Separators of Arithmetically Cohen-Macaulay fat points in P^1 x P^1

TL;DR

This paper provides a method to compute the degree of separators for fat points in P^1 x P^1 that are arithmetically Cohen-Macaulay, extending previous results for reduced points.

Contribution

It introduces a formula to determine separator degrees of fat points in ACM sets in P^1 x P^1 using only numerical data, generalizing prior work on reduced points.

Findings

Derived a formula for separator degrees of fat points in ACM sets

Extended known results from reduced points to fat points

Provided a computational method based on numerical descriptions

Abstract

Let Z be a set of fat points in P^1 x P^1 that is also arithmetically Cohen-Macaulay (ACM). We describe how to compute the degree of a separator of a fat point of multiplicity m for each point in the support of Z using only a numerical description of Z. Our formula extends the case of reduced points which was previously known.