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=[x1,…,xn] over a field , 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≥ 2, an ideal I⊂ [x1,…,xn] generated by a general set of r≥ n+2 forms of degree d≥ 2 is Gorenstein if and only if d=2 and r= n+1 2-1. We prove the `only if' implication of this conjecture when n=3. For arbitrary n≥ 2, we prove that if d=2 and r≥ (n+2)(n+1)/6 then the ideal is Gorenstein if and only if r=n+1 2-1, which settles the `if' assertion of the conjecture for n≤ 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 (1m,…,nm):f is equigenerated, where 1,…,n are independent linear forms and f is a form, is given a solution in some important cases.