Combinatorics of line arrangements and characteristic varieties

射影平面上的三角排列

Triangular arrangements on the projective plane

研究三角排列的组合学性质，证明其总能由单位根排列实现，并给出自由与非自由三角排列的示例。

Deletion theorem and combinatorics of hyperplane arrangements

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.

Resonance varieties, admissible line combinatorics and combinatorial pencils

In this paper we define the combinatorial analogous of a pencil, and show its relationship with the concept of admissibility. Such an object is usefull to study the isomorphisms between fundamental groups of the complements of line arrangement with the same combinatorial type. This definition generalizes the idea of net given by Yuzvinsky and others. The main theorem in this paper states that there is a correspondence between components of the resonance variety and combinatorial pencils.

Schubert Varieties, Linear Codes and Enumerative Combinatorics

We consider linear error correcting codes associated to higher dimensional projective varieties defined over a finite field. The problem of determining the basic parameters of such codes often leads to some interesting and difficult questions in combinatorics and algebraic geometry. This is illustrated by codes associated to Schubert varieties in Grassmannians, called Schubert codes, which have recently been studied. The basic parameters such as the length, dimension and minimum distance of these codes are known only in special cases. An upper bound for the minimum distance is known and it is …

String cone and Superpotential combinatorics for flag and Schubert varieties in type A

We study the combinatorics of pseudoline arrangements and their relation to the geometry of flag and Schubert varieties. We associate to each pseudoline arrangement two polyhedral cones, defined in a dual manner. We prove that one of them is the weighted string cone by Littelmann and Berenstein-Zelevinsky. For the other we show how it arises in the framework of cluster varieties and mirror symmetry by Gross-Hacking-Keel-Kontsevich: for the flag variety the cone is the tropicalization of their superpotential while for Schubert varieties a restriction of the superpotential is necessary. We prove…

Translated tori in the characteristic varieties of complex hyperplane arrangements

We give examples of complex hyperplane arrangements for which the top characteristic variety contains positive-dimensional irreducible components that do not pass through the origin of the character torus. These examples answer several questions of Libgober and Yuzvinsky. As an application, we exhibit a pair of arrangements for which the resonance varieties of the Orlik-Solomon algebra are (abstractly) isomorphic, yet whose characteristic varieties are not isomorphic. The difference comes from translated components, which are not detected by the tangent cone at the origin.

Strongly symmetric smooth toric varieties

We investigate toric varieties defined by arrangements of hyperplanes and call them strongly symmetric. The smoothness of such a toric variety translates to the fact that the arrangement is crystallographic. As a result, we obtain a complete classification of this class of toric varieties. Further, we show that these varieties are projective and describe associated toric arrangements in these varieties.

Freeness and near freeness are combinatorial for line arrangements in small degrees

We prove Terao conjecture saying that the freeness is determined by the combinatorics for arrangements of 13 lines in the complex projective plane and that the property of being nearly free is combinatorial for line arrangements of up to 12 lines in the complex projective plane.

Local models of Shimura varieties, I. Geometry and combinatorics

We survey the theory of local models of Shimura varieties. In particular, we discuss their definition and illustrate it by examples. We give an overview of the results on their geometry and combinatorics obtained in the last 15 years. We also exhibit their connections to other classes of algebraic varieties such as nilpotent orbit closures, affine Schubert varieties, quiver Grassmannians and wonderful completions of symmetric spaces.

Arithmetics and combinatorics of tropical Severi varieties of univariate polynomials

We give a description of the tropical variety of univariate polynomials of degree n having two double roots. As a set, it is given as the union of three types of maximal cones of dimension n-1, where only cones of two of these types are cones of the secondary fan of {0,...,n}. Through Kapranov's theorem, this goal is achieved by a careful study of the possible valuations of the elementary symmetric functions of the roots of a polynomial with two double root. Despite its apparent simplicity, the computation of the tropical Severi variety has both combinatorial and arithmetic ingredients.

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.

