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

Over an infinite field $K$ with $\mathrm{c}\mathrm{h}\mathrm{a}\mathrm{r}(K)\mathrm{\ne }2,3$, 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 ${\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}}_{16}^7$ has at least $5$ irreducible components. As a byproduct of our study about ${\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}}_{16}^7$ we also find a new elementary component in ${\mathrm{H}\mathrm{i}\mathrm{l}\mathrm{b}}_{15}^7$. 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 $(1,5,5,1)$ 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.