UNEXPECTED CURVES IN ℙ2, LINE ARRANGEMENTS, AND MINIMAL DEGREE OF JACOBIAN RELATIONS

Abstract

We reformulate a fundamental result due to Cook, Harbourne, Migliore and Nagel on the existence and irreducibility 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.