拟阵与超平面排列的共振簇及李代数

Resonance varieties and Lie algebras of matroids and hyperplane arrangements

本文综述拟阵 Orlik-Solomon 代数上的共振簇、完整李代数与完整 Chen 李代数，比较可实现与不可实现拟阵的异同并提出开放问题。

Bow varieties---geometry, combinatorics, characteristic classes

Cherkis bow varieties are believed to be the set of spaces where 3d mirror symmetry for characteristic classes can be observed. We describe geometric structures on a large class of Cherkis bow varieties by developing the necessary combinatorial presentations, including binary contingency tables and skein diagrams. We make the first steps toward the sought after statement for 3d mirror symmetry for characteristic classes by conjecturing a formula for cohomological stable envelopes. Additionally we provide an account of the full statement, with examples, for elliptic stable envelopes.

一般排列簇的最小对数歧变的上界

Bounding minimal log discrepancies of general arrangement varieties

本文证明了在一般排列簇上，最小对数歧变可由簇的维数上界限制，扩展了关于该猜想的先前结果。

超平面配置与向量群的紧化

Hyperplane Arrangements and Compactifications of Vector Groups

本文研究了超平面配置的 Schubert 簇作为仿射空间等变紧化的充分必要条件，将理论推广至部分紧化及态射，揭示了其与环面簇和多面体扇对应关系的相似性。

子空间配置的组合平坦与Schubert簇

Combinatorial flats and Schubert varieties of subspace arrangements

本文通过定义多拟阵的组合平坦格$\mathcal L_P$，解决其缺乏类似拟阵平坦格几何模型的问题，并证明在特征为0的域上可实现时，该格可由子空间配置的Schubert簇建模，推广了已有结果。

Reflection groups, reflection arrangements, and invariant real varieties

Let $X$ be a nonempty real variety that is invariant under the action of a reflection group $G$. We conjecture that if $X$ is defined in terms of the first $k$ basic invariants of $G$ (ordered by degree), then $X$ meets a $k$-dimensional flat of the associated reflection arrangement. We prove this conjecture for the infinite types, reflection groups of rank at most $3$, and $F_4$ and we give computational evidence for $H_4$. This is a generalization of Timofte's degree principle to reflection groups. For general reflection groups, we compute nontrivial upper bounds on the minimal dimension of …

某个7次圆锥曲线-直线排列的实现空间与一个π₁等价Zariski对

The realization space of a certain conic line arrangement of degree 7 and a $π_1$-equivalent Zariski pair

本文通过构造一个π₁等价Zariski对，确定了具有特定固定组合结构的7次圆锥曲线-直线排列实现空间的连通分支数。

Tate properties, polynomial-count varieties, and monodromy of hyperplane arrangements

The order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given. It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over $\Q$, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial. We construct a hyperplane arrangement defined over $\Q$, whose Milnor fiber has a nontrivial monodromy o…

The Geometry and Combinatorics of Some Hessenberg Varieties Related to the Permutohedral Variety

We construct a concrete isomorphism from the permutohedral variety to the regular semisimple Hessenberg variety associated to the Hessenberg function $h_+(i)=i+1$, $1\le i\le n-1$. In the process of defining the isomorphism, we introduce a sequence of varieties which we call the prepermutohedral varieties. We first determine the toric structure of these varieties and compute the Euler characteristics and the Betti numbers using the theory of toric varieties. Then, we describe the cohomology of these varieties. We also find a natural way to encode the one-dimensional components of the cohomolog…

具有二重和三重交点的极小加一生成线排列

Minimal plus-one generated line arrangements with double and triple intersection points

该论文在弱组合学框架下，完整分类了复射影平面中具有二重和三重交点的极小加一生成线排列。

Special arrangements of lines: codimension two ACM varieties in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$

In this paper we investigate special arrangements of lines in multiprojective spaces. In particular, we characterize codimensional two arithmetically Cohen-Macaulay (ACM) varieties in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$, called varieties of lines. We also describe their ACM property from combinatorial algebra point of view.

abelian varieties的退化组合学与Hodge理论：Mumford构造的综述

Combinatorics and Hodge theory of degenerations of abelian varieties: A survey of the Mumford construction

综述Mumford构造退化的abelian varieties，分析其与toric几何的关系，并研究与regular matroids相关的多变量退化几何与Hodge理论，扩展Clemens关于1参数半稳定退化的基本结果。

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.

