On the Gorenstein locus of the punctual Hilbert scheme of degree 11

Abstract

Let k be an algebraically closed field of characteristic 0 and let Hilb_d^G(P_k^N) be the open locus of the Hilbert scheme Hilb_d(P_k^N) corresponding to Gorenstein subschemes. We proved in several previous papers that Hilb_d^G(P_k^N) is irreducible for d⩽10 and N⩾1, characterizing its singular locus. In the present paper we prove that also Hilb_{11}^G(P_k^N) is irreducible for each N⩾1. We also give some results about its singular locus

On the cactus rank of cubic forms

“…This work is close in line to [Iarrobino 1994], [Iarrobino, Kanev 1999] and [Elias, Rossi 2011], in their study of apolarity and the local Gorenstein algebra associated to a polynomial. Applications to higher secant varieties can be found in [Chiantini, Ciliberto, 2002], [Buczynska, Buczynski 2010] and [Landsberg, Ottaviani 2011], while the papers [Landsberg, Teitler 2010], [Brachat et al 2010], [Bernardi et al 2011] and [Carlini et al 2011] concentrate on effective methods to compute the rank and to compute an explicit decomposition of a form. In a different direction, the rank of cubic forms associated to canonical curves has been computed in [De Poi, Zucconi 2011a] and [De Poi, Zucconi 2011b].…”

Apolarity and direct sum decomposability of polynomials

“…Note however that for limits of direct sums of type (1, n − 1) one cannot expect a similar result to Theorem 5.16. This is because for n = 14, the polynomial F presented in Example 5.15 is a limit of direct sums of type (1,13) and it has been proved it is not an apolar limit. 6.…”

“…Finally, we remark that for general G ∈ S 3 C 6 , the scheme R is the shortest nonsmoothable Gorenstein scheme. See [27,Lemma 6.21], where it is shown that R is non-smoothable, and [13], where it is shown that all shorter Gorenstein schemes are smoothable.…”

A structure theorem for $2$-stretched Gorenstein algebras

“…an embedding X ⊆ A 4 k ⊆ P 4 k . In [5] we proved the irreducibility of Hi lb G 11 (P n k ) studying the locus of singular X such that X ∼ = spec(A), where A is not local with H A = (1, 4, 4, 1, 1). Thus a point X ∈ Hi lb 11 (P n k ) is singular (i.e.…”

“…In [5] a similar analysis has been carried out in the case d = 11. In that paper we were also able to deal with the singular nature of all X ∈ Hi lb G 11 (P n k ), but those ones isomorphic to spec(A) where A is a local, Artinian, Gorenstein algebra with H A = (1, 4, 4, 1, 1).…”