Hilbert functions of Gorenstein algebras associated to a pencil of forms

Let R be a polynomial ring in r variables over an infinite field K, and denote by D a corresponding dual ring, upon which R acts as differential operators. We study type two graded level Artinian algebras A = R/I , having socle degree j. For each such algebra A, we consider the family of Artinian Gorenstein [AG] quotients of A having the same socle degree. By Macaulay duality, A corresponds to a unique 2-dimensional vector space WA of forms in Dj , and each such AG quotient of A corresponds to a form in WA up to non-zero multiple. For WA = 〈F,G〉, each such AG quotient Aλ corresponds to an element of the pencil of forms (one dimensional subspaces) of WA: given Fλ = F+λG,λ ∈ K∪∞ we have Aλ = R/Ann(Fλ). Our main result is a lower bound for the Hilbert function H(Aλgen) of the generic Gorenstein quotient, in terms of H(A), and the pair HF = H(R/Ann F ) and HG = H(R/Ann G). This result restricts the possible sequences H that may occur as the Hilbert function H(A) for a type two level algebra A.