The geometry and combinatorics of Springer fibers

This survey paper describes Springer fibers, which are used in one of the earliest examples of a geometric representation. We will compare and contrast them with Schubert varieties, another family of subvarieties of the flag variety that play an important role in representation theory and combinatorics, but whose geometry is in many respects simpler. The end of the paper describes a way that Springer fibers and Schubert varieties are related, as well as open questions.

Rationally smooth Schubert varieties and inversion hyperplane arrangements

We show that an element $w$ of a finite Weyl group $W$ is rationally smooth if and only if the hyperplane arrangement $I$ associated to the inversion set of $w$ is inductively free, and the product $(d_1+1) \cdots (d_l+1)$ of the coexponents $d_1,\ldots,d_l$ is equal to the size of the Bruhat interval $[e,w]$, where $e$ is the identity in $W$. As part of the proof, we describe exactly when a rationally smooth element in a finite Weyl group has a chain Billey-Postnikov decomposition. For finite Coxeter groups, we show that chain Billey-Postnikov decompositions are connected with certain modular…

Intersection cohomology of hypertoric varieties

A hypertoric variety is a quaternionic analogue of a toric variety. Just as the topology of toric varieties is closely related to the combinatorics of polytopes, the topology of hypertoric varieties interacts richly with the combinatorics of hyperplane arrangements and matroids. Using finite field methods, we obtain combinatorial descriptions of the Betti numbers of hypertoric varieties, both for ordinary cohomology in the smooth case and intersection cohomology in the singular case. We also introduce a conjectural ring structure on the intersection cohomology of a hypertoric variety.

Kähler-Einstein metrics on symmetric general arrangement varieties

We calculate Chow quotients of some families of symmetric \(T\)-varieties. In complexity two we obtain new examples of Kähler-Einstein metrics by bounding the symmetric alpha invariant of their orbifold quotients. As an additional application we determine the homeomorpism class of the orbit space of the compact torus action.

Symmetric subgroup orbit closures on flag varieties: Their equivariant geometry, combinatorics, and connections with degeneracy loci

We give explicit formulas for torus-equivariant fundamental classes of closed $K$-orbits on the flag variety $G/B$ when $G$ is one of the classical groups $SL(n,\C)$, $SO(n,\C)$, or $Sp(2n,\C)$, and $K$ is a symmetric subgroup of $G$. We describe parametrizations of each orbit set and the combinatorics of its weak order, allowing us to compute formulas for the equivariant classes of all remaining orbit closures using divided difference operators. In each of the cases in type A, we realize the $K$-orbit closures as universal degeneracy loci of a certain type, involving a vector bundle $V$ over …

Resonance varieties over fields of positive characteristic

This is a revision and update of the part of the preprint math.CO/0405210 concerning field coefficients, line complexes, and the Hessian arrangement. The material from that paper concerning coefficients in arbitrary commutative rings and the deleted B_3 arrangement will appear separately.

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 …

Torus actions and combinatorics of polytopes

An n-dimensional polytope P^n is called simple if exactly n codimension-one faces meet at each vertex. The lattice of faces of a simple polytope P^n with m codimension-one faces defines an arrangement of even-dimensional planes in R^{2m}. We construct a free action of the group R^{m-n} on the complement of this arrangement. The corresponding quotient is a smooth manifold Z_P invested with a canonical action of the compact torus T^m with the orbit space P^n. For each smooth projective toric variety M^{2n} defined by a simple polytope P^n with the given lattice of faces there exists a subgroup T…

