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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?

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I have just found out (thanks to Prof. Iarrobino) that there are usually divisorial components (e.g. 15 points on high dimensional affine space) by the Erman, Daniel(1-CA); Velasco, Mauricio(1-CA) A syzygetic approach to the smoothability of zero-dimensional schemes. (English summary) Adv. Math. 224 (2010), no. 3, 1143–1166. Still I would be happy to know the answer for small number of points. –  Wajcha Jul 11 '13 at 19:09
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1 Answer 1

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.

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I am very sorry but there are a few things I do not see: 1) I am interested only in smoothable part, I am rather sure that there are components that consists of nonsmoothable schemes supported in one point. For example here: mathoverflow.net/questions/20288/… - those a priori could have a big dimension? –  Wajcha Jul 11 '13 at 8:06
    
2) There seems to be a high (k-1) codimension component defined inductively: first a divisor, when the support has $k-1$ points, then in these divisor a divisor, when the support has $k-2$ points etc up to a point when we obtain codimension $k-1$ and support in one point. Thus we parameterize schemes that can be decomposed one by one point. But why this should be a big one? Maybe there are greater components? –  Wajcha Jul 11 '13 at 8:12
    
3) Could you plese provide a more detailed reference to Gusein-Zade. (Also I am interested more in big $n$ and $k$ and on lower bounds on codimension, so I would be very happy if this is not a divisor) Thank you! –  Wajcha Jul 11 '13 at 8:16
    
4) Indeed, for n=2, the codimension is $k-1$. –  Wajcha Jul 11 '13 at 8:27
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