On the Hilbert function of general fat points in $\mathbb{P}^1 \times \mathbb{P}^1$

TL;DR

This paper investigates the bi-graded Hilbert function of general fat points in ^1 ^1, introducing new methods to compute the function for specific cases of multiplicity and bi-degree.

Contribution

It applies multiprojective-affine-projective and differential Horace methods to compute the Hilbert function for fat points with multiplicities one, two, and three.

Findings

Computed Hilbert function for double points using multiprojective-affine-projective method.

Determined the entire bi-graded Hilbert function for triple points.

Extended understanding of fat points in bi-projective spaces.

Abstract

We study the bi-graded Hilbert function of ideals of general fat points with same multiplicity in $\mathbb{P}^1\times\mathbb{P}^1$. Our first tool is the multiprojective-affine-projective method introduced by the second author in previous works with A.V. Geramita and A. Gimigliano where they solved the case of double points. In this way, we compute the Hilbert function when the smallest entry of the bi-degree is at most the multiplicity of the points. Our second tool is the differential Horace method introduced by J. Alexander and A. Hirschowitz to study the Hilbert function of sets of fat points in standard projective spaces. In this way, we compute the entire bi-graded Hilbert function in the case of triple points.