Unexpected curves arising from special line arrangements

TL;DR

This paper characterizes supersolvable line arrangements whose dual configurations admit unexpected curves, expanding understanding of special geometric configurations and their properties in algebraic geometry.

Contribution

It provides a characterization of supersolvable line arrangements with unexpected curves and introduces new infinite families of such arrangements.

Findings

Identifies conditions under which dual configurations admit unexpected curves.

Characterizes supersolvable line arrangements with this property.

Provides new infinite families of arrangements with unexpected curves.

Abstract

In a recent paper arXiv:1602.02300v2, Cook II, Harbourne, Migliore and Nagel related the splitting type of a line arrangement in the projective plane to the number of conditions imposed by a general fat point of multiplicity $j$ to the linear system of curves of degree $j+1$ passing through the configuration of points dual to the given arrangement. If the number of conditions is less than the expected, we say that the configuration of points admits unexpected curves. In this paper, we characterize supersolvable line arrangements whose dual configuration admits unexpected curves and we provide other infinite families of line arrangements with this property.