Logarithmic stable toric varieties and their moduli

TL;DR

This paper establishes an isomorphism between the Chow quotient of a toric variety by a subtorus and the moduli space of stable toric varieties when both are viewed as logarithmic stacks, providing a new perspective on their structure.

Contribution

It constructs the Chow quotient stack and proves its isomorphism with the moduli space of stable toric varieties as logarithmic stacks, clarifying their relationship.

Findings

Chow quotient and moduli space are isomorphic as logarithmic stacks.

Construction of the Chow quotient stack with key properties.

Demonstration of the isomorphism between these spaces.

Abstract

The Chow quotient of a toric variety by a subtorus, as defined by Kapranov-Sturmfels-Zelevinsky, coarsely represents the main component of the moduli space of stable toric varieties with a map to a fixed projective toric variety, as constructed by Alexeev and Brion. We show that, after endowing both spaces with the structure of a logarithmic stack, the resulting spaces are isomorphic. Along the way, we construct the Chow quotient stack and demonstrate several properties that it satisfies.