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

Abstract

Over an infinite field K with we investigate smoothable Gorenstein K-points in a punctual Hilbert scheme and obtain the following results: (i) every K-point defined by local Gorenstein K-algebras with Hilbert function is smoothable (this is the only case non treated in the range considered by Iarrobino and Kanev in 1999; (ii) the Hilbert scheme has at least five irreducible components. As a byproduct of our study about we also find a new elementary component in We face the problem from a new point of view, that is based on properties of double-generic initial ideals and of marked schemes. 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. We also test our tools to find the already known result that K-points defined by local Gorenstein K-algebras with Hilbert function are smoothable. The problem that we consider is strictly related to the study of the irreducibility of the Gorenstein locus in a Hilbert scheme and, more generally, of the irreducibility of a Hilbert scheme, which is a very open question.