On the number of generators of ideals defining Gorenstein Artin algebras with Hilbert function 1,n+1,1+n+12,…,n+12+1,n+1,1

Let R = k [ w , x 1 , … , x n ] / I be a graded Gorenstein Artin algebra. I = ann F for some F in the divided power algebra k [ W , X 1 , … , X n ] . Suppose that R I 2 is a height one ideal generated by n quadrics so that I 2 ⊂ ( w ) after a possible change of variables. Let J = I ∩ k [ x 1 , … , x n ] . Then μ ( I ) ≤ μ ( J ) + n + 1 and I is said to be μ -generic if μ ( I ) = μ ( J ) + n + 1 . In this article we prove necessary conditions, in terms of F , for an ideal to be μ -generic. With some extra assumptions on the exponents of terms of F , we obtain a characterization for height four ideals I to be μ -generic.