Polarizations on $M_{0,n}$ from Kapranov's quotient constructions

In Kapranov's marvelous paper Chow quotients of Grassmannian I, he proves that $\overline{M}_{0,n}$ is isomorphic to both the Hilbert quotient and Chow quotient $(\mathbb{P}^1)^n//\text{SL}_2$. These quotients are by definition closed subvarieties of the Hilbert scheme and Chow variety (respectively) of $(\mathbb{P}^1)^n$. Every Chow variety has a natural polarization coming from Chow forms, and Hilbert schemes have a family of polarizations. Is it known what ample divisors on $\overline{M}_{0,n}$ we get by restricting these polarizations via Kapranov's construction?

-
There are some papers on this question. For starters, have you seen Alexeev and Swinarski's paper math.uga.edu/~davids/0812.0778.pdf? – J.C. Ottem Apr 5 '11 at 20:55
Indeed, it is a nice paper. The Chow quotient maps to all the GIT quotients, so the polarizations on the latter pull back to the former, namely M_{0,n}, but the Chow-GIT morphisms are contractions (for n > 6), so come from nef but not ample divisors. So the Alexeev-Swinarski divisors cannot be the Hilbert or Chow polarizations (though perhaps their convex hull contains these ample divisors?...) – Noah Giansiracusa Apr 5 '11 at 22:37