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 M 0,n of stable genus-zero n-pointed curves as a subvariety of a smooth toric variety defined via tropical methods.

Moduli Spaces of Rational Weighted Stable Curves and Tropical Geometry

“…However, the connection between the algebraic and tropical moduli spaces runs much deeper. Building on previous work of Tevelev [24], Gibney and Maclagan [11] exhibit M trop 0,n as a tropicalization. They find an embedding of M 0,n into a torus, such that the tropicalization of M 0,n is a balanced fan Σ, such that Σ ∼ = M trop 0,n .…”

“…Fix this fan structure. Then we have the following result, due to Gibney and Maclagan [11,Theorem 5.7], as well as Tevelev [24,Theorem 5.5]. THEOREM 3.1.…”

Combinatorics of tropical Hurwitz cycles

“…This also rules out rational equivalence (as defined in [1]): two cycles are equivalent, if and only if their recession fans are equal.But there is another, coarser equivalence on M trop 0,n that comes from toric geometry. As was shown in[20], the classical M 0,n can be embedded in the toric variety X (M n )) ∼ = Pic(M 0,n ) D I → δ I ,…”

“…A much stronger relation was proven in [20], where it is shown that (for the right embedding) the tropicalization of M 0,n is M The ith Psi class is the subset ψ i of M 0,n , consisting of the locus of all n-marked curves such that the ith leaf is attached to a vertex that is at least four-valent.…”

Computing Cox rings

“…. , T r2 ] and vice versa, we use the following operations, compare also [21]. Consider a homomorphism π : T r → T n of tori and its kernel H ⊆ T r .…”