Isoresidual fibrations and the birational geometry of moduli of pointed stable curves

We show that the pseudo-effective cone of divisors of ${\bar{M}}_{0,n}$ is not polyhedral for $n\ge 8$ by constructing an extremal non-polyhedral ray of the dual cone of moving curves via maps on meromorphic strata of differentials returning the residues at the poles of the parameterised differentials. An immediate corollary is that these spaces are not Mori Dream Spaces.