The Grothendieck-Knudsen moduli space $\bar M_{0,n}$ of stable, $n$-pointed rational curves has a natural stratification given by topological type. It is a natural question whether the boundary generates the effective cone of divisors, or whether the one-dimensional strata generate the effective cone of curves (the Fulton Conjecture). We construct many (non-boundary) divisors that are generators of the effective cone, as well as rigid curves intersecting the interior. The main technique is to identify $M_{0,n}$ with the Brill-Noether locus of a reducible curve given by a hypergraph. This is joint work with Jenia Tevelev.