Multigraded regularity: syzygies and fat points

TL;DR

This paper explores multigraded regularity in algebraic geometry, focusing on its relation to fat point schemes in product projective spaces, aiming to enhance understanding of their Hilbert functions.

Contribution

It investigates connections between recent multigraded regularity definitions and their implications for fat point schemes in product projective spaces.

Findings

Establishes links between different multigraded regularity notions.

Provides insights into the multigraded Hilbert function of fat points.

Enhances understanding of syzygies in multigraded contexts.

Abstract

The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understanding of the multigraded Hilbert function of fat point schemes in P^{n_1} x ... x P^{n_k}.