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 \(R={\mathbbm{k}}\,[x_{1},\ldots ,x_{n}]\) over an infinite field \({\mathbbm{k}}\), assuming that I is generated in a fixed degree d. For such an ideal I, there is a simple formula relating this degree, the minimal number of generators of I, and the degree of the entries of the associated skew-symmetric matrix. 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 satisfying them. We conjecture that, for arbitrary \(n\ge 2\), an ideal \(I\subset {\mathbbm{k}}[x_1,\ldots ,x_n]\) 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 \(r= {{n+1}\atopwithdelims ()2}-1\). 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 \(r={{n+1}\atopwithdelims ()2}-1\), which settles the ‘if’ assertion of the conjecture for \(n\le 5\). We also elaborate around one of the questions of Fröberg–Lundqvist. In a different direction, we show 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. We give a solution in some special cases.