Isomorphism classes of short Gorenstein local rings via Macaulay's inverse system (Q2841363)

Let \(K\) be an algebraically closed field of characteristic zero. The authors study Artinian Gorenstein local \(K\)-algebras \((A,\mathfrak m)\) such that \(\mathfrak m^4 = 0\) (i.e., such that the socle degree is \(\leq 3\)), and specifically the isomorphism class of such algebras. Their main result is that the classification of such algebras is equivalent to the projective classification of the cubic hypersurfaces in \(\mathbb P^n_K\). The main step is to prove 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). This is false for higher socle degree, and is surprising because it reduces the study of this class of local rings to the graded case. This blend of classical results on cubic hypersurfaces in projective space and the algebraic methods for studying local Gorenstein algebras allows the authors to recover recent results by several authors (see the introduction). A central tool is the use of Macaulay's inverse systems, giving a one-to-one correspondence between Artinian Gorenstein algebras and suitable polynomials. They carefully describe the background for inverse systems in section 2, before proceeding to their main results.