# $Hilb_{lines}^{x}(X)$ and $Hilb_{lines}^{x}(X_{red})$

Let $X$ be a irreducible closed subscheme of $\mathbb{P}^N_{\mathbb{C}}$, and $U$ is a nonempty open where $X$ is smooth and moreover for every $x\in U$ and for every line $l\subseteq X$ with $x\in l$ assume $l \subseteq X\setminus \mathrm{sing}(X)$. If $x\in U$, clearly $$Hilb_{lines}^{x}(X)=Hilb_{lines}^{x}(X_{red})$$ as the identity of sets, but is it true as identity of schemes?

Thanks.

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If you let $V = X \setminus \mathrm{sing}(X)$, then assuming $x\in U$ we see that the Hilbert scheme of lines in $X$ passing through $x$ is the same as the Hilbert scheme of lines in $V$ passing through $x$. Indeed, the flat families of lines through $x$ in $X$ are exactly the same as the flat families of lines through $x$ in $V$, so the representing schemes are the same as well (if somebody has a better way of saying this please comment!). Furthermore, $V$ is reduced, so this is also the same as the Hilbert scheme of lines in $X_{\mathrm{red}}$ passing through $x$.
I thought what you said, but object $Hilb_{lines}^x(V)$ is not nice because $V$ is not projective... – gio Jul 2 '11 at 23:24