On free line arrangements with double, triple and quadruple points
Marek Janasz, Izabela Leśniak
TL;DR
The paper investigates free real line arrangements with only double, triple and quadruple points, establishing finiteness results and enumerating admissible weak-combinatorics; it shows a global bound $d\le18$ and isolates realizable types, all non-pencil cases being simplicial. It then analyzes $M$-line arrangements, proving they are simplicial in the real non-pencil case and listing the admissible weak-combinatorics, while using pseudoline tests and wiring diagrams to separate realizable from non-realizable types. A parallel algebraic-surface perspective is developed by attaching log-surfaces $U_{\mathcal{L}}$ to $M$-line arrangements and deriving sharp inequalities for $c_1^2(U_{\mathcal{L}})/c_2(U_{\mathcal{L}})$, with equality at the dual Hesse case. The results illuminate the tight interplay between combinatorial data, real realizability, and the geometry of associated log-surfaces, advancing understanding of freeness in restricted incidence settings.
Abstract
We show that there are only finitely many combinatorial types of free real line arrangements with only double, triple and quadruple intersection points, and we enlist all admissible weak-combinatorics of them. Then we classify all real $M$-line arrangements. In particular, we show that real $M$-line arrangements are simplicial.