TROPICALIZATION OF THE MODULI SPACE OF STABLE MAPS
Tony Yue Yu · 2014
Let X be an algebraic variety and let S be a tropical variety associated to X. We study the tropicalization map from the moduli space of stable maps into X to the moduli space of tropical curves in S. We prove that it is a continuous map and that its image is a compact finite polyhedral complex. In other words, the tropical curves deform continuously when we deform the algebraic curves; moreover, the locus of realizable tropical curves inside the space of all tropical curves is compact and polyhedral. Our main tools are Berkovich spaces, formal models, balancing conditions, vanishing cycles and quantifier elimination.