  Basepoint free cycles on the moduli space $\overline{M}_{0,n}$ of stable
n-pointed rational curves, defined using Gromov-Witten invariants of smooth
projective homogeneous spaces X are studied. Intersection formulas to find
classes are given, with explicit examples for X a projective space, and X a
smooth projective quadric hypersurface. When X is projective space, divisors
are shown equivalent to conformal blocks divisors for type A at level one,
giving maps from $\overline{M}_{0,n}$ to birational models constructed as GIT
quotients, parametrizing configurations of weighted points supported on
(generalized) Veronese curves.
