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 $Q$?

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    $\begingroup$ The dimension of the space of automorphisms is $n(n+1)/2$, and the dimension of $Q$ is $n-1$, so the blow-up of $Q$ at $k$ general points is well-defined if and only if $k \leq n(n+1)/2(n-1)$. One could play around with taking the largest such $k$, but since the upper bound is never an integer for $n>3$ this might not behave nicely. $\endgroup$ – Will Sawin Feb 9 '15 at 21:47
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    $\begingroup$ If you generalize $P^1\times P^1$ to higher dimensions in a different way, the answer will be positive. In fact, $M_{0,n}$ can be realized as a blowup of $(P^1)^{n-3}$. $\endgroup$ – Sasha Feb 11 '15 at 6:59

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.


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