GIT Compactifications of $M_{0,n}$ from Conics

TL;DR

This paper explores GIT quotients parametrizing n-pointed conics, establishing morphisms from the moduli space of stable n-pointed rational curves to these quotients, generalizing known results for simpler cases.

Contribution

It introduces new GIT quotients for n-pointed conics and shows how the moduli space _{0,n} maps to them, extending Kapranov's classical results.

Findings

_{0,n} admits morphisms to GIT quotients of n-pointed conics.

These morphisms factor through Hassett's moduli spaces with weighted points.

The work generalizes known GIT constructions from _{0,n} _{0,n} to conics.

Abstract

We study GIT quotients parametrizing n-pointed conics that generalize the GIT quotients $(\mathbb{P}^1)^n//SL2$. Our main result is that $\overline{M}_{0,n}$ admits a morphism to each such GIT quotient, analogous to 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.