Equations for Chow and Hilbert quotients

Gibney, Angela; Maclagan, Diane

doi:10.2140/ant.2010.4.855

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Equations for Chow and Hilbert quotients

Angela Gibney and Diane Maclagan

Vol. 4 (2010), No. 7, 855–885

DOI: 10.2140/ant.2010.4.855

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 ${\bar{M}}_{0, n}$ 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

Milestones

Received: 29 May 2009

Revised: 17 February 2010

Accepted: 5 May 2010

Published: 29 January 2011

Authors

Angela Gibney
Department of Mathematics University of Georgia Athens, GA 30602 United States

Diane Maclagan
Mathematics Institute Zeeman Building University of Warwick Coventry CV4 7AL United Kingdom

Vol. 4, No. 7, 2010