Some analogs of Zariski's Theorem on nodal line arrangements

For line arrangements in P^2 with nice combinatorics (in particular, for those which are nodal away the line at infinity), we prove that the combinatorics contains the same information as the fundamental group together with the meridianal basis of the abelianization. We consider higher dimensional analogs of the above situation. For these analogs, we give purely combinatorial complete descriptions of the following topological invariants (over an arbitrary field): the twisted homology of the complement, with arbitrary rank one coefficients; the homology of the associated Milnor fiber and Alexan…

关于仅含二重、三重和四重点的自由直线排列

On free line arrangements with double, triple and quadruple points

本文研究仅含二重、三重和四重点的自由实直线排列，证明其组合类型有限并列出全部容许弱组合类型，进而分类实$M$-直线排列并证明其为单纯排列。

可分解排列的拓扑与组合学

On the topology and combinatorics of decomposable arrangements

研究可分解排列的拓扑与组合性质，证明其关联格里ev代数分解为自由代数的直积，并探讨alexander不变量的分解及chern秩的局部贡献。

Companion varieties for root systems and Fermat arrangements

Unexpected hypersurfaces are a brand name for some special linear systems. They were introduced around 2017 and are a field of intensive study since then. They attracted a lot of attention because of their close tights to various other areas of mathematics including vector bundles, arrangements of hyperplanes, geometry of projective varieties. Our research is motivated by the what is now known as the BMSS duality, which is a new way of deriving projective varieties out of already constructed. The last author coined the concept of companion surfaces in the setting of unexpected curves admitted …

线与一个光滑圆锥的排列的自由性

Freeness of arrangements of lines and one conic with ordinary quasi-homogeneous singularities

本文研究由线和一个光滑圆锥组成的自由排列，通过弱组合学方法对具有准齐次普通奇点的排列进行不完全分类。

C型Schubert簇半环面退化的组合学

Combinatorics of semi-toric degenerations of Schubert varieties in type C

本文利用带自相交的pipe dream为C型辛Gelfand-Tsetlin多面体的Schubert类推广引入组合模型，并证明该推广可由斜有丝分裂算子构造。

Invariants of Combinatorial Line Arrangements and Rybnikov's Example

Following the general strategy proposed by G.Rybnikov, we present a proof of his well-known result, that is, the existence of two arrangements of lines having the same combinatorial type, but non-isomorphic fundamental groups. To do so, the Alexander Invariant and certain invariants of combinatorial line arrangements are presented and developed for combinatorics with only double and triple points. This is part of a more general project to better understand the relationship between topology and combinatorics of line arrangements.

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.

Arrangements of rational sections over curves and the varieties they define

We introduce arrangements of rational sections over curves. They generalize line arrangements on P^2. Each arrangement of d sections defines a single curve in P^{d-2} through the Kapranov's construction of \bar{M}_{0,d+1}. We show a one-to-one correspondence between arrangements of d sections and irreducible curves in M_{0,d+1}, giving also correspondences for two distinguished subclasses: transversal and simple crossing. Then, we associate to each arrangement A (and so to each irreducible curve in M_{0,d+1}) several families of nonsingular projective surfaces X of general type with Chern numb…

Schubert geometry and combinatorics

This chapter combines an introduction and research survey about Schubert varieties. The theme is to combinatorially classify their singularities using a family of polynomial ideals generated by determinants.

关于具有奇数重数的线构形

On line arrangements with odd multiplicities

本文利用4-流形中局部平坦球面的拓扑论证，对具有奇数重数奇点的线构形的弱组合学给出限制，并证明不存在仅含三重点的13条线构形。

热带Matroid Schubert变种与分级Möbius代数

Tropical matroid Schubert varieties and the graded Möbius algebra

本文引入热带Matroid Schubert变种，证明其热带上同调环与Matroid的分级Möbius代数同构，扩展了Arrangement Schubert变种到任意Matroid。

On some freeness-type properties for line arrangements

By way of Ziegler restrictions we study the relation between nearly free plane arrangements and combinatorics and we give a Yoshinaga-type criterion for plus-one generated plane arrangements.

