Skeletons of stable maps I: rational curves in toric varieties
Dhruv Ranganathan · Journal of the London Mathematical Society · 2017
We study the Berkovich analytification of the space of genus 0 logarithmic stable maps to a toric variety X and present applications to both algebraic and tropical geometry. On algebraic side, insights from tropical geometry give two new geometric descriptions of this space of maps — (1) as an explicit toroidal modification of M ¯ 0 , n × X and (2) as a tropical compactification in a toric variety. On the combinatorial side, we prove that the tropicalization of the space of genus 0 logarithmic stable maps coincides with the space of tropical stable maps, giving a large new collection of examples of faithful tropicalizations for moduli. Moreover, we identify the optimal settings in which the tropicalization of the moduli space of maps is faithful. The Nishinou–Siebert correspondence theorem is shown to be a consequence of this geometric connection between the algebraic and tropical moduli.