Artinian level algebras of low socle degree

TL;DR

This paper investigates the Hilbert functions and classification of Artinian level local algebras with low socle degree, establishing existence results and structural properties for socle degree up to three.

Contribution

It provides explicit constructions for all admissible Hilbert functions in socle degree at most three and proves that certain maximal cases are graded.

Findings

Existence of level local algebras for all admissible Hilbert functions with socle degree ≤ 3.

Maximal Hilbert function level algebras of socle degree three are graded.

Parametrization of extremal strata in the graded case by Cho and Iarrobino.

Abstract

In this paper we study Hilbert functions and isomorphism classes of Artinian level local algebras via Macaulay's inverse system. Upper and lower bounds concerning numerical functions admissible for level algebras of fixed type and socle degree are known. For each value in this range we exhibit a level local algebra with that Hilbert function, provided that the socle degree is at most three. Furthermore we prove that level local algebras of socle degree three and maximal Hilbert function are graded. In the graded case the extremal strata have been parametrized by Cho and Iarrobino.