Multinets, resonance varieties, and pencils of plane curves

We show that a line arrangement in the complex projective plane supports a nontrivial resonance variety if and only if it is the underlying arrangement of a "multinet," a multi-arrangement with a partition into three or more equinumerous classes which have equal multiplicities at each inter-class intersection point, and satisfy a connectivity condition. We also prove that this combinatorial structure is equivalent to the existence of a pencil of plane curves, also satisfying a connectivity condition, whose singular fibers include at least three products of lines, which comprise the arrangement…

Characteristic varieties of arrangements

The k-th Fitting ideal of the Alexander invariant B of an arrangement A of n complex hyperplanes defines a characteristic subvariety, V_k(A), of the complex algebraic n-torus. In the combinatorially determined case where B decomposes as a direct sum of local Alexander invariants, we obtain a complete description of V_k(A). For any arrangement A, we show that the tangent cone at the identity of this variety coincides with R^1_k(A), one of the cohomology support loci of the Orlik-Solomon algebra. Using work of Arapura and Libgober, we conclude that all positive-dimensional components of V_k(A) a…

Compactifications defined by arrangements II: locally symmetric varieties of type IV

We define a new class of completions of locally symmetric varieties of type IV which interpolates between the Baily-Borel compactification and Mumford's toric compactifications. An arithmetic arrangement in a locally symmetric variety of type IV determines such a completion canonically. This completion admits a natural contraction that leaves the complement of the arrangement untouched. The resulting completion of the arrangement complement is very much like a Baily-Borel compactification: it is the proj of an algebra of meromorphic automorphic forms. When that complement has a moduli space in…

7次和8次圆锥-直线构型的强Ziegler对示例

Examples of strong Ziegler pairs of conic-line arrangements of degree 7 and 8

本文证明了某些具有相同组合结构但嵌入拓扑不同的平面曲线对（Zariski对）同时也是Milnor代数不同构的强Ziegler对。

Cox rings and combinatorics II

We study varieties with a finitely generated Cox ring. In a first part, we generalize a combinatorial approach developed in earlier work for varieties with a torsion free divisor class group to the case of torsion. Then we turn to modifications, e.g., blow ups, and the question how the Cox ring changes under such maps. We answer this question for a certain class of modifications induced from modifications of ambient toric varieties. Moreover, we show that every variety with finitely generated Cox ring can be explicitly constructed in a finite series of toric ambient modifications from a combin…

P-Grothendieck多项式的次数与Pfaffian簇的正则性

Degrees of P-Grothendieck polynomials and regularity of Pfaffian varieties

本研究利用移位杨表组合数学证明P-Grothendieck多项式次数公式，结合已有结果得到vexillary斜对称矩阵Schubert簇的Castelnuovo-Mumford正则性上界，还得到Grassmannian相关多项式次数与簇正则性的新公式。

Notes on the Hodge Conjecture for Fermat Varieties

We review a combinatoric approach to the Hodge Conjecture for Fermat Varieties and announce new cases where the conjecture is true.

Multinets, parallel connections, and Milnor fibrations of arrangements

The characteristic varieties of a space are the jump loci for homology of rank 1 local systems. The way in which the geometry of these varieties may vary with the characteristic of the ground field is reflected in the homology of finite cyclic covers. We exploit this phenomenon to detect torsion in the homology of Milnor fibers of projective hypersurfaces. One tool we use is the interpretation of the degree 1 characteristic varieties of a hyperplane arrangement complement in terms of orbifold fibrations and multinets on the corresponding matroid. Another tool is a polarization construction, ba…

从帕斯卡定理到齐格勒线排布的几何

From Pascal's Theorem to the geometry of Ziegler's line arrangements

本文研究 Ziegler 与 Yuzvinsky 用以说明同调不变量和形式性不由组合学决定的六边形线排布，揭示其与帕斯卡定理的几何联系，并给出 Jacobian syzygy 最小次数及形式性的一般结论。