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

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 $({\mathbb{P}}^1{)}^n\mathrm{/}\mathrm{/}PGL2$ are isomorphic to ${\bar{M}}_{0,n}$. 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.