Hilbert depth of graded modules over polynomial rings in two variables

TL;DR

This paper provides an arithmetic criterion for determining when a Hilbert series corresponds to a finitely generated graded module of positive depth over a two-variable polynomial ring, especially in the generic coprime degree case.

Contribution

It introduces a new arithmetic criterion based on numerical semigroups for Hilbert series of graded modules over polynomial rings in two variables.

Findings

Criterion characterizes Hilbert series of modules with positive depth

In the generic case, the criterion involves numerical semigroups

Applicable to polynomial rings over arbitrary fields

Abstract

In this article we mainly consider the positively Z-graded polynomial ring R=F[X,Y] over an arbitrary field F and Hilbert series of finitely generated graded R-modules. The central result is an arithmetic criterion for such a series to be the Hilbert series of some R-module of positive depth. In the generic case, that is, the degrees of X and Y being coprime, this criterion can be formulated in terms of the numerical semigroup generated by those degrees.