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 $Q$?



Quadrics and Moduli Spaces
The moduli space $\overline{M}\_{0,6}$ can be realized by blowing-up in $\mathbb{P}^3$ five points $p\_i$ and then the strict transforms of the ten lines $l\_{i,j} = \left\langle p\_i,p\_j\right\rangle$ through two of them. This is the Kapranov's construction of $\overline{M}\_{0,n}$.


Now, consider in $\mathbb{P}^3$ the unique smooth quadric surface $Q$ containing the five points and four lines of the type $l\_{i,k}, l\_{i,t}, l\_{j,k}, l\_{j,t}$. The strict transform of $Q$ in $\overline{M}\_{0,6}$ has a modular interpretation, namely this is the Keel-Vermeire divisor. An example of an extremal divisor in $\overline{M}\_{0,6}$ which can not be written in terms of boudary divisors. 


For a generalization of this construction for $n\geq 7$ you may take a look to this paper:


A. M. Castravet and J. Tevelev, *Hypertrees, projections, and moduli of stable rational curves*, J. Reine Angew. Math. 675 (2013), 121-180. 


