# Laplace equations, Lefschetz properties and line arrangements

## TL;DR

This paper extends previous results linking Lefschetz properties, Laplace equations, and line arrangements, providing new characterizations and improving existing propositions related to algebraic and geometric properties of ideals and hypersurfaces.

## Contribution

It generalizes key results on Lefschetz properties, characterizes ideals generated by powers of linear forms via dual point configurations, and establishes equivalences among failing SLP, Laplace equations, and unexpected curves.

## Key findings

- Generalized main results on Lefschetz properties and Laplace equations.
- Characterized ideals generated by powers of linear forms through dual point configurations.
- Proved equivalence among failing SLP, Laplace equations, and unexpected curves.

## Abstract

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].