On Kapranov's description of $\overline{M}_{0,n}$ as a Chow quotient

TL;DR

This paper provides a characteristic-independent proof that the moduli space of stable n-pointed rational curves can be realized as a Chow quotient, extending Kapranov's complex-analytic result to arbitrary characteristic.

Contribution

It offers a direct, characteristic-free proof of the isomorphism between the Chow quotient of $(P^1)^n$ by PGL2 and $ar{M}_{0,n}$, including explicit constructions of the universal family.

Findings

The Chow quotient $(P^1)^n//PGL2$ is isomorphic to $ar{M}_{0,n}$ in arbitrary characteristic.

Explicit universal family construction confirms the isomorphism.

Reduction to the case $n=4$ simplifies the proof and relates to operad formalism.

Abstract

We provide a direct proof, valid in arbitrary characteristic, of the result originally proven by Kapranov over $\mathbb{C}$, that the Hilbert and Chow quotients $(\mathbb{P}^1)^n//PGL2$ are isomorphic to $\overline{M}_{0,n}$. In both cases this is done by explicitly constructing the universal family and then showing that the induced morphism is an isomorphism onto its image. The proofs of these results in many ways reduce to the case $n = 4$; in an appendix we outline a formalism of this phenomenon relating to certain operads.