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