Hadamard products of symbolic powers and Hadamard fat grids

TL;DR

This paper investigates the algebraic properties of Hadamard products of symbolic powers of ideals of points in projective space, introduces Hadamard fat grids, and computes key invariants for these configurations.

Contribution

It establishes conditions for Hadamard products of symbolic powers of point ideals and introduces Hadamard fat grids, analyzing their algebraic invariants.

Findings

Derived conditions for Hadamard products of symbolic powers of point ideals.

Defined and studied Hadamard fat grids and their algebraic invariants.

Computed minimal resolution, Waldschmidt constant, and resurgence for Hadamard fat grids.

Abstract

In this paper we address the question if, for points $P, Q \in \mathbb{P}^{2}$, $I(P)^{m} \star I(Q)^{n}=I(P \star Q)^{m+n-1}$ and we obtain different results according to the number of zero coordinates in $P$ and $Q$. Successively, we use our results to define the so called Hadamard fat grids, which are the result of the Hadamard product of two sets of collinear points with given multiplicities. The most important invariants of Hadamard fat grids, as minimal resolution, Waldschmidt constant and resurgence, are then computed.