On Strong Lefschetz Property of 0-dimensional complete intersections

TL;DR

This paper establishes a criterion linking the Strong Lefschetz Property of 0-dimensional complete intersections to the non-vanishing of the Hessian of their associated form, providing a new proof of a known result.

Contribution

It offers a novel proof for the equivalence between the SLP in degree 1 and the nonzero Hessian condition for 0-dimensional complete intersections.

Findings

SLP in degree 1 is equivalent to nonzero Hessian of the associated form

Provides a different proof approach from existing literature

Confirms the known characterization of SLP for 0-dimensional complete intersections

Abstract

We prove that a homogeneous 0-dimensional complete intersection satisfies the Strong Lefschetz Property (SLP) in degree 1 if and only if its associated form has nonzero Hessian. The result is essentially known in the literature, but our proof is different compared with the previous ones.