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Let $X$ be a smooth projective variety over an algebracally closed field $k$. (In my case $k=\mathbb{C}, X=\mathbb{P}^n$.)

Let us consider the subset of $k$-points of the Hilbert scheme $Hilb(X)$ consisting of smooth closed subvarieties of $X$. Is this subset Zariski open in $Hilb(X)$?

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    $\begingroup$ Yes. There is a universal subscheme $Z\subset X\times \mathrm{Hilb}(X)$, hence a proper map $Z\rightarrow \mathrm{Hilb}(X)$, and you are looking at the open subset of the base where the fiber is smooth. $\endgroup$
    – abx
    Commented Mar 10, 2017 at 15:56

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