Consider the component R of the Hilbert scheme of $k$ points in $P^n$, that contains smoothable schemes. By the punctual Hilbert scheme I mean a subscheme Z of the Hilbert scheme that consits of schemes supported at one point. Let $X$ be the intersection of $Z$ and $R$ (that is smoothable schemes supported at one point).
What is the dimension of $X$? Is it usually codimension one in R? (notice that $X$ may be reducible, so I am not interested in components of small dimension)
I am also interested in the same question, when I consider only closures of Gorenstein schemes. Can it desrease the dimension of $X$ or is there a maximal dimensional component with a generic scheme being Gorenstein?