Topological line arrangements with high multiplicities

Abstract

We investigate constraints on the existence of topological and smooth realisations of combinatorial line arrangements and $(n_k)$-configurations in the complex projective plane. By replacing complex lines with locally-flatly or smoothly embedded 2-spheres, we explore the extent to which classical geometric results, such as Hirzebruch's inequality, persist in the topological or smooth category. We introduce two classes of special line arrangements that we call odd and even. We provide constraints for any smoothly realised, non-trivial, odd arrangement via Furuta's 10/8-Theorem. By looking at branched double covers and using the G-signature theorem, we study topologically realised, non-trivial, even arrangement. Finally, we establish a new lower bound for $(n_k)$-configurations, showing that for any topologically realised configuration we have $n \geq k^2-5$, which implies the non-existence of topological realisations for finite projective planes.