A structure theorem for $2$-stretched Gorenstein algebras

Abstract

In this paper we study the isomorphism classes of local, Artinian, Gorenstein k-algebras A whose maximal ideal M satisfies dim k (M 3 /M 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 k ).

Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

“…The following Theorem 6.14 generalises numerous earlier smoothability results on stretched (by Sally, see [Sal79]), 2-stretched (by Casnati and Notari, see [CN13]) and almost-stretched (by Elias and Valla, see [EV11]) algebras. It is important to understand that, in contrast with the mentioned papers, we avoid a full classification of algebras.…”

“…It is worth mentioning that these results are rather easy consequences of the introduced machinery. In this section we also prove the following general smoothability result (see Thm 6.14), which has no restriction on the length of the algebra and generalises the smoothability results from [Sal79], [CN13] and [EV11].…”

“…The following proposition was proved in [CN13] under the assumption that k is algebraically closed of characteristic zero and in [Jel13, Thm 5.1] under the assumption that k = C. For completeness we include the proof (with no further assumptions on k other than the ones listed at the beginning of this section).…”

On the rationality of Poincaré series of Gorenstein algebras via Macaulay's correspondence

“…Since the Poincaré series of each local Artinian, Gorenstein ring with embedding dimension at most four is rational (see [17], [19], [20], [15]) we also obtain the following corollary. [7], Remark 4.2). Thus the statement follows from Corollary 4.3.…”

“…F = F ≥1 . Moreover, thanks to Lemma 2.2 of [7] we know that, if s ≥ 2 and Ann(F ) ⊆ S[n] 2 + , then we can also assume F 1 = 0, i.e. F = F ≥2 : we will always make such an assumption in what follows.…”