On the Hilbert function of Artinian local complete intersections of codimension three

TL;DR

This paper characterizes the Hilbert functions of quadratic Artinian complete intersections of codimension three, establishing criteria for their admissibility and providing constructive methods and structural insights.

Contribution

It offers a complete characterization of Hilbert functions for quadratic Artinian complete intersections of codimension three and links their symmetry properties to the Hilbert function.

Findings

Hilbert functions are characterized by admissibility criteria.

A constructive method for realizing given Hilbert functions as complete intersections.

Symmetric decomposition of the ideal is determined by the Hilbert function.

Abstract

In singularity theory or algebraic geometry, it is natural to investigate possible Hilbert functions for special algebras $A$ such as local complete intersections or more generally Gorenstein algebras. The sequences that occur as {the} Hilbert functions of standard graded complete intersections are well understood classically thanks to Macaulay and Stanley. Very little is known in the local case except in codimension two. In this paper we characterise the Hilbert functions of quadratic Artinian complete intersections of codimension three. Interestingly we prove that a Hilbert function is admissible for such a Gorenstein ring if and only if is admissible for such a complete intersection. We provide an effective construction of a local complete intersection for a given Hilbert function. We prove that the symmetric decomposition of such a complete intersection ideal is determined by its Hilbert function.