Abstract
In this paper we study the 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 J. Elias and M.E. Rossi. Then we use such results in order to complete the description of the singular locus of the Gorenstein locus of $\Hilb_{11}(\p n)$
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