Isoresidual fibrations and the moduli space of pointed rational curves

After surveying the results of the last 20 years on the structure of effective divisors on M0,n, we show that the pseudo-effective cone of divisors is not polyhedral for n≥8. Using ideas from Teichmüller dynamics and birational geometry, we construct an extremal non-polyhedral ray of the dual cone of moving curves using residue maps of strata of meromorphic differentials.