# Punctual Hilbert Schemes

Let $H$ be the Hilbert scheme of Artin local rings (quotients of a power series ring $R$ in $e$ variables over $\mathbb{C}$) of length $n$. Consider the set $G\subset H$ of rings $A$ with the property that the kernel of $d:A\to \Omega^1_A$ is just the constants. Is $G$ dense in $H$?

This is true if $e\leq 2$, thanks to the irreducibility of $H$. In general, I have no idea and pointers to relevant information would be appreciated.

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Can you give an example of a special ring for which this fails? (Even in the dimension 2 case might be useful to see what goes wrong) – Jack Huizenga Apr 25 '13 at 22:13
Here is an example, I am ashamed to say have forgotten due to whom. Take $f=x^2y^2+x^5+y^5$ in two variable power series. Take the quotient by the ideal $(f_x,f_y)$ (the partial derivatives). By Briancon-Skoda, $f^2$ belongs to this ideal. In this case $f$ does not and since $df=0$ in the module of differentials we have such an example. – Mohan Apr 26 '13 at 12:27