  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.
