# Lefschetz property and powers of linear forms in $\mathbb{K}[x,y,z]$

## TL;DR

This paper proves that in the polynomial ring [x,y,z], multiplication by the square of a general linear form has maximal rank on any graded component for ideals generated by arbitrary powers of general linear forms, confirming a conjecture.

## Contribution

It confirms the conjecture that multiplication by the square of a general linear form has maximal rank for ideals generated by arbitrary powers of general linear forms.

## Key findings

- Maximal rank property holds for arbitrary powers of general linear forms.
- Confirms the conjecture for all sets of general linear forms.
- Extends previous results from uniform powers to arbitrary powers.

## Abstract

In [9], Migliore, Mir\'o-Roig and Nagel, proved that if $R = \mathbb{K}[x,y,z]$, where $\mathbb{K}$ is a field of characteristic zero, and $I=(L_1^{a_1},\dots,L_r^{a_4})$ is an ideal generated by powers of 4 general linear forms, then the multiplication by the square $L^2$ of a general linear form $L$ induces an homomorphism of maximal rank in any graded component of $R/I$. More recently, Migliore and Mir\'o-Roig proved in [8] that the same is true for any number of general linear forms, as long the powers are uniform. In addition, they conjecture that the same holds for arbitrary powers. In this paper we will solve this conjecture and we will prove that if $I=(L_1^{a_1},\dots,L_r^{a_r})$ is an ideal of $R$ generated by arbitrary powers of any set of general linear forms, then the multiplication by the square $L^2$ of a general linear form $L$ induces an homomorphism of maximal rank in any graded component of $R/I$.