On the exponents of free and nearly free projective plane curves

TL;DR

This paper demonstrates that all integer pairs can serve as exponents for free or nearly free irreducible plane curves and line arrangements, providing simple example families and analyzing their topological properties.

Contribution

It introduces two simple families of examples showing all possible exponent pairs for free/nearly free curves and studies their complement topologies.

Findings

All integer pairs occur as exponents for free/nearly free curves.

Many complements are not $K(1)$ spaces.

Explicit example families are constructed.

Abstract

We show that all the possible pairs of integers occur as exponents for free or nearly free irreducible plane curves and line arrangements, by producing only two types of simple families of examples. The topology of the complements of these curves and line arrangements is also discussed, and many of them are shown not to be $K(\pi,1)$ spaces.