We construct imaginary cycles as Lefschetz cycles for hypergeometric integrals associated with a hypersphere arrangement and discuss the relation between the twisted rational de Rham cohomology. We also discuss it in degenerate cases where several hyperspheres contact with each other. We pose two geometric problems involved in it.

We describe dualities and complexes of logarithmic forms and differentials for central affine and corresponding projective arrangements. We generalize the Borel–Serre formula from vector bundles to sheaves on $\mathbb{P}^d$ with locally free resolutions of length one. Combining these results we present a generalization of a formula due to Mustaţă and Schenck, relating the Poincaré polynomial of an arrangement in $\mathbb{P}^3$ (or a locally tame arrangement in $\mathbb{P}^d$ with zero-dimensional non-free locus) to the total Chern polynomial of its sheaf of logarithmic 1-forms.

We consider the average diameter of a bounded cell of a simple arrangement defined by $n$ hyperplanes in dimension $d$. In particular, we investigate the conjecture stating that the average diameter is no more than the dimension $d$. Previous results in dimensions 2 and 3 suggested that specific extensions of the cyclic arrangement might achieve the largest average diameter. We show that the suggested arrangements do not always achieve the largest diameter and disprove a related conjecture dealing with the minimum number of facets belonging to exactly one bounded cell. In addition, we computationally determine the largest possible average diameter in dimensions 3 and 4 for arrangements defined by no more than 8 hyperplanes via the associated uniform oriented matroids. These new entries substantiate the hypothesis that the largest average diameter is achieved by an arrangement minimizing the number of facets belonging to exactly one bounded cell. The computational framework to generate specific arrangements, and to compute the average diameter and the number of facets belonging to exactly one bounded cell is presented.

Let $\mathcal{A}$ be an affine hyperplane arrangement in $\mathbb{C}^{\ell}$ with complement $U$. Let $f_1, \dots, f_n$ be linear polynomials defining the hyperplanes of $\mathcal{A}$, and $A^{\cdot}$ the algebra of differential forms generated by the one-forms $\mathrm{d} \log f_1 , \dots, \mathrm{d} \log f_n$. To each $\lambda \in \mathbb{C}^n$ we associate the master function $\Phi = \prod^{n}_{i=1} f^{\lambda_i}_i$ on $U$ and the closed logarithmic one-form $\omega = \mathrm{d} \log \Phi$. We assume $\omega$ is a general element of a rational linear subspace $D$ of $A^1$ of dimension $q \gt 1$ such that the map $\bigwedge^k(D) \rightarrow A^k$ given by multiplication in $A^{\cdot}$ is zero for all $p \lt k \leq q$, and is nonzero for $k = p$. With this assumption, we prove the critical locus $\mathrm{crit}(\Phi)$ of $\Phi$ has components of codimension at most $p$, and these are intersections of level sets of $p$ rational master functions. We give conditions that guarantee $\mathrm{crit}(\Phi)$ is nonempty and every component has codimension equal to $p$, in terms of syzygies among polynomial master functions. If $\mathcal{A}$ is $p$-generic, then $D$ is contained in the degree $p$ resonance variety $\mathcal{R}^p(\mathcal{A})$—in this sense the present work complements previous work on resonance and critical loci of master functions. Any arrangement is 1-generic; we give a precise description of $\mathrm{crit}(\Phi_{\lambda})$ in case $\lambda$ lies in an isotropic subspace $D$ of $A^1$, using the multinet structure on $\mathcal{A}$ corresponding to $D \subseteq \mathcal{R}^1 (\mathcal{A})$. This is carried out in detail for the Hessian arrangement. Finally, for arbitrary $p$ and $\mathcal{A}$, we establish necessary and sufficient conditions for a set of integral one-forms to span such a subspace, in terms of nested sets of $\mathcal{A}$, using tropical implicitization.

The middle convolution for completely integrable systems with logarithmic singularities along hyperplane arrangements is defined as a natural generalization of the middle convolution for Fuchsian ordinary differential equations. Additivity of the generalized middle convolution is proved. It is observed that the singular locus may increase by the generalized middle convolution. Examples concerning with hypergeometric series in several variables are given.

The general hypergeometric integrals on the complex matrix space are considered. We study the twisted algebraic de Rham cohomology group associated with the integrand of the general hypergeometric integrals. After reviewing some fundamental facts, we discuss the exterior product structure of the cohomology group.

We review some aspects of the homology of a local system on the complement of a hyperplane arrangement. We describe a relationship between linear representations of the braid groups due to R. Lawrence, D. Krammer and S. Bigelow and the holonomy representations of the KZ connection. We explain a method to describe such representations by the iterated integrals of logarithmic 1-forms.

We discuss combinatorial invariance of the betti numbers of the Milnor fiber for arrangements of lines with points of multiplicity two and three and describe a link between this problem and enumeration of solutions of the Catalan equation over function field in the case when its coefficients are products of linear forms and the equation defines an elliptic curve.

