Unexpected curves in $\mathbb{P}^2$, line arrangements, and minimal degree of Jacobian relations

TL;DR

This paper links unexpected plane curves to the minimal degree of Jacobian syzygies of dual line arrangements, providing new insights and restrictions on when irreducible unexpected quintics can occur.

Contribution

It reformulates a key result on unexpected curves using Jacobian syzygies and explores specific cases where irreducible unexpected quintics appear.

Findings

Irreducible unexpected quintics occur only when Z has 11 or 12 points.

Five specific cases of unexpected quintics are identified.

A new approach connects unexpected curves with Jacobian syzygies.

Abstract

We reformulate a fundamental result due to Cook, Harbourne, Migliore and Nagel on the existence and irreduciblity of unexpected plane curves of a set of points $Z$ in $\mathbb{P}^2$, using the minimal degree of a Jacobian syzygy of the defining equation for the dual line arrangement $\mathcal A_Z$. Several applications of this new approach are given. In particular, we show that the irreducible unexpected quintics may occur only when the set $Z$ has the cardinality equal to 11 or 12, and describe five cases where this happens.