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

Over an infinite field <i>K</i> with char(K)≠2,3, we investigate smoothable Gorenstein <i>K</i>-points in a punctual Hilbert scheme and obtain the following results: (i) every <i>K</i>-point defined by local Gorenstein <i>K</i>-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 Hilb167 has at least five irreducible components. As a byproduct of our study about Hilb167, we also find a new elementary component in Hilb157. 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 <i>K</i>-point, which is very useful in this context. We also test our tools to find the already known result that <i>K</i>-points defined by local Gorenstein <i>K</i>-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.