Gorenstein points in $\mathbb{P}^3$ via Hadamard product of projective varieties

TL;DR

This paper introduces a method to construct Gorenstein sets of points in projective 3-space using Hadamard products, enabling explicit coordinate determination and advancing algebraic geometry techniques.

Contribution

It presents a novel construction of Gorenstein points via Hadamard products and applies existing results to explicitly build and analyze these configurations.

Findings

Constructed Gorenstein point sets with specified h-vectors.

Provided explicit coordinate formulas for points in the Gorenstein sets.

Linked Hadamard products with algebraic geometry constructions.

Abstract

We show how to construct a stick figure of lines in $\mathbb{P}^3$ using the Hadamard product of projective varieties. Then, applying the results of Migliore and Nagel, we use such stick figure to build a Gorenstein set of points with given $h-$vector ${\mathbf h}$. Since the Hadamard product is a coordinate-wise product, we show, at the end, how the coordinates of the points, in the Gorenstein set, can be directly determined.