Skeletons of stable maps I: Rational curves in toric varieties

TL;DR

This paper explores the structure of genus 0 logarithmic stable maps to toric varieties through algebraic and tropical geometry, revealing new geometric descriptions and faithfulness conditions for tropicalization.

Contribution

It provides two novel geometric descriptions of the moduli space of maps and establishes conditions for faithful tropicalization, linking algebraic and tropical geometries.

Findings

Tropicalization of the moduli space matches the space of tropical stable maps.

Identifies conditions for faithful tropicalization of the moduli space.

Shows the Nishinou--Siebert correspondence as a consequence of these geometric insights.

Abstract

We study the Berkovich analytification of the space of genus $0$ logarithmic stable maps to a toric variety $X$ and present applications to both algebraic and tropical geometry. On algebraic side, insights from tropical geometry give two new geometric descriptions of this space of maps -- (1) as an explicit toroidal modification of $\overline M_{0,n}\times X$ and (2) as a tropical compactification in a toric variety. On the combinatorial side, we prove that the tropicalization of the space of genus $0$ logarithmic stable maps coincides with the space of tropical stable maps, giving a large new collection of examples of faithful tropicalizations for moduli. Moreover, we identify the optimal settings in which the tropicalization of the moduli space of maps is faithful. The Nishinou--Siebert correspondence theorem is shown to be a consequence of this geometric connection between the algebraic and tropical moduli spaces.