# Birational Geometry of Moduli spaces of rational curves (Castravet)

The Grothendieck-Knudsen moduli space M_0,n of stable rational curves

with n markings is arguably the simplest among the moduli spaces of stable

curves: it is a smooth projective variety that can be described explicitly

as a blow-up of projective space, with strata corresponding to nodal curves

similar to the torus invariant strata of a toric variety. However, in contrast

with toric varieties, the cones of effective cycles are much more mysterious

and only partial results are known for n ≥ 8. For n ≥ 13, there are negative

results: in characteristic zero, M_0,n is not a Mori Dream Space.

In these lectures, I will discuss known results about cones of effective cy-

cles on the moduli space and prove that M_0,n is not a Mori Dream Space by

reducing the question to a study of blow-ups of weighted projective planes.

Finally, I will describe an approach to Kuznetsov’s conjecture on the derived

category of M 0,n and other related moduli spaces.