Line arrangements and configurations of points with an unusual geometric property

TL;DR

This paper investigates unexpected algebraic curves arising from specific point configurations in the plane, analyzing their structure, conditions for occurrence, and implications for line arrangements and Terao's conjecture.

Contribution

It introduces a new problem related to conditions imposed by fat points on linear systems, characterizes unexpected curves, and connects these to line arrangement properties and Terao's conjecture.

Findings

Unexpected curves do not exist for points in linear general position.

Criteria for the occurrence and irreducibility of unexpected curves are established.

A new interpretation of the splitting type of line arrangements is provided.

Abstract

The SHGH conjecture proposes a solution to the question of how many conditions a general union of fat points imposes on the complete linear system of curves in $\mathbb P^2$ of fixed degree $d$, and it is known to be true in many cases. We propose a new problem, namely to understand the number of conditions imposed by a general union of fat points on the incomplete linear system defined by the condition of passing through a given finite set of points $Z$ (not general). Motivated by work of Di Gennaro-Ilardi-Vall\`es and Faenzi-Vall\`es, we give a careful analysis for the case where there is a single general fat point, which has multiplicity $d-1$. There is an expected number of conditions imposed by this fat point, and we study those $Z$ for which this expected value is not achieved. We show, for instance, that if $Z$ is in linear general position then such unexpected curves do not exist. We give criteria for the occurrence of such unexpected curves and describe the range of values of $d$ for which they occur. Unexpected curves have a very particular structure, which we describe, and they are often unique for a given set of points. In particular, we give a criterion for when they are irreducible, and we exhibit examples both where they are reducible and where they are irreducible. Furthermore, we relate properties of $Z$ to properties of the arrangement of lines dual to the points of $Z$. In particular, we obtain a new interpretation of the splitting type of a line arrangement. Finally, we use our results to establish a Lefschetz-like criterion for Terao's conjecture on the freeness of line arrangements.