On the number of generators of ideals defining Gorenstein Artin algebras with Hilbert function $(1,n+1,1+(\genfrac{}{}{0ex}{}{n+1}{2}),...,(\genfrac{}{}{0ex}{}{n+1}{2})+1,n+1,1)$

Abstract

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 $R{I}_2$ 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.

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References in corpus (1)

• A characterization of Gorenstein Hilbert functions in codimension four with small initial degree