Inverse system of Gorenstein points in ${\mathbb P^n_{\res}}$

TL;DR

This paper characterizes Gorenstein point sets in projective space via their inverse systems, providing a geometric description and applications to algebraic and tensor decomposition problems.

Contribution

It introduces a new characterization of Gorenstein point sets using inverse systems and describes the parameter space, with implications for Artinian Gorenstein rings and related areas.

Findings

Characterization of Gorenstein points via inverse systems

Description of the parameter space of Gorenstein point sets

Applications to Artinian Gorenstein rings and tensor decompositions

Abstract

Given a set of distinct points $X=\{P_1, \dots,P_r\} $ in $ \mathbb P^n_{\res}, $ in this paper we characterize being $X$ arithmetically Gorenstein through the ``special" structure of the inverse system of the defining ideal $I(X) \subseteq R= \res[X_0, \dots, X_n].$ We describe the corresponding Zariski locally closed subset of $\mathbb (\mathbb P_{\res}^{n})^r$ parametrized by the coordinates of the points. Several examples are given to show the effectiveness of the results. As a consequence of the main result we characterize the Artinian Gorenstein rings which are the Artinian reduction of Gorenstein points. The problem is of interest in several areas of research, among others the $G$-linkage, the Waring decomposition, the smoothability of Artinian algebras and the identifiability of symmetric tensors.