central hyperplane arrangement o…

Morse theory, Milnor fibers and minimality of hyperplane arrangements

Through the study of Morse theory on the associated Milnor fiber, we show that complex hyperplane arrangement complements are minimal. That is, the complement of a complex hyperplane arrangement has the homotopy type of a CW complex in which the number of p-cells equals the p-th betti number.

New characterizations of freeness for hyperplane arrangements

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.

Decomposition of D-modules over a hyperplane arrangement in the plane

We consider the D-module defined as the push-forward of a rank one linear system on the complement of a central plane hyperplane arrangement, and calculate its decomposition series, using algebraic calculations in the Weyl algebra.

Free hyperplane arrangements over arbitrary fields

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 $\mathbb{Q}$.

Hodge spectrum of hyperplane arrangements

In this article there are two main results. The first result gives a formula, in terms of a log resolution, for the graded pieces of the Hodge filtration on the cohomology of a unitary local system of rank one on the complement of an arbitrary divisor in a smooth projective complex variety. The second result is an application of the first. We give a combinatorial formula for the spectrum of a hyperplane arrangement. M. Saito recently proved that the spectrum of a hyperplane arrangement depends only on combinatorics. However, a combinatorial formula was missing. The formula is achieved by a dif…

Length and multiplicity of the local cohomology with support in a hyperplane arrangement

Let $R$ be the polynomial ring in $n$ variables with coefficients in a field $K$ of characteristic zero. Let $D_n$ be the $n$-th Weyl algebra over $K$. Suppose that $f \in R$ defines a hyperplane arrangement in the affine space $K^n$. Then the length and the multiplicity of the 1st local cohomology group $H^1_{(f)}(R)$ as left $D_n$-module coincide and are explicitly expressed in terms of the Poincaré polynomial or the Möbius function of the arrangement.

Combinatorially equivalent hyperplane arrangements

We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong $σ$-Gröbner bases. Moreover, we prove that the Terao's conjecture over finite fields implies the conjecture over the rationals.

Freeness of hyperplane arrangements and related topics

This is the expanded notes of the lecture by the author in "Arrangements in Pyrenees", June 2012. We are discussing relations of freeness and splitting problems of vector bundles, several techniques proving freeness of hyperplane arrangements, K. Saito's theory of primitive derivations for Coxeter arrangements, their application to combinatorial problems and related conjectures.

On the explicit calculation of Hirzebruch-Milnor classes of hyperplane arrangements

The Hirzebruch-Milnor class is given by the difference between the homology Hirzebruch characteristic class and the virtual one. It is known that the Hirzebruch-Milnor class for a certain singular hypersurface can be calculated by using the Hodge spectrum of each stratum of singular locus. So far there is no explicit calculation of this invariant for any non-trivial examples, and we calculate this invariant by two different ways for low dimmensional hyperplane arrangements.

Gysin morphisms for non-transversal hyperplane sections with an application to line arrangements

We prove the existence of Gysin morphisms for hyperplane sections that may not satisfy the usual hypotheses of the Lefschetz hyperplane theorem. As an application, we show the triviality of the Alexander polynomial of a particular class of non-symmetric line arrangements, thus providing positive evidence for a conjecture of Papadima and Suciu.

Degeneration of pole order spectral sequences for hyperplane arrangements of 4 variables

For essential reduced hyperplane arrangements of 4 variables, we show that the pole order spectral sequence degenerates almost at $E_2$, and completely at $E_3$, generalizing the 3 variable case where the complete $E_2$-degeneration is known. These degenerations are useful to determine the roots of Bernstein-Sato polynomials supported at the origin. For the proof we improve an estimate of the Castelnuovo-Mumford regularity of logarithmic vector fields which was studied by H. Derksen and J. Sidman.

关于某些Cynk-Szemberg八次超平面排列的同调性质

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。

移动超平面排列的微分方程

Differential Equations for Moving Hyperplane Arrangements

本文研究超平面乘积的Mellin积分（组合关联子）在移动超平面时的行为，并旨在仅基于超平面排列构造全纯湮灭$D$-理想以编码这些函数。

Positivity of GIT heights of zero-cycles and hyperplane arrangements

In 1996 as part of the development of arithmetic intersection theory and Arakelov theory, Zhang defined a "GIT height function" for semi-stable algebraic cycles in projective space. In the same work, Zhang conjectured that this height function was positive. We prove this conjecture for zero-cycles and hyperplane arrangements.

Hyperplane arrangements, M-tame polynomials and twisted cohomology

A new relation between a class of complex polynomials with a good behavior at infinity studied by A. Némethi and A. Zaharia and the cohomology groups of affine complex hyperplane arrangement complements with rank one local system coefficients is introduced and explored. This approach gives in particular new upper-bounds for the dimension of the twisted cohomology groups of line arrangement complements in the complex affine plane.

On the Falk invariant of hyperplane arrangements attached to gain graphs

The fundamental group of the complement of a hyperplane arrangement in a complex vector space is an important topological invariant. The third rank of successive quotients in the lower central series of the fundamental group was called Falk invariant of the arrangement since Falk gave the first formula and asked to give a combinatorial interpretation. In this article, we give a combinatorial formula for the Falk invariant of hyperplane arrangements attached to certain gain graphs.

