Extremality of Rational Tails Boundary Strata in $\overline{\mathcal{M}}_{g,n}$

We review and develop some techniques used to investigate the effective cones of higher codimension classes. Our results show that a large collection of boundary strata of rational tails type are extremal in their effective cones on $\overline{\mathcal{M}}_{g,n}$ and provide evidence for the conjecture that all boundary strata of $\overline{\mathcal{M}}_{g,n}$ are extremal. As a corollary, we show that all boundary strata are extremal in genus zero.