It is well known that $\overline{M}_{0,5}$, the moduli space of $5$-pointed rational curves, can be realized as the blow-up of $\mathbb{P}^2$ in four general points. Therefore, we may interpret $\overline{M}_{0,5}$ as the blow-up of a smooth quadric surface $Q\subset\mathbb{P}^3$ in three general points.

Now let $Q\subset\mathbb{P}^n$ be a smooth quadric hypersurface with $n\geq 4$. Does there exist a modular interpretation for a variety obtained by blowing-up a certain number of general points in $Q$?