Mori Dream Spaces and $\barM_{0,n}$
Ana-Maria Castravet (Arizona)

Mori Dream spaces are varieties for which there is a nice combinatorial description for rational maps to other varieties. A theorem of Hu and Keel asserts that a variety $X$ is a Mori Dream space if and only if its total coordinate ring is finitely generated.

Keel and Hu posed the question whether the moduli space $\barM_ {0,n}$ of $n$-pointed, stable, genus $0$ curves is a Mori Dream space. The underlying open question is to understand the effective cone of $\barM_{0,n}$. We have a method for constructing extremal rays of the effective cone of $\barM_{0,n}$ for any $n\geq6$. For $n=6$ we recover the Keel-Vermeire divisors. It is a result of Hassett and Tschinkel that the Keel- Vermeire divisors together with the boundary divisors generate the effective cone of $\barM_{0,6}$. We show that the corresponding sections generate the total coordinate ring of $\barM_{0,6}$.