Quadratically presented Gorenstein ideals

TL;DR

This paper provides explicit, computationally friendly formulas for the presentation matrices of certain Gorenstein ideals in three variables, leveraging Macaulay inverse systems and applying to ideals with the weak Lefschetz property.

Contribution

It introduces new formulas for the presentation matrices of quadratically presented Gorenstein ideals, especially those satisfying the weak Lefschetz property, using Macaulay inverse systems.

Findings

Presented explicit formulas involving matrix multiplication for Gorenstein ideals' presentation matrices.

Derived the presentation matrix for a specific class of ideals J_1 in characteristic zero.

Improved formulas for linearly presented Gorenstein ideals' presentation matrices.

Abstract

Let $J$ be a quadratically presented grade three Gorenstein ideal in the standard graded polynomial ring $R= k[x,y,z]$, where $k$ is a field. Assume that $R/J$ satisfies the weak Lefschetz property. We give the presentation matrix for $J$ in terms of the coefficients of a Macaulay inverse system for $J$. (This presentation matrix is an alternating matrix and $J$ is generated by the maximal order Pfaffians of the presentation matrix.) Our formulas are computer friendly; they involve only matrix multiplication; they do not involve multilinear algebra or complicated summations. As an application, we give the presentation matrix for $J_1=(x^{n+1},y^{n+1},z^{n+1}):(x+y+z)^{n+1}$, when $n$ is even and the characteristic of $k$ is zero. Generators for $J_1$ had been identified previously; but the presentation matrix for $J_1$ had not previously been known. The first step in our proof is to give improved formulas for the presentation matrix of a linearly presented grade three Gorenstein ideal $I$ in terms of the coefficients of the Macaulay inverse system for $I$.