Moduli of algebraic and tropical curves

Abstract. Moduli spaces are a geometer’s obsession. A cele-brated example in algebraic geometry is the space M̄g,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 de-fined 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 be-tween these moduli spaces. In joint work [ACP12] with Caporaso and Payne, we exhibit a connection, which passes through a third type of geometry- nonarchimedean analytic geometry. 1. The moduli bug Geometers of all kinds are excited, one may say obsessed, with mod-uli spaces; these are the spaces which serve as parameter spaces for the basic spaces geometers are most interested in. It was none other than Riemann who introduced the moduli bug into geometry, when he noted that Riemann surfaces of genus g> 1 “depend on 3g − 3 Moduln”. This is a consequence of his famous “Riemann existence theorem”, which tells us how to put together a Riemann surfaces by slitting a number of copies of the Riemann sphere and sewing them together. The number 3g−3 is simply the number of “effective complex parameters ” necessary for obtaining every Riemann surface this way. 1 This brings us to the classic case of the moduli phenomenon, and the obsession that comes with it, the spaceMg of Riemann surfaces of genus g: fixing a compact oriented surface S, each point on Mg cor-responds uniquely to a complex structure C on S. Being an algebrai