Quadric surfaces with a rational point

Question

If I parameterize the space of quadric surfaces defined over $\mathbb{Q}$ in $\mathbb{P}^3_{\mathbb{Q}}$ as $\mathbb{P}_{\mathbb{Q}}^{\binom{3+2}{2}-1}=\mathbb{P}_{\mathbb{Q}}^9$, what does the locus of the quadric surfaces with a rational point look like?

Answer 1

If you pick a rational point in $P^3$, the quadrics containing it form a hyperplane in $P^9$. Thus the locus you are interested in is a countable union of hyperplanes.