On the number of generators of ideals defining Gorenstein Artin algebras with Hilbert function $$ \left( 1,n+1 , 1+{n+1\atopwithdelims ()2}, \ldots ,{n+1\atopwithdelims ()2}+1, n+1 ,1\right) $$ 1 , n + 1 , 1 + n + 1 2 , … , n + 1 2 + 1 , n + 1 , 1

Abstract

Let \(R = k[w, x_1, \ldots , x_n]/I\) be a graded Gorenstein Artin algebra. \(I = \mathrm{ann }F\) for some \(F\) in the divided power algebra \(k[W, X_1,\ldots , X_n]\). Suppose that \(RI_2\) is a height one ideal generated by \(n\) quadrics so that \(I_2 \subset (w)\) after a possible change of variables. Let \(J = I \cap k[x_1, \ldots , x_n]\). Then \(\mu (I) \le \mu (J)+n+1\) and \(I\) is said to be \(\mu \)-generic if \(\mu (I) = \mu (J) + n+1\). In this article we prove necessary conditions, in terms of \(F\), for an ideal to be \(\mu \)-generic. With some extra assumptions on the exponents of terms of \(F\), we obtain a characterization for height four ideals \(I\) to be \(\mu \)-generic.