On the Triangles in Certain Types of Line Arrangements

Abstract

In this article we combinatorially describe the triangles that are present in two types of line arrangements, those which have global cyclicity and those which are infinity type line arrangements. A combinatorial nomenclature has been described for both the types and some properties of the nomenclature have been proved. Later using the nomenclature we describe the triangles present in both types of line arrangements in Theorems $A,B$. We also prove that the set of triangles uniquely determine, in a certain precise sense, the line arrangements with global cyclicity and not the infinity type line arrangements where counter examples have been provided. In Theorem $9.1$, given a nomenclature, we characterize when a particular line symbol in the nomenclature is a line at infinity for the arrangement determined by the nomenclature.