Equigenerated Gorenstein ideals of codimension three

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 $n\ge 2$, an ideal $I\subset \kk[x_1,\ldots,x_n]$ generated by a general set of $r\ge n+2$ forms of degree $d\ge 2$ is Gorenstein if and only if $d=2$ and $r=\left(\genfrac{}{}{0ex}{}{n+1}{2}\right)-1$. We prove the `only if' implication of this conjecture when $n=3$. For arbitrary $n\ge 2$, we prove that if $d=2$ and $r\ge (n+2)(n+1)/6$ then the ideal is Gorenstein if and only if $r=\left(\genfrac{}{}{0ex}{}{n+1}{2}\right)-1$, which settles the `if' assertion of the conjecture for $n\le 5$. Finally, we elaborate around one of the questions of Fröberg--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 $({\ell }_1^m,\dots ,{\ell }_n^m):\mathfrak{f}$ is equigenerated, where ${\ell }_1,\dots ,{\ell }_n$ are independent linear forms and $\mathfrak{f}$ is a form, is given a solution in some important cases.