Geometry of Arrangements

Bergman complexes are intertwined with arrangement theory in various ways.
They figure as tropicalizations of arrangement complements and, at the same time,
provide combinatorial core structure for certain arrangement compactifications.
They add to the collection of abstract (simplicial, resp. polyhedral) complexes
associated with matroids, which makes them interesting objects from the viewpoint
of algebraic and topological combinatorics. We review old and new results on these
beautiful complexes.

Resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial and geometric contexts. I will discuss their structure in an algebraic setting and will present a sharp formula for the Hilbert series of the Koszul module associated to the resonance variety in question. I will then explain how this can be applied to prove Suciu's Conjecture on the Chen invariants of hyperplane arrangements. Based on joint work with Aprodu, Raicu and Suciu.

In this talk I will describe novel tools for computing the likelihood correspondence of an arrangement of hypersurfaces in a projective space. This uses the module of logarithmic derivations. As application, I will focus on the case of graphic arrangements.

Graphic hyperplane arrangements are an interesting example of arrangements, since they are the subarrangements of the well-studied braid arrangement and have a strong connection to graph theory. This makes it possible to use graph theoretical tools to study them and specifically their module of logarithmic derivations. This talk will highlight some of these tools in the hyperplane arrangement setting, showcase some (recent) results and suggest further research directions to explore.

The main aim of my talk is to present the notion of Poincare polynomials of plane curves with quasi-homogeneous singularities. At the first step we show how to define such an object for conic-line arrangements with ordinary quasi-homogeneous singularities, and then we focus on arrangements of conics with some ADE singularities. Then I report on the addition technique that can be used to construct new examples of free plane curves.

The talk is based on the recent preprints by the author.

The combinatorics of a central arrangement of real hyperplanes is encoded in a
zonotope, the Minkowski sum of segments perpendicular to the hyperplanes. An
arrangement is strongly inscribable if it has a zonotope all whose vertices
lie on the unit sphere. The study of inscribed zonotopes and inscribable
arrangements arises from efforts to extend Rivin’s solution to Steiner’s
inscribability problem beyond three dimensions. Strongly inscribable
arrangements reveal new connections to reflection groups and Grünbaum’s quest
for simplicial arrangements. This talk is based on joint work with Sebastian
Manecke.

Many questions on arrangements of lines in the projective plane remain open. For instance, the complete classification of complex arrangements without double points is still unknown, as is the full list of real simplicial arrangements, where the regions formed by the lines are exclusively triangles.

In the hop to generate new line arrangements with interesting properties from known ones, we introduce operators acting on the set of plane line arrangements.

In this talk, I will present examples of realization spaces of line arrangements that are preserved by certain such operators. In particular, we will see that the elliptic surface (\Xi_1(n)) over the modular elliptic curve (X_1(n))—which parametrizes elliptic curves with a torsion point of order (n > 3)—can also be interpreted as (the compactification of) the realization space of certain line arrangements.

We will explore the existence of an operator acting on these arrangements and, consequently, on the elliptic surface (\Xi_1(n)), and describe its action on the surface.

Work in collaboration with Lukas Kühne.

Recently, Cuntz and Kühne introduced the class of a connected subgraph arrangements. This encompasses such classical families as the Shi arrangements in type A and the braid arrangements. Cuntz and Kühne were able to classify all free, supersolvable, factored and simplicial members among them. We revisit this class and investigate some additional properties. In particular, we study the question of asphericity within this class. This is a report on recent joint work with Giordani, Möller and Mücksch.