On the boundedness of some real line arrangements of type at most one

TL;DR

This paper proves that real line arrangements with intersection points of multiplicity at most five are bounded in size, with a maximum of 522 lines, and there are finitely many such combinatorial types.

Contribution

It establishes a new boundedness result for real line arrangements with restricted intersection multiplicities, showing finiteness in their combinatorial types.

Findings

Maximum of 522 lines in such arrangements

Finiteness of combinatorial types of arrangements

Boundedness property for arrangements with intersection multiplicity at most five

Abstract

In this note, we show that real line arrangements of type at most one, admitting only intersection points of multiplicity at most five, satisfy certain boundedness properties. In particular, we prove that a free real arrangement of $d$ lines with intersection multiplicities bounded by $5$ can have at most $522$ lines and consequently there exist only finitely many combinatorial types of such arrangements.