#### Vol. 4, No. 7, 2010

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Equations for Chow and Hilbert quotients

### Angela Gibney and Diane Maclagan

Vol. 4 (2010), No. 7, 855–885
##### 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.

##### Keywords
Chow quotient, Hilbert quotient, moduli of curves, space of phylogenetic trees
##### Mathematical Subject Classification 2000
Primary: 14L30
Secondary: 14M25, 14L24, 14H10