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Equations for Chow and Hilbert quotients

Angela Gibney, Diane Maclagan

2010
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Algebra & Number Theory
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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 M 0,n of stable genus-zero n-pointed curves as a subvariety of a smooth toric variety defined via tropical methods.
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... C2000: primary 14L30; secondary 14M25, 14L24, 14H10. Keywords: Chow quotient, Hilbert quotient, moduli of curves, space of phylogenetic trees. 855 856 Angela Gibney and Diane Maclagan nonvanishing Plücker coordinates, and the quotient U/T n−1 is the moduli space M 0,n of smooth n-pointed genus-zero curves. In this case the desired compactification is the celebrated moduli space M 0,n of stable n-pointed genus-zero curves. [Kapranov 1993] showed that M 0,n is isomorphic to both the Chow and Hilbert quotients of G(2, n) by the T n−1 -action. We give explicit equations for M 0,n as a subvariety of a smooth toric variety X whose fan is the well-studied space of phylogenetic trees. We show that the equations for M 0,n in the Cox ring S of X are generated by the Plücker relations homogenized with respect to the grading of S. We now describe our results in more detail. The notation X/ / T d is used to refer to either the Chow or the Hilbert quotient. We assume that no irreducible component of X lies in any coordinate subspace. This means that X/ / T d is a subscheme of ސ m / / T d . The quotient ސ m / / T d is a not necessarily normal toric variety [Kapranov et al. 1992 ] whose normalization we denote by X . By X/ / n T d we mean the pullback of X/ / T d to X . Our main theorem, in slightly simplified form, is the following. This is proved in Theorems 3.2 and 4.6 and Proposition 4.3. (1) (Equations) The ideal I of the Hilbert or Chow quotient X/ / n T d in the Cox ring S = k[y 1 , . . . y r ] of X can be computed effectively. Explicitly, I is obtained by considering the f i as polynomials in y 1 , . . . , y m+1 , homogenizing them with respect to the Cl(X )-grading of S, and then saturating the result by the product of all the variables in S. (2) (GIT) There is a GIT construction of the Chow and Hilbert quotients of X , and these are related to the GIT quotients of X by variation of the GIT quotient. This gives equations for all quotients in suitable projective embeddings. Let H = Hom(Cl(X ), k × ). There is a nonzero cone Ᏻ ⊂ Cl(X ) ⊗ ޒ for which X/ / n T d is the GIT quotient X/ / n T d = Z (I )/ / α H for any rational α ∈ relint(Ᏻ), where Z (I ) is the subscheme of ށ r defined by I . For any GIT quotient X/ / β T d of X , there are choices of α outside Ᏻ for which Z (I )/ / α H = X/ / β T d . A more precise formulation of the homogenization is given in Theorem 3.2 and Remark 3.3. We explain in Corollary 4.4 how each choice of α ∈ relint(Ᏻ) gives an embedding of X/ / n T d into some projective space. We use tropical algebraic geometry in the spirit of [Tevelev 2007 ] to embed M 0,n in a smooth toric variety X . The combinatorial data describing and the simple Equations for Chow and Hilbert quotients 857 equations for M 0,n in the Cox ring of X are described in the following theorem. Let [n] = {1, . . . , n} and set Ᏽ = {I ⊂ [n] : 1 ∈ I, |I | ≥ 2, |[n] \ I | ≥ 2}. The set Ᏽ indexes the boundary divisors of M 0,n . Theorem 1.2. Let be the fan in ޒ ( n 2 )−n described in Section 5 (the space of phylogenetic trees). The rays of are indexed by the set Ᏽ. (1) (Equations) Equations for M 0,n in the Cox ring S = k[x I : I ∈ Ᏽ] of X are obtained by homogenizing the Plücker relations with respect to the grading of S and then saturating by the product of the variables of S. Specifically, the ideal is where the generating set runs over all {i, j, k, l} with 1 ≤ i < j < k < l ≤ n, and x I = x [n]\I if 1 ∈ I . (2) (GIT) There is a nonzero cone Ᏻ ⊂ Cl(X ) ⊗ ޒ ∼ = Pic(M 0,n ) ⊗ ޒ for which for rational α ∈ int ( Ᏻ) we have the GIT construction of M 0,n as M 0,n = Z (I M 0,n )/ / α H, where Z (I M 0,n ) ⊂ ށ |Ᏽ| is the affine subscheme defined by I M 0,n , and H is the torus Hom(Cl(X ), k × ) ∼ = (k × ) |Ᏽ|−( n 2 )+n . (3) (VGIT) Given β ∈ ޚ n there is α ∈ ޚ |Ᏽ|−( n 2 )+n for which Z (I M 0,n )/ / α H = G(2, n)/ / β T n−1 , so all GIT quotients of G(2, n) by T n−1 can be obtained from M 0,n by variation of the GIT. [Howard et al. 2009] , where GIT quotients of G(2, n) by T n−1 , or equivalently of ސ( 1 ) n by Aut(ސ 1 ), were studied. Statement (3) relates to Keel and Tevelev, in an article titled "Equations for M 0,n " [2009], studied the image of the particular embedding of M 0,n into a product of projective spaces given by the complete linear series of the very ample divisor κ = K M 0,n + I ∈Ᏽ δ I . Theorem 1.2 concerns projective embeddings of M 0,n corresponding to a full-dimensional subcone of the nef cone of M 0,n , including that given by κ. A key idea of this paper is to work in the Cox ring of sufficiently large toric subvarieties of X . This often allows one to give equations in fewer variables. Also, a truly concrete description of X may be cumbersome or impossible, as in the case of M 0,n , but a sufficiently large toric subvariety such as X can often be obtained.

doi:10.2140/ant.2010.4.855
fatcat:36egmdszfvdmnbul7g2aecepxu