On Strong Lefschetz Property of 0-dimensional complete intersections and Veronese varieties

TL;DR

The paper investigates Strong Lefschetz Property (SLP) in degree 1 for 0-dimensional complete intersections through the lens of Macaulay inverse systems and the associated form $A_{f f}$. It proves that nonvanishing discriminant $\,\Delta(A_{f f})\neq 0$ is equivalent to a smooth projective hypersurface $A_{f f}=0$ and to the nonintersection $\,\mathbb{P}(J({\bf f})_{T-1})\cap \mathcal{V}^{T-1}_n=\emptyset$, providing a computable criterion for SLP; this also translates to generic $J({\bf f})$ of fixed multidegree yielding SLP in degree $1$, and to generic Milnor algebras $M(f)$ under the smoothness and discriminant conditions. The authors construct 1-dimensional almost complete intersections from generic $g_j\in K_{d_j}$, show their associated varieties are smooth, and analyze saturation and dual modules to establish a deformation path that transfers properties from the almost complete intersection to the 0-dimensional case, culminating in proofs of Theorems A3 and SLP1. This approach links Veronese geometry, Macaulay duality, and Lefschetz properties, yielding a concrete discriminant-based criterion that guarantees SLP in degree $1$ for a broad class of algebras. The results advance understanding of when SLP holds in these algebras and illuminate deep connections between inverse systems and geometric properties of associated varieties.

Abstract

We show that the Strong Lefschetz Property in degree 1 for a homogeneous 0-dimensional complete intersection holds if the corresponding associated form, the Macaulay inverse systems, has a non-zero discriminant.

Theorems & Definitions (8)

• Conjecture 1.1
• Remark 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Corollary 1.6
• Theorem 2.1
• Theorem 2.2