UNEXPECTED CURVES IN ℙ2, LINE ARRANGEMENTS, AND MINIMAL DEGREE OF JACOBIAN RELATIONS
Alexandru Dimca · Journal of Commutative Algebra · 2023
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 ℙ2, using the minimal degree of a Jacobian syzygy of the defining equation for the dual line arrangement 𝒜Z. 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.