Hypergraphic divisors, curves, and morphisms

The birational geometry of the moduli space of stable, n-pointed, rational
curves is still largely unknown. For example, it is not known if the cone
of effective divisors is polyhedral, if the cone of effective curves is
generated by 1-strata (Fulton's conjecture), or if it has only finitely
many small modifications. In joint work with Jenia Tevelev, we find that
some of this complexity can be explained by the fact that this moduli
space can be interpreted as the Brill-Noether locus of some very special
reducible curves related to hypergraphs. Using this we construct many new
extremal divisors and birational morphisms with unexpected exceptional
loci.