Reducibility of a family of local Artinian Gorenstein algebras

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 (non-graded) 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 three 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. We also compare the Jordan type of the AG algebra A and of its associated graded algebra A * , showing that they can be different. We explore a problem of realizability of partial symmetric decompositions.