A structure theorem for $2$-stretched Gorenstein algebras

Abstract

In this paper, we study isomorphism classes of local, Artinian, Gorenstein $k$-algebras $A$ whose maximal ideal $\frak M$ satisfies $\dim _k(\fM ^3/\fM ^4)=1$ by means of Macaulay's inverse system generalizing a recent result by Elias and Rossi. Then we use such results in order to complete the description of the singular locus of the Gorenstein locus of $\Hilb _{11}(\p n)$.