Moduli of algebraic and tropical curves

Moduli spaces are a geometer’s obsession. A celebrated example in algebraic geometry is the space M̄ g,n of stable n -pointed algebraic curves of genus g , due to Deligne—Mumford and Knudsen. It has a delightful combinatorial structure based on weighted graphs . Recent papers of Branetti, Melo, Viviani and of Caporaso defined an entirely different moduli space of tropical curves , which are weighted metrized graphs. It also has a delightful combinatorial structure based on weighted graphs. One is led to ask whether there is a geometric connection between these moduli spaces. In joint work [1] with Caporaso and Payne, we exhibit a connection, which passes through a third type of geometry — nonarchimedean analytic geometry. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.