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

TL;DR

This paper investigates the minimal number of generators of ideals defining Gorenstein Artin algebras with specific Hilbert functions, providing necessary conditions and characterizations for when such ideals are considered generic.

Contribution

It introduces necessary conditions for ideals to be generic in the context of Gorenstein Artin algebras and characterizes generic ideals in codimension four under certain assumptions.

Findings

Necessary conditions for genericity in terms of the defining polynomial F.

A characterization of generic ideals in codimension four with additional assumptions.

Bounds on the minimal number of generators for these ideals.

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 $RI_2$ 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.