[Paper Review] Chow Quotients of Toric Varieties as Moduli of Stable Log Maps

Qile Chen, Matthew Satriano|arXiv (Cornell University)|Jan 17, 2012

Algebraic Geometry and Number Theory12 references4 citations

TL;DR

This paper establishes that the normalization of the Chow quotient $X//T_0$ for a projective normal toric variety $X$ and a rank-one subtorus $T_0$ is isomorphic to the coarse moduli space of stable log maps from rational curves to $X$ with fixed curve class and contact orders. The construction uses orbit closures $Z_x = \overline{T_0 x}$ to define a map from $T' = T/T_0$ to the Kontsevich space $\mathfrak{M}_{0,2}(X,\beta_0)$, and the closure of this image in the moduli space yields the coarse moduli space of stable log maps, providing a new geometric interpretation of Chow quotients via log Gromov-Witten theory.

ABSTRACT

Let $X$ be a projective normal toric variety and $T_0$ a rank one subtorus of the defining torus of $X$. We show that the normalization of the Chow quotient $X//T_0$, in the sense of Kapranov-Sturmfels-Zelevinsky, coarsely represents the moduli space of stable log maps to $X$ with discrete data given by $T_0\subset X$.

Motivation & Objective

• To establish a geometric correspondence between Chow quotients of toric varieties and moduli spaces of stable log maps.
• To show that the normalization of the Chow quotient $X//T_0$ coarsely represents the moduli stack $\mathcal{K}_{\Gamma_0}(X)$ of stable log maps with discrete data $\Gamma_0 = (0,\beta_0,2,\{c_0,c_\infty\})$.
• To provide an alternative construction of $\mathcal{K}_{\Gamma_0}(X)$ via the closure of $T'$ in the Kontsevich space $\mathfrak{M}_{0,2}(X,\beta_0)$, analogous to the Chow quotient construction.
• To extend Olsson’s result on toric Hilbert schemes to the Kontsevich space setting, showing that the normalization of the closure of $T'$ in $\mathfrak{M}_{0,2}(X,\beta_0)$ carries a moduli interpretation in terms of stable log maps.

Proposed method

• Define the Chow quotient $X//T_0$ as the closure of $T' = T/T_0$ in the Chow variety $C(X)$ of cycles of fixed dimension and homology class.
• Construct a morphism $T' \to \mathfrak{M}_{0,2}(X,\beta_0)$ by associating to each $x \in T$ the stable map $f_x: \mathbb{P}^1 \to X$ whose image is the normalization of the orbit closure $Z_x = \overline{T_0 x}$.
• Define $\mathfrak{M}$ as the closure of $T'$ in the Kontsevich space $\mathfrak{M}_{0,2}(X,\beta_0)$, and show that $\mathfrak{M}$ is a birational model of $X//T_0$.
• Prove that $\mathcal{K}_{\Gamma_0}(X)$ is the normalization of $\mathfrak{M}$, using log smoothness and irreducibility of the moduli stack for genus 0.
• Use tropical geometry to interpret contact orders $c_0, c_\infty$ as slopes of unbounded edges in tropical curves associated to stable log maps.
• Show that the standard log structure $\mathcal{M}_X$ on a projective normal toric variety is generalized Deligne-Faltings, ensuring compatibility with the log Gromov-Witten theory of Abramovich-Chen and Gross-Siebert.

Experimental results

Research questions

• RQ1 Is the normalization of the Chow quotient $X//T_0$ isomorphic to the coarse moduli space of stable log maps to $X$ with fixed discrete data $\Gamma_0 = (0,\beta_0,2,\{c_0,c_\infty\})$?
• RQ2 Can the construction of the Chow quotient be reinterpreted via the Kontsevich space of stable maps, replacing the Chow variety with the moduli space $\mathfrak{M}_{0,2}(X,\beta_0)$?
• RQ3 Does the normalization of the closure of $T'$ in $\mathfrak{M}_{0,2}(X,\beta_0)$ carry a moduli interpretation in terms of stable log maps, analogous to Olsson’s result for the toric Hilbert scheme?
• RQ4 How do tropical curves encode the contact orders $c_0$ and $c_\infty$ in the context of stable log maps to toric varieties?
• RQ5 Is the standard log structure $\mathcal{M}_X$ on a projective normal toric variety generalized Deligne-Faltings, ensuring compatibility with the log Gromov-Witten theory?

Key findings

• The normalization of the Chow quotient $X//T_0$ is isomorphic to the coarse moduli space of the stack $\mathcal{K}_{\Gamma_0}(X)$ of stable log maps from rational curves with two marked points and fixed curve class $\beta_0$ and contact orders $c_0, c_\infty$.
• The stack $\mathcal{K}_{\Gamma_0}(X)$ is irreducible, as a consequence of the normalization of $X//T_0$ being irreducible and the moduli space being a coarse moduli space.
• The closure $\mathfrak{M}$ of $T'$ in the Kontsevich space $\mathfrak{M}_{0,2}(X,\beta_0)$ is isomorphic to the normalization of $\mathfrak{M}$, and this normalization is isomorphic to $\mathcal{K}_{\Gamma_0}(X)$, establishing a new moduli interpretation.
• The contact orders $c_0$ and $c_\infty$ correspond to the slopes of the unbounded edges in the tropical curves associated to stable log maps, providing a combinatorial interpretation of tangency conditions.
• The standard log structure $\mathcal{M}_X$ on a projective normal toric variety is generalized Deligne-Faltings, as shown by constructing a strict smooth cover from $[A_P/\mathbb{G}_m]$ and verifying the existence of a suitable chart.
• The moduli stack $\mathcal{K}_{\Gamma_0}(X)$ is log smooth and hence normal, which is a key ingredient in proving that its coarse moduli space is the normalization of $\mathfrak{M}$.