Gluing and Hilbert functions of monomial curves

TL;DR

This paper introduces a method using gluing semigroups to construct numerous 1-dimensional local rings with non-decreasing Hilbert functions, including Gorenstein rings, providing evidence for Rossi's conjecture.

Contribution

It presents an effective technique to generate large families of 1-dimensional Gorenstein local rings associated with monomial curves, expanding understanding of Hilbert functions in this context.

Findings

Constructed infinitely many families of local rings with non-decreasing Hilbert functions.

Provided examples of Gorenstein local rings supporting Rossi's conjecture.

Showed that associated graded rings need not be Cohen-Macaulay.

Abstract

In this article, by using the technique of gluing semigroups, we give infinitely many families of 1-dimensional local rings with non-decreasing Hilbert functions. More significantly, these are local rings whose associated graded rings are not necessarily Cohen-Macaulay. In this sense, we give an effective technique to construct large families of 1-dimensional Gorenstein local rings associated to monomial curves, which support Rossi's conjecture saying that every Gorenstein local ring has non-decreasing Hilbert function.