Abstract
We contribute to the foundations of tropical geometry with a view toward formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework, the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves. Our approach to tropical geometry permits tropical moduli problems—moduli of curves or otherwise—to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.
Highlights
We develop a stack-theoretic approach to the moduli spaces of tropical curves that naturally resolves the same issue in tropical geometry
We introduce forgetful morphisms and show they realize the universal family over the moduli space
This is an example of a remarkable rigidification that occurs in logarithmic geometry
Summary
Let g and n be nonnegative integers such that 2g − 2 + n > 0. In [ACP15], Abramovich, Caporaso, and Payne describe a natural continuous tropicalization map tropAg,CnP : Mga,nn → Mgtr,onp from the Berkovich analytification Mga,nn to the set-theoretic tropical moduli space Mgtr,nop (see [Viv13]) Their map is given by associating a tropical curve with edge lengths in R>0 to a stable one-parameter degeneration of a smooth algebraic curve over a valuation ring extending a trivially valued field k. Theorem 4 shows that a∗Mtgr,onp functions as a lift of the moduli stack Mtgr,onp to LogSch, thereby realizing both of these objects within the same category From this point of view, the tropicalization map tropg,n is the map that associates to a metrized curve complex its underlying tropical curve. Expanding on Corollary 1, in [Uli19] the third author is giving a new proof of the main result of [ACP15] comparing the Artin fan of Mlgo,gn with the Artin fan of Mtgr,onp
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