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It is well known that if $X$ is a smooth surface, then the Hilbert scheme of points $X^{[n]}$ is also smooth. What about the subscheme $S_p$ of $X^{[n]}$ consisting of all schemes of finite length $Z$ such that $Supp \ Z=\{p\}$? (Here $p\in X$ is a fixed point.) Is this scheme still smooth?

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    $\begingroup$ That too is smooth. Fine analysis fo these are done in particular by Briancon and Granger. $\endgroup$
    – Mohan
    May 2, 2020 at 17:47
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    $\begingroup$ Hmmm... I guess you have to define what you mean. If you want $S_p$ to represent some functor then my impression is that this is not very well behaved, e.g. probably not a variety. I think the more natural thing is to take $S_p$ to be the fiber of the Hilbert-Chow morphism $X^{[n]}\rightarrow X^{(n)}$ over the point $[np]$. This is known to be reduced irreducible and Cohen Macaulay, but unfortunately (or not depending on what you like) is singular once $n>2$. The first singular example is $S_p\subset X^{[3]}$ which is isomorphic to the cone over the twisted cubic (singular at the cone point!). $\endgroup$ May 2, 2020 at 21:44
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    $\begingroup$ I think about 50% of people call $S_p$ the punctual Hilbert scheme and 50% of people call $X^{[n]}$ the punctual Hilbert scheme. But you might find some relevant results in the literature if you search for "punctual Hilbert scheme". $\endgroup$ May 2, 2020 at 21:46
  • $\begingroup$ Thanks Mohan! Thanks also @YosemiteStan I'm interested in both interpretations. Do you have a reference for the fiber interpretation? $\endgroup$ May 3, 2020 at 11:51

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