Dimension of smoothable punctual Hilbert scheme 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?
 A: The locus $X$ is not typically a divisor in $R$.  You are asking about the dimensions of the fibers of the Hilbert-Chow morphism, $$\text{FC}:\text{Hilb}^k(\mathbb{P}^n)\to \text{Chow}_k(\mathbb{P}^n),$$ when restricted to the "good" component $R$.  This has been studied quite a bit.  For instance, when $n=2$, the good component is all of $\text{Hilb}^k(\mathbb{P}^2)$, and the Hilbert-Chow morphism is "semismall" -- I learned of this from my colleague Mark de Cataldo.  In particular, since the small diagonal in $\text{Chow}_k(\mathbb{P}^2)$ has codimension $2(k-1)$, the inverse image $X$ of the small diagonal in $\text{Hilb}^k(\mathbb{P}^2)$ has codimension at least $(k-1)$.
In fact, for $n$ general, the "curvilinear" locus inside $X$ (which is contained in the Gorenstein locus, and even in the LCI locus) has codimension $(k-1)$ inside all of $R$.  My guess is, if you check the literature, then this is the biggest component of $X$.  One place to start is the articles of Gusein-Zade and his coauthors: they often write down the classes of the fibers of the Hilbert-Chow morphism in the Grothendieck group of varieties.  That class certainly has the dimension encoded in it.
