  In this note we generalize the main result in [DIV: R. Di Gennaro, G. Ilardi,
J. Valles, Singular hypersurfaces characterizing the Lefschetz properties J.
Lond. Math. Soc. (2) 89 (2014), no. 1, 194-212] on artinian ideals failing
Lefschetz properties, varieties satisfying Laplace equations and existence of
suitable singular hypersurfaces. Moreover we characterize the minimally
generation of ideals generated by power of linear forms by the configuration of
their dual points in the projective plane and we use this result to improve
some propositions on line arrangments and Strong Lefschetz property (SLP) at
range 2 in [DIV]. The starting point was an example in [CHMN: D. Cook II, B.
Harbourne, J. Migliore, U. Nagel, Line arrangements and configurations of
points with an unusual geometric property (2017), arXiv:1602.02300v2]. Finally
we show the equivalence among failing SLP, Laplace equations and some
unexpected curves in [CHMN].
