Isomorphism classes of short Gorenstein local rings via Macaulay’s inverse system

Abstract

Abstract. Let K be an algebraically closed field of characteristc zero. In this paper we study the isomorphism classes of Artinian Gorenstein local Kalgebras with socle degree three by means of Macaulay's inverse system. We prove that their classification is equivalent to the projective classification of cubic hypersurfaces in P n K . This is an unexpected result because it reduces the study of this class of local rings to the graded case. The result has applications in problems concerning the punctual Hilbert scheme Hilb d (P n K ) and in relation to the problem of the rationality of the Poincaré series of local rings.

On the irreducibility and the singularities of the Gorenstein locus of the punctual Hilbert scheme of degree 10

“…In this case we will not exploit any explicit classification up to isomorphisms for such algebras as we did in the case dim k (M 2 /M 3 ) ≤ 3, but we will make use of some classical results about Artinian Gorenstein k-algebras combined with a recent structure theorem for such algebras A with h A = (1, N, N, 1) (see [8]). …”

“…Indeed on the one hand Theorem 3.3 of [8] states that each Artinian, Gorenstein k-algebras with h A = (1, N, N, 1) is canonically graded, i.e. there exists a homogeneous ideal…”

“…In order to prove the above theorem we will also make use of the classification results proved in [6,4] and the structure theorem proved in [8]. The proof of the Main Theorem is given in Section 3.…”

“…Indeed, in our proofs we make wide use of some results stated only in the 0-characteristic case (see, e.g. [4,[7][8][9]). …”

Classifying local Artinian Gorenstein algebras

“…The isomorphism Apolar( f ) Apolar( f 3 ) was first proved by Elias and Rossi, see [16,Thm 3.3]. See Example 2.22 for a more conceptual proof for k = C and Sect.…”