Geometry of 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.

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.