Correspondence theorems via tropicalizations of moduli spaces (Q2800064)

The modern enumerative geometry is based on the intersection theory of moduli spaces of stable maps to algebraic varieties. In turn, the tropical enumerative geometry started with an actual enumeration of tropical curves matching given constraints. Later Mikhalkin proposed an idea to develop the intersection theory of moduli spaces of tropical curves and solve enumerative problems in analogy with the algebraic-geometric approach. For enumeration of rational curves in toric varieties, this idea was elaborated by \textit{T. Nishinou} and \textit{B. Siebert} [Duke Math. J. 135, No. 1, 1--51 (2006; Zbl 1105.14073)]. The present paper continues this line. The author constructs embeddings of the moduli spaces of labeled marked rational curves in toric varieties into algebraic tori and shows that their tropicalizations are just the moduli spaces of appropriate marked rational tropical curves. Moreover, these embeddings commute with tropicalization, which finally yields a correspondence theorem between rational curves in toric varieties passing through appropriate number of points and intersecting some toric divisors with prescribed multiplicities on one side and appropriate rational tropical curves in the corresponding tropical toric varieties on the other side, and, furthermore, the tropical intersection multiplicities of the above tropical curves appear to be the numbers of the enumerated algebraic curves that tropicalize to the given tropical curve.