Equations for Chow and Hilbert quotients

Abstract

We give explicit equations for the Chow and Hilbert quotients of a projective scheme $X$ by the action of an algebraic torus $T$ in an auxiliary toric variety. As a consequence we provide geometric invariant theory descriptions of these canonical quotients, and obtain other GIT quotients of $X$ by variation of GIT quotient. We apply these results to find equations for the moduli space ${\overline{M}}_{0,n}$ of stable genus-zero $n$-pointed curves as a subvariety of a smooth toric variety defined via tropical methods.