  We define a class of surfaces corresponding to the ADE root lattices and
construct compactifications of their moduli spaces as quotients of projective
varieties for Coxeter fans, generalizing Losev-Manin spaces of curves. We
exhibit modular families over these moduli spaces, which extend to families of
stable pairs over the compactifications. One simple application is a geometric
compactification of the moduli of rational elliptic surfaces that is a finite
quotient of a projective toric variety.
