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

Basepoint free cycles on $\overline{M}_{0,n}$ from Gromov-Witten theory