On connected subgraph arrangements

• Applicant: Professor Dr. Gerhard Röhrle
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 539865068

The theory of hyperplane arrangements has been and continues to be a driving force in mathematics over many decades. It naturally lies at the crossroads of algebra, combinatorics, algebraic geometry, and topology. This proposal in turn is concerned with the interplay of combinatorial and geometric aspects of these subject matters. In their recent paper in 2022, Cuntz and Kühne introduced a special class of hyperplane arrangements stemming from a given (connected) graph G, so called connected subgraph arrangements A(G). They include such prominent classes such as the braid arrangements, Shi arrangements, and resonance arrangements among countless many others. The aims of this proposal are fourfold. Firstly, we intend to answer some of the questions raised in the work of Cuntz and Kühne. Secondly we aim to strengthen several of the results from their paper. Thirdly we hope to prove some new results for this class of arrangements, e.g. over finite fields. In their work, Cuntz and Kühne classified all connected subgraph arrangements over the rationals which are free, factored, simplicial or supersolvable. In this proposal we aim to extend and strengthen these results as follows. We aim to show that a connected subgraph arrangement A(G) over the rationals is free if and only if it is inductively free; that it is factored if and only if it is inductively factored. Cuntz and Kühne specifically raise the question of classifying members among the arrangements A(G) over the rationals that are aspherical. This is probably a hopeless task, as determining asphericity for a given set of arrangements is a notoriously difficult undertaking. However, our further aim is to compile a comprehensive list of graphs G for which A(G) is not aspherical. As asphericity is a local property, this then allows us to conclude that large classes of connected subgraph arrangements are not aspherical. These in particular encompass the aforementioned resonance arrangements of rank at least 5.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2458: Combinatorial Synergies