Tropicalization of the moduli space of stable maps

TL;DR

This paper investigates the tropicalization map from the moduli space of stable algebraic maps into a variety to the space of tropical curves, establishing its continuity, compactness, and polyhedral structure.

Contribution

It proves the tropicalization map is continuous with a compact, polyhedral image, linking algebraic and tropical geometry through new structural insights.

Findings

The tropicalization map is continuous.

The image of the map is compact and polyhedral.

Deformations of algebraic curves correspond to continuous deformations of tropical curves.

Abstract

Let $X$ be an algebraic variety and let $S$ be a tropical variety associated to $X$. We study the tropicalization map from the moduli space of stable maps into $X$ to the moduli space of tropical curves in $S$. We prove that it is a continuous map and that its image is compact and polyhedral. Loosely speaking, when we deform algebraic curves in $X$, the associated tropical curves in $S$ deform continuously; moreover, the locus of realizable tropical curves inside the space of all tropical curves is compact and polyhedral. Our main tools are Berkovich spaces, formal models, balancing conditions, vanishing cycles and quantifier elimination for rigid subanalytic sets.