2014/06/30 by Andreas Gross · 1 citation
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Algebraic curve #Algebraic surface #Algebraic variety #Cryptography and Residue Arithmetic #Moduli of algebraic curves #Moduli space #Morphism #Polynomial and algebraic computation #Toric variety #Tropical geometry #Variety (cybernetics) #math.AG #msc:14N10 #msc:14T05
paper · pdf · doi:10.1142/s0219199715500431
published as Commun. Contemp. Math. (2016) 18 · published version
openalex publication_date 2015/04/09 · openalex created_date 2016/06/24 · arxiv created 2016/11/29 · arxiv updated 2016/12/01 · openalex updated_date 2026/08/05
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 show that the tropicalizations of the curves in a given toric variety which intersect the boundary divisors in their interior and with prescribed multiplicities, and pass through an appropriate number of generic points are precisely the tropical curves in the corresponding tropical toric variety satisfying the analogous condition. Moreover, the intersection-theoretically defined multiplicities of the tropical curves are equal to the numbers of algebraic curves tropicalizing to them.