arXiv research

Generalizes cohomology ring result for combinatorial line arrangements.

Study restricts line arrangements with odd points using topological arguments.

Study shows certain toric arrangements have minimal topological complements.

We note an intimate connection between the Lefschetz Theorem for c-arrangements, and a theorem of Hironaka relating the complement of an arrangement to its boundary manifold. This results in a generalization of Hironaka's result.

We show that certain aspherical manifolds arising from hyperplane arrangements in negatively curved manifolds have relatively hyperbolic fundamental group.

New method determines arrangement combinatorics from Milnor fiber boundary.

Let A be a line arrangement in the complex projective plane CP2. We define and describe the inclusion map of the boundary manifold --the boundary of a close regular neighborhood of A-- in the exterior of the arrangement. We obtain two explicit descriptions of the map induced on the fundamental groups. These computation…

Bordifications of hyperplane arrangements yield complexes with homotopy type of wedges of spheres.

We consider arrangements of n connected codimensional one submanifolds in closed d-dimensional manifold M. Let f be the number of connected components of the complement in M to the union of submanifolds. We prove the sharp lower bound for f via n and homology group H_{d-1}(M). The sets of all possible f-values for give…

The study finds PK cone metrics on complex manifolds near hyperplane arrangements.

Researchers reinterpret complex hyperbolic orbifolds using line arrangements.

There are several topological spaces associated to a complex hyperplane arrangement: the complement and its boundary manifold, as well as the Milnor fiber and its own boundary. All these spaces are related in various ways, primarily by a set of interlocking fibrations. We use cohomology with coefficients in rank 1 loca…

We study the topology of the boundary manifold of a line arrangement in CP^2, with emphasis on the fundamental group G and associated invariants. We determine the Alexander polynomial Delta(G), and more generally, the twisted Alexander polynomial associated to the abelianization of G and an arbitrary complex representa…

New invariant identifies complex line arrangements with same combinatorics but different embeddings.

The paper introduces relative objects and proves a transversality theorem.

New potential functions reveal Frobenius-like structure in weighted hyperplane arrangements.

Euclidean systems and real PK arrangements linked via geometry.

We investigate several topological and combinatorial properties of line arrangements. We associate to a line arrangement a link obtained by intersecting the arrangement with some sphere. Several topics are discussed: (a) some link configurations can be realized by complex line arrangements but not by real line arrangem…

New infinite $K(π,1)$ arrangements found in higher dimensions.

Study shows asphericity isn't inherited in certain arrangements.

Icosidodecahedral arrangement has torsions in homology, and is a K(π,1).

This paper studies topological properties of line arrangements in complex projective plane.

Study a specific line arrangement and compute its fundamental group via braid monodromy.

Research examines arrangements of hyperplanes in real projective spaces, focusing on specific cases.

A line arrangement of $3n$ lines in $\mathbb CP^2$ satisfies Hirzebruch property if each line intersect others in $n+1$ points. Hirzebruch asked if all such arrangements are related to finite complex reflection groups. We give a positive answer to this question in the case when the line arrangement in $\mathbb CP^2$ is…

Study of Alexander modules for hyperplane arrangements, distinguishing complements.

Researchers classify K-stability of log Fano hyperplane arrangements.

Proves lattice isomorphic arrangements can have equivalent complements but non-homeomorphic embeddings.

Using the invariant developed in [6], we differentiate four arrangements with the same combinatorial information but in different deformation classes. From these arrangements, we construct four other arrangements such that there is no orientation-preserving homeomorphism between them. Furthermore, some couples of arran…

Study on links formed by pseudocircle arrangements, focusing on three unavoidable cases.

We define several homology theories for central hyperplane arrangements, categorifying well-known polynomial invariants including the characteristic polynomial, Poincare polynomial, and Tutte polynomial. We consider basic algebraic properties of such chain complexes, including long-exact sequences associated to deletio…

Survey on hyperplane arrangements and their topology.

Study conic-line arrangements of degree 7, finding their topology and components.

The fundamental group of the complement of a hyperplane arrangement plays an important role in studying the corresponding arrangements. In particular, for large families of hyperplane arrangements, this fundamental group, being isomorphic to the fundamental group of a complement of a line arrangement, has some remarkab…

Our aim is to generalize the result that two generic complex line arrangements are equivalent. In fact for a line arrangement A we associate its defining polynomial, the product of a_ix+b_iy+c_i, so that A = (f=0). We prove that the defining polynomials of two generic line arrangements are, up to a small deformation, t…

Two unique conic-line arrangements with degree 9 are found.

Link projections with the same circle arrangement can be transformed by specific moves.

Church-Ellenberg-Farb used the language of FI-modules to prove that the cohomology of certain sequences of hyperplane arrangements with S_n-actions satisfies representation stability. Here we lift their results to the level of the arrangements themselves, and define when a collection of arrangements is "finitely genera…

Geometric arguments show simplicial arrangements with few double points can't have an irreducible cubic curve dual.

An arrangement of pseudocircles is a finite set of oriented closed Jordan curves each two of which cross each other in exactly two points. To describe the combinatorial structure of arrangements on closed orientable surfaces, in (Linhart, Ortner 2004) so-called *intersection schemes* were introduced. Building up on res…

Topology of space line arrangements depends on line count and multiple points.

A toric arrangement is a finite set of hypersurfaces in a complex torus, every hypersurface being the kernel of a character. In the present paper we build a CW-complex homotopy equivalent to the arrangement complement, with a combinatorial description similar to that of the well-known Salvetti complex. If the toric arr…

New bounds on embeddability of pseudocircle and graph arrangements on surfaces.

New maximal acyclic matchings prove minimality of hyperplane arrangements.

In arrangements of pseudocircles (Jordan curves) the weight of a vertex (intersection point) is the number of pseudocircles that contain the vertex in its interior. We give improved upper bounds on the number of vertices of weight <=k in certain arrangements of pseudocircles in the plane. In particular, forbidding cert…

Study of first homology group of Milnor fiber boundary for generic hyperplane arrangements in C^3.

The invariant $\mathcal{I}(\mathcal{A},ξ,γ)$ was first introduced by E. Artal, V. Florens and the author. Inspired by the idea of G. Rybnikov, we obtain a multiplicativity theorem of this invariant under the gluing of two arrangements along a triangle. An application of this theorem is to prove that the extended Rybnik…

Let F be R or C, d the dimension of F over R. Denote by P(F) either the affine plane A(F) or the hyperbolic plane H(F) over F. An arrangement L of k lines in P(F) (pairwise non-parallel in the hyperbolic case) has a link at infinity K(L) comprising k unknotted (d-1)-spheres in the (2d-1)-sphere, whose topology reflects…