Smoothable Gorenstein points via marked schemes and double-generic initial ideals

Over an infinite field K with char(K)<> 2, 3, 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 (1; 7; 7; 1) is smoothable (this is the only case non treated in the range considered by Iarrobino and Kanev in 1999); (ii) the Hilbert scheme Hilb^7_{16} has at least 5 irreducible components. As a byproduct of our study about Hilb^7_{16} we also nd a new elementary component in Hilb^7_{15}. 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 dened by local Gorenstein K-algebras with Hilbert function (1; 5; 5; 1) 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.