Equigenerated Gorenstein ideals of codimension three

TL;DR

This paper investigates the structure and classification of Gorenstein ideals of codimension three in polynomial rings, providing formulas, conjectures, and proofs about their generation and properties, and exploring related algebraic dualities and linkage questions.

Contribution

It introduces a simple formula linking degree, number of generators, and matrix entries for Gorenstein ideals, and proves parts of a conjecture characterizing such ideals generated by general forms.

Findings

Existence of Gorenstein ideals matching given data via a characteristic-free argument.

Proof of the 'only if' part of the conjecture for n=3.

Validation of the 'if' part of the conjecture for n≤5 and degree d=2.

Abstract

We focus on the structure of a homogeneous Gorenstein ideal $I$ of codimension three in a standard polynomial ring $R=\kk[x_1,\ldots,x_n]$ over a field $\kk$, assuming that $I$ is generated in a fixed degree $d$. For such an ideal $I$ this degree comes along with the minimal number of generators of $I$ and the degree of the entries of the associated skew-symmetric matrix in a simple formula. We give an elementary characteristic-free argument to the effect that, for any such data linked by this formula, there exists a Gorenstein ideal $I$ of codimension three filling them. We conjecture that, for arbitrary $n\geq 2$, an ideal $I\subset \kk[x_1,\ldots,x_n]$ generated by a general set of $r\geq n+2$ forms of degree $d\geq 2$ is Gorenstein if and only if $d=2$ and $r= {{n+1}\choose 2}-1$. We prove the `only if' implication of this conjecture when $n=3$. For arbitrary $n\geq 2$, we prove that if $d=2$ and $r\geq (n+2)(n+1)/6$ then the ideal is Gorenstein if and only if $r={{n+1}\choose 2}-1$, which settles the `if' assertion of the conjecture for $n\leq 5$. Finally, we elaborate around one of the questions of Fr\"oberg--Lundqvist. In a different direction, we reveal a connection between the Macaulay inverse and the so-called Newton dual, a matter so far not brought out to our knowledge. Finally, we consider the question as to when the link $(\ell_1^m,\ldots,\ell_n^m):\mathfrak{f}$ is equigenerated, where $\ell_1,\ldots,\ell_n$ are independent linear forms and $\mathfrak{f}$ is a form, is given a solution in some important cases.