Q:

Quadrics and Moduli Spaces

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 $Q$? 

A:

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. 