Smooth quadric over p-adic integers

Question

Let $k$ be a $p$-adic field with ring of integers $\mathcal{O}_K$ and residue field $\mathbb{F}$. Say I have a (projective) quadric $Q$ which is smooth over $\mathcal{O}_K$, such that the reduction $\bar{Q}$ (smooth over $\mathbb{F}$) contains a line $\cong \mathbb{P}^1$ defined over $\mathbb{F}$. Does it follow that $Q$ contains a line $\cong \mathbb{P}^1$ defined over $\mathcal{O}_K$?

If yes, I guess this would somehow follow from Hensel's lemma, but I'm not sure how exactly.

Answer 1

Yes, for the reason you give. The Hilbert scheme of lines in $Q$ is smooth over $\mathcal O_K$ (the obstruction to smoothness lies in $H^1$ of a normal bundle $\mathcal N$; the homogeneity of $Q$ shows that $\mathcal N$ is generated by $H^0$, so has $H^1=0$ from the classification of bundles on $\mathbb P^1$). Now apply Hensel's lemma to this Hilbert scheme.