Chow quotients of toric varieties as moduli of stable log maps

Abstract

Let $X$ be a projective normal toric variety and $T_0$ a rank-$1$ subtorus of the defining torus $T$ of $X$. We show that the normalization of the Chow quotient $X∕∕T_0$, in the sense of Kapranov, Sturmfels, and Zelevinsky, coarsely represents the moduli space of stable log maps to $X$ with discrete data given by $T_0\subset X$. We also obtain similar results when $T_0\to T$ is a homomorphism that is not necessarily an embedding.