GIT Compactifications of ℳ0,n from Conics

We study geometric invariant theory (GIT) quotients parameterizing n-pointed conics that generalize the GIT quotients $(\mathbb P^1)^n//{\mathrm SL}_2$. Our main result is that $\bar {\mathcal M}_{0,n}$ admits a morphism to each such GIT quotient, generalizing the well-known result of Kapranov for the simpler $(\mathbb P^1)^n$ quotients. Moreover, these morphisms factor through Hassett’s moduli spaces of weighted pointed rational curves, where the weight data comes from the GIT linearization data.