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.