Hilbert functions of Gorenstein algebras associated to a pencil of forms
  Let R be a polynomial ring in r variables and D a dual ring upon which R acts as partial differential operators (classical apolarity). For a type two graded level Artinian algebras A=R/I, of socle degree j 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 W(A)=<F,G> of degree-j homogeneous elements (forms) in D, and each such AG quotient of A corresponds to an element $F_\lambda=F+\lambda G$ in W(A), up to non-zero constant multiple. Our main result is a lower bound for the Hilbert function of the generic AG quotient of A in terms of the Hilbert functions of A and the AG quotients of R determined by F and by G. This result restricts the possible sequences H that may occur as the Hilbert function for a type two level algebra A.   This result may also be viewed as obtaining an optimal lower bound on the dimensions of the spaces of degree-u partial derivatives of $F+\lambda G$ for a generic lambda in terms of the corresponding dimensions for the forms F and G. 
Anthony Iarrobino