Correspondence Theorems via Tropicalizations of Moduli Spaces

TL;DR

This paper establishes a correspondence between algebraic and tropical moduli spaces of rational curves in toric varieties, enabling a new proof of Nishinou and Siebert's theorem through tropical intersection theory.

Contribution

It introduces embeddings of moduli spaces into algebraic tori that preserve tropicalizations and evaluation maps, leading to a general correspondence theorem for enumerative problems.

Findings

Embedded moduli spaces' tropicalizations match their algebraic counterparts.

Evaluation maps commute with tropicalization, preserving enumerative data.

Reproved Nishinou and Siebert's correspondence theorem using tropical intersection theory.

Abstract

We show that the moduli spaces of irreducible labeled parametrized marked rational curves in toric varieties can be embedded into algebraic tori such that their tropicalizations are the analogous tropical moduli spaces. These embeddings are shown to respect the evaluation morphisms in the sense that evaluation commutes with tropicalization. With this particular setting in mind we prove a general correspondence theorem for enumerative problems which are defined via "evaluation maps" in both the algebraic and tropical world. Applying this to our motivational example we reprove Nishinou and Siebert's correspondence theorem using tropical intersection theory.