Ana-Maria Castravet

Title: Blown-up toric surfaces with non-polyhedral effective cone

Abstract:
I will report on recent joint work with Antonio Laface, Jenia Tevelev and Luca Ugaglia.
We construct examples of projective toric surfaces whose blow-up at a general point 
has a non-polyhedral pseudoeffective cone, both in characteristic 0 and in prime 
characteristic. As a consequence, we prove that the pseudo-effective cone of the
Grothendieck-Knudsen moduli space of stable, n-pointed, rational stable curves, 
is not polyhedral if n>=10 in characteristic 0 and in positive characteristic for an
infinite set of primes of positive density. In particular, these moduli spaces are not 
Mori dream spaces even in positive characteristic.