We show that the KZ system has a topological interpretation in the sense that it may be understood as a variation of complex mixed Hodge structure whose successive pure weight quotients are polarized. This in a sense completes and elucidates work of Schechtman–Varchenko done in the early 1990's. A central ingredient is a new realization of the irreducible highest weight representations of a Lie algebra of Kac–Moody type, namely on an algebra of rational polydifferentials on a countable product of Riemann spheres. We also obtain the kind of properties that in the $\mathfrak{sl} (2)$ case are due to Ramadas and are then known to imply the unitarity of the WZW system in genus zero.

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.

This paper provides an overview of selected results and open problems in the theory of hyperplane arrangements, with an emphasis on computations and examples. We give an introduction to many of the essential tools used in the area, such as Koszul and Lie algebra methods, homological techniques, and the Bernstein–Gelfand–Gelfand correspondence, all illustrated with concrete calculations. We also explore connections of arrangements to other areas, such as De Concini–Procesi wonderful models, the Feichtner–Yuzvinsky algebra of an atomic lattice, fatpoints and blowups of projective space, and plane curve singularities.

The Dwyer–Fried invariants of a finite cell complex $X$ are the subsets $\Omega^i_r (X)$ of the Grassmannian of $r$-planes in $H^1 (X, \mathbb{Q})$ which parametrize the regular $\mathbb{Z}^r$-covers of $X$ having finite Betti numbers up to degree $i$. In previous work, we showed that each $\Omega$-invariant is contained in the complement of a union of Schubert varieties associated to a certain subspace arrangement in $H^1 (X, \mathbb{Q})$. Here, we identify a class of spaces for which this inclusion holds as equality. For such “straight” spaces $X$, all the data required to compute the $\Omega$-invariants can be extracted from the resonance varieties associated to the cohomology ring $H^{\ast} (X, \mathbb{Q})$. In general, though, translated components in the characteristic varieties affect the answer.

Arrangement theory plays an essential role in the study of the unfolding model used in many fields. This paper describes how arrangement theory can be usefully employed in solving the problems of counting (i) the number of admissible rankings in an unfolding model and (ii) the number of ranking patterns generated by unfolding models. The paper is mostly expository but also contains some new results such as simple upper and lower bounds for the number of ranking patterns in the unidimensional case.

The aim of this paper is to make sample computations with the Salvetti complex of the "center of mass" arrangement introduced in [CK07] by Cohen and Kamiyama. We compute the homology of the Salvetti complex of these arrangements with coefficients in the sign representation of the symmetric group on $\mathbb{F}_p$ in the case of four particles. We show, when $p$ is an odd prime, the homology is isomorphic to the homology of the configuration space $F(\mathbb{C}, 4)$ of distinct four points in $\mathbb{C}$ with the same coefficients. When $p = 2$, we show the homology is different from the equivariant homology of $F(\mathbb{C}, 4)$, hence we obtain an alternative and more direct proof of a theorem of Cohen and Kamiyama in [CK07].

Let $X$ be a hypersurface in the projective space $\mathbf{P}^{n+1}$ and $G(n + 1, 1)$ be the Grassmann variety $G(n+ 1, 1)$ of lines in $\mathbf{P}^{n+1}$. The subvariety $F(X)$ of $G(n+ 1, 1)$ consisting of lines contained in $X$ is called the Fano variety of $X$. We study a detailed structure of the Fano variety of the Fermat hypersurface $X$ of degree $d$ for $n \geq d$. More precisely, we show that a certain open subset $F^0 (X)$ of $F(X)$ has a fibration structure over a moduli space of marked pointed rational curves and that the fibers are complete intersections of Fermat hypersurfaces introduced in [T]. We also study singularities of $F(X)$.

We consider the $\mathfrak{gl}_N$ Gaudin model of a tensor power of the standard vector representation. The geometric Langlands correspondence in the Gaudin model relates the Bethe algebra of the commuting Gaudin Hamiltonians and the algebra of functions on a suitable space of $N$-th order differential operators. In this paper we introduce a third side of the correspondence: the algebra of functions on the critical set of a master function. We construct isomorphisms of the third algebra and the first two. Our main technical tool is the Bethe vector averaging maps, which is a new object.

This note is an exposition on the results from [SiW] related to Lyubeznik's conjecture on local cohomology. We use the method of Bockstein homomorphisms, which to our knowledge has not been employed before in commutative algebra. As an example, we strengthen the classical Stanley–Reisner correspondence by relating Bockstein homomorphisms on local cohomology in monomial rings to the topological Bockstein homomorphisms of simplicial complexes.

We introduce the necessary language in Section 1, then present the motivation with our main theorem in Section 2, and finally illustrate our results in the case of Stanley–Reisner ideals in Section 3.

The first part of this paper is a survey on algebro-geometric aspects of sheaves of logarithmic vector fields of hyperplane arrangements. In the second part we prove that the relative de Rham cohomology (of degree two) of ADE-type adjoint quotient map is naturally isomorphic to the module of certain multiderivations. The isomorphism is obtained by the Gauss–Manin derivative of the Kostant–Kirillov form.