Hilbert-Samuel sequences of homogeneous finite type

TL;DR

This paper classifies all Hilbert-Samuel sequences of homogeneous finite type for graded local algebras generated by two degree-one elements over an algebraically closed field of characteristic zero.

Contribution

It provides a complete list of Hilbert-Samuel sequences of homogeneous finite type for a specific class of graded local algebras.

Findings

List of all such sequences for algebras generated by two degree-one elements.

Characterization of sequences with finite isomorphism classes.

Clarification of the structure of these algebras.

Abstract

This paper deals with the problem of the classification of the local graded Artinian quotients $\mathbb{K}[x,y]/I$ where $\mathbb{K}$ is an algebraically closed field of characteristic $0$. They have a natural invariant called Hilbert-Samuel sequence. We say that a Hilbert-Samuel sequence is of homogeneous finite type, if it is the Hilbert-Samuel sequence of a finite number of isomorphism classes of graded local algebras. We give the list of all the Hilbert-Samuel sequences of homogeneous finite type in the case of algebras generated by $2$ elements of degree $1$.