Smoothable Gorenstein Points Via Marked Schemes and Double-generic Initial Ideals

Abstract

Over an infinite field K with char(K) = 2, 3, we investigate smoothable Gorenstein Kpoints in a punctual Hilbert scheme from a new point of view, which is based on properties of double-generic initial ideals and of marked schemes. We obtain the following results: (i) points defined by graded Gorenstein K-algebras with Hilbert function (1, 7, 7, 1) are smoothable, in the further hypothesis that K is algebraically closed; (ii) the Hilbert scheme Hilb 7 16 has at least three irreducible components. The properties of marked schemes give us a simple method to compute the Zariski tangent space to a Hilbert scheme at a given K-point, which is very useful in this context. Over an algebraically closed field of characteristic 0, we also test our tools to find the already known result that points defined by graded Gorenstein K-algebras with Hilbert function (1, 5, 5, 1) are smoothable. In characteristic zero, all the results about smoothable points also hold for local Artin Gorenstein K-algebras.

On almost revlex ideals with Hilbert function of complete intersections

“…We can then identify J with the point Proj(S/(JS)) of the Hilbert scheme Hilb n D , which parameterizes flat families of closed subschemes in P n K with Hilbert polynomial D. Hence, we will say that J is (or corresponds to) a point of Hilb n D . Our aim is to give a lower bound for the dimension of the Zariski tangent space T J to Hilb n D at the point J, using techniques and results that have been developed in [5,6]. A similar investigation has been given in [10, Lemma 6.1 and Theorem 6.2] under the more restrictive hypotheses that JS is a hilb-segment ideal with respect to a suitable term order and the field K has characteristic zero.…”

“…Referring to [5,6,22], first we briefly recall how one can obtain a set of equations defining the Zariski tangent space to Hilb n D at J, but more generally at any point of a suitable open subset of Hilb n D . Also recall that J is an Artinian quasi-stable ideal and P(J) denotes its Pommaret basis.…”

“…For every x γ ∈ P(J) and x j ≻ min(x γ ), we consider the polynomial x j f γ and, by a suitable Noetherian confluent reduction process that is based on property (⋆) (see [5,Definition 4.2]), compute the polynomials that are involved in the following equality…”

“…Using this formula, and a lower bound on the dimension of the Zariski tangent space to a Hilbert scheme at a point corresponding to a stable ideal (Corollary 5.4), we exhibit several cases in which the point corresponding to an Artinian almost reverse lexicographic ideal with the Hilbert function of a complete intersection is singular in the Hilbert scheme (see Section 6). The main tools for this result are taken from the more general study of marked schemes over quasi-stable ideals [5], similarly to [10, Section 6] for reverse lexicographic ideals with the Hilbert function of general points.…”

Smoothable zero dimensional schemes and special projections of algebraic varieties

“…Then is smoothable if and only if the corresponding point lies in . Whether a given R is smoothable is a difficult question, see . It is connected with the search for equations of secant varieties .…”