Let $K$ be a field of characteristic different from two. Conics over $K$ (that is curves of degree two in $\mathbb{P}^2_K$) are parametrized by $\mathbb{P}(k[x,y,z]_2) = \mathbb{P}^5_K$.

Conisider the set $R = \{C \in \mathbb{P}(k[x,y,z]_2) \: | \: C \text{ has a $K$-point}\}\subseteq\mathbb{P}^5$.

If $K$ is algebraically closed then $R = \mathbb{P}^5$ but if not there are conics without points.

When $K$ is not algebraically closed does $R$ have any meaningful geometric structure?