Effective cones of quotients of moduli spaces of stable $n$-pointed curves of genus zero

Let $X_n := \overline{M}_{0,n}$, the moduli space of $n$-pointed stable genus zero curves, and let $X_{n,m}$ be the quotient of $X_n$ by the action of $\mathcal{S}_{n-m}$ on the last $n-m$ marked points. The cones of effective divisors $\overline{NE}^1(X_{n,m})$, $m = 0,1,2$, are calculated. Using this, upper bounds for the cones $Mov(X_{n,m})$ generated by divisors with moving linear systems are calculated, $m = 0,1$, along with the induced bounds on the cones of ample divisors of $\overline{M}_g$ and $\overline{M}_{g,1}$. As an application, the cone $\overline{NE}^1(\overline{M}_{2,1})$ is analyzed in detail.