$\DeclareMathOperator\Hilb{Hilb}\DeclareMathOperator\Spec{Spec}$Let $X$ be a smooth surface over a field $k$. Fogarty proved that the Hilbert scheme of points $\Hilb^n(X)$ is regular. My primary question is: if $X$ is a smooth (quasi-projective) surface over $\mathbb{Z}$, is $\Hilb^n(X)$ smooth?

Since both the Hilbert scheme and smoothness are compatible with base change, this would imply that $\Hilb^n(X_k)$ is smooth even when $\operatorname{char}(k)>0$. Every proof of Fogarty's theorem that I've seen proves regularity by computing the dimension of the tangent spaces, which won't suffice to prove smoothness in positive characteristic. But are there known examples of surfaces where $\Hilb^n(X)$ is regular but not smooth, or is this an open question?

Edit: as Will Sawin pointed out in the comments, we can base change (using eg Tag 038X of the Stacks Project) to deduce that $\Hilb^n(X_k)$ is smooth for any field. Since $\Hilb^n(X)\to\Spec(\mathbb{Z})$ is locally of finite type and has smooth fibers, $\Hilb^n(X)\to\Spec(\mathbb{Z})$ is smooth if it is flat.

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    $\begingroup$ In positive characteristic, can't you just base change to a perfect field and check smoothness there using regularity? $\endgroup$
    – Will Sawin
    Mar 1 at 1:48
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    $\begingroup$ Oh yes, thank you! Tag 038X of the Stacks Project works for smooth over a field $\endgroup$ Mar 1 at 2:43

1 Answer 1


I found a reference: this is proved in Proposition 7.27 of "Recovering the good component of the Hilbert scheme," by Ekedahl and Skjelnes.


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