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

2011-07-02T19:58:13Z

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.

Answer by Jack Huizenga for $Hilb_{lines}^{x}(X)$ and $Hilb_{lines}^{x}(X_{red})$

2011-07-02T22:38:47Z

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 $x$.

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

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

Answer by Jack Huizenga for $Hilb_{lines}^{x}(X)$ and $Hilb_{lines}^{x}(X_{red})$