Reducibility of a family of local Artinian Gorenstein algebras

TL;DR

This paper investigates the structure and reducibility of families of Artinian Gorenstein algebras, focusing on Jordan types and symmetric decompositions of their Hilbert functions, revealing multiple irreducible components in certain cases.

Contribution

It provides the first examples of Gorenstein sequences with families of algebras having multiple irreducible components distinguished by symmetric decompositions.

Findings

Families of AG algebras can have multiple irreducible components.

Jordan type and symmetric decomposition filtrations interact to determine component structure.

Examples are constructed in codimension three, the lowest with such complexity.

Abstract

The Jordan type of an Artinian algebra is the Jordan block partition associated to multiplication by a generic element of the maximal ideal. We study the Jordan type for Artinian Gorenstein (AG) local algebras A, and the interaction of Jordan type with the symmetric decomposition of the Hilbert function H(A). We give examples of Gorenstein sequences H for which the family Gor(H) of AG algebras having Hilbert function H has several irreducible components, each corresponding to a symmetric decomposition of H. The component structure results from the intersection of two opposing filtrations of the family Gor(H) of AG algebras: that by Jordan type satisfies the usual dominance property; the second filtration, by symmetric decomposition, satisfies a known semicontinuity property. Our examples are in codimension three -- the lowest codimension of such an example, as Gor(H) is irreducible in codimension two.