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.