On the Hilbert functions of sets of points in P^1 x P^1 x P^1

TL;DR

This paper investigates how the Hilbert function of a set of points in the product of three projective lines reveals geometric properties, generalizing previous results from two-dimensional cases.

Contribution

It extends the understanding of Hilbert functions from P^1 x P^1 to P^1 x P^1 x P^1, providing methods to extract geometric info from algebraic data.

Findings

Methods to determine geometric properties from Hilbert functions

Generalization of previous results to three-dimensional product spaces

Enhanced understanding of point configurations in P^1 x P^1 x P^1

Abstract

Let H_X be the trigraded Hilbert function of a set X of reduced points in P^1 x P^1 x P^1. We show how to extract some geometric information about X from H_X. This note generalizes a similar result of Giuffrida, Maggioni, and Ragusa about sets of points in P^1 x P^1.