On Hadamard products of linear varieties

TL;DR

This paper explores the Hadamard product of linear varieties, providing a complete description in projective plane and insights into the structure and Hilbert function of such products in three-dimensional space.

Contribution

It offers a comprehensive classification of Hadamard products of linear varieties in 2 and 3, including conditions for specific geometric configurations and Hilbert function calculations.

Findings

In 2, characterized possible outcomes of Hadamard products.

For disjoint collinear point sets, identified conditions for collinearity or grid formation.

In 3, shown that Hadamard product points lie on rulings of a quadric and computed the Hilbert function.

Abstract

In this paper we address the Hadamard product of linear varieties not necessarily in general position. In $\mathbb{P}^2$ we obtain a complete description of the possible outcomes. In particular, in the case of two disjoint finite sets X and X' of collinear points, we get conditions for their hadamard product to be either a collinear finite set of points or a grid of | X|| X'| points. In $\mathbb{P}^3,$ under suitable conditions (which we prove to be generic), we show that the hadamard product of X and X' consists of | X||X'| points on the two different rulings of a non-degenerate quadric and we compute its Hilbert function in the case |X| = |X'|.