A structure theorem for \(2\)-stretched Gorenstein algebras (Q311780)

Let \(K\) be an algebraically closed field of characteristic zero. The study of Artinian \(k\)-algebras is a classical topic in commutative algebra. It is well known that each Artinian \(k\)-algebra is a direct sum of local ones; hence, one can restrict attention to the local case.NEWLINENEWLINEIn the paper under review, the authors study isomorphism classes of local, Artinian, Gorenstein \(k\)-algebras \(A\) whose maximal ideal \(\mathfrak{m}\) satisfies \(\dim_k(\mathfrak{m}^3/\mathfrak{m}^4) = 1\). A recent result by \textit{J. Elias} and \textit{M. E. Rossi} [Trans. Am. Math. Soc. 364, No. 9, 4589--4604 (2012; Zbl 1281.13015)] says that an Artinian Gorenstein local ring \(A\) with Hilbert function \((1,n,n,1)\) is isomorphic to its own associated graded ring (with respect to the maximal ideal). In this paper the authors extend this result to Artinian, Gorenstein algebras \(A\) with Hilbert function \((1, n, m, 1, \cdots)\).NEWLINENEWLINEA central tool in this paper is Macaulay's inverse system, which establishes a one-to-one correspondence between Artinian Gorenstein algebras and suitable polynomials.NEWLINENEWLINEThe authors use such results in order to complete the description of the singular locus of the Gorenstein locus of \(\mathrm{Hilb}_{11}(\mathbb{P}^n_k)\).