Integral points and effective cones of moduli spaces of stable maps

Consider the Fulton-MacPherson configuration space of $n$ points on ${\mathrm{\P }}^1$, which is isomorphic to a certain moduli space of stable maps to ${\mathrm{\P }}^1$. We compute the cone of effective ${\mathfrak{S}}_n$-invariant divisors on this space. This yields a geometric interpretation of known asymptotic formulas for the number of integral points of bounded height on compactifications of ${\text{SL}}_2$ in the space of binary forms of degree $n\ge 3$.