First Milnor cohomology of hyperplane arrangements

We show a combinatorial formula for a lower bound of the dimension of the non-unipotent monodromy part of the first Milnor cohomology of a hyperplane arrangement satisfying some combinatorial conditions. This gives exactly its dimension if a stronger combinatorial condition is satisfied. We also prove a non-combinatorial formula for the dimension of the non-unipotent part of the first Milnor cohomology, which apparently depends on the position of the singular points. The latter generalizes a formula previously obtained by the second named author.

Grothendieck classes and Chern classes of hyperplane arrangements

We show that the characteristic polynomial of a hyperplane arrangement can be recovered from the class in the Grothendieck group of varieties of the complement of the arrangement. This gives a quick proof of a theorem of Orlik and Solomon relating the characteristic polynomial with the ranks of the cohomology of the complement of the arrangement. We also show that the characteristic polynomial can be computed from the total Chern class of the complement of the arrangement; this has also been observed by Huh. In the case of free arrangements, we prove that this Chern class agrees with the Chern…

Complements and higher resonance varieties of hyperplane arrangements

Hyperplane arrangements form the geometric counterpart of combinatorial objects such as matroids. The shape of the sequence of Betti numbers of the complement of a hyperplane arrangement is of particular interest in combinatorics, where they are known, up to a sign, as Whitney numbers of the first kind, and appear as the coefficients of chromatic, or characteristic, polynomials. We show that certain combinations, some nonlinear, of these Betti numbers satisfy Schur positivity. At the same time, we study the higher degree resonance varieties of the arrangement. We draw some consequences, using …

Freeness and multirestriction of hyperplane arrangements

Generalizing a result of Yoshinaga in dimension 3, we show that a central hyperplane arrangement in 4-space is free exactly if its restriction with multiplicities to a fixed hyperplane of the arrangement is free and its reduced characteristic polynomial equals the characteristic polynomial of this restriction. We show that the same statement holds true in any dimension when imposing certain tameness hypotheses.

Resonance webs of hyperplane arrangements

Each irreducible component of the first resonance variety of a hyperplane arrangement naturally determines a codimension one foliation on the ambient space. The superposition of these foliations define what we call the resonance web of the arrangement. In this paper we initiate the study of these objects with emphasis on their spaces of abelian relations.

Suciu关于超平面排列的猜想

Suciu's Conjecture on hyperplane arrangements

本文证明任意超平面排列的共振可分离，并借助arXiv:2512.10160的主要结果，以最有效形式证明了Suciu关于超平面排列基本群的猜想。

Some characterizations of freeness of hyperplane arrangement

We will consider some characterizations of freeness of a hyperplane arrangement, in terms of the following properties: locally freeness, factorization of characteristic polynomial and freeness of restricted multiarrangement. In the case of 3-arrangement, freeness is characterized by factorization of characteristic polynomial and coincidence of its roots with exponents of restricted multiarrangement. In the case of higher dimension, it is characterized by a kind of locally freeness and freeness of restricted multiarrangement. As an application, we prove the freeness of certain arrangements whic…

Hyperplane arrangements and mixed Hodge numbers of the Milnor fiber

For each complex central essential hyperplane arrangement $\mathcal{A}$, let $F_{\mathcal{A}}$ denote its Milnor fiber. We use Tevelev's theory of tropical compactifications to study invariants related to the mixed Hodge structure on the cohomology of $F_{\mathcal{A}}$. We prove that the map taking each arrangement $\mathcal{A}$ to the Hodge-Deligne polynomial of $F_{\mathcal{A}}$ is locally constant on the realization space of any loop-free matroid. When $\mathcal{A}$ consists of distinct hyperplanes, we also give a combinatorial description for the homotopy type of the boundary complex of an…

The Hilbert series of Hodge ideals of hyperplane arrangements

Given a reduced effective divisor D on a smooth variety X, we describe the generating function for the classes of the Hodge ideals of D in the Grothendieck group of coherent sheaves on X in terms of the motivic Chern class of the complement of the support of D. As an application, we compute the generating function for the Hilbert series of Hodge ideals of a hyperplane arrangement in terms of the Poincare polynomial of the arrangement.

A survey of combinatorial aspects in the topology of complex hyperplane arrangements

We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras, higher homotopy groups, cohomology rings, twisted homology with rank 1 complex coefficients, and Milnor fibers.

超平面排列的周期与多重 polylogarithms

Periods of hyperplane arrangements and multiple polylogarithms

研究通过组合条件证明超平面排列的周期可表示为多重 polylogarithms 特殊值的线性组合，并应用于全单项群的反射排列。

Addition-deletion theorem for free hyperplane arrangements and combinatorics

In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this atricle, we prove that Terao's celebrated addition-deletion theorem for free arrangements is combinatorial, i.e., whether you can apply it depends only on the intersection lattice of arrangements. The proof is based on a classical technique. Since some parts are already completed recently, we prove the rest part, i.e., the combinatoriality of the addition theorem. As a corollary, we can define a new class of free arrangements called the additionally …

$k$-Lefschetz properties, sectional matrices and hyperplane arrangements

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.