Limits of graded Gorenstein algebras of Hilbert function \((1,3^k,1)\)

Let , the polynomial ring over a field \(\textsf {k}\) $\mathsf{k}$ . Several of the authors previously classified nets of ternary conics and their specializations over an algebraically closed field, Abdallah et al. (Eur J Math 9(2), Art. No. 22, 2023). We here show that when \(\textsf {k}\) $\mathsf{k}$ is algebraically closed, and considering the Hilbert function sequence , \(k\geqslant 2\) $k⩾2$ (i.e. \(T=(1, 3,3,\ldots , 3,1)\) $T=(1,3,3,\dots ,3,1)$ where k is the multiplicity of 3), then the family \(G_T\) $G_T$ parametrizing graded Artinian algebra quotients \(A=R/I\) $A=R/I$ of R having Hilbert function T is irreducible, and \(G_T\) $G_T$ is the closure of the family of Artinian Gorenstein algebras of Hilbert function T. We then classify up to isomorphism the elements of these families and of \(G_T\) $G_T$ . Finally, we give examples of codimension 3 Gorenstein sequences, such as (1, 3, 5, 3, 1), for which \(G_T\) $G_T$ has several irreducible components, one being the Zariski closure of .