Correspondence Theorems via Tropicalizations of Moduli Spaces

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.

published version

Cited by in corpus (6)

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• Intersection Theory on Tropicalizations of Toroidal Embeddings
• Moduli of stable maps in genus one and logarithmic geometry II
• Tropical geometry of genus two curves
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• Moduli Spaces of Rational Graphically Stable Curves