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.

中文翻译：

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热带曲线的模数堆栈

我们为热带几何学的基础做出了贡献，以制定热带模量问题，并以曲线的模量空间作为我们的主要示例。我们为曲线的模空间提出了一个模函子，并表明它可以通过有理多面体锥类别上的几何堆叠来表示。在这个框架中，带有标记点的曲线模空间之间的自然遗忘态射起到了通用曲线的作用。我们对热带几何的方法允许将热带模量问题（曲线模量或其他）扩展到对数方案。我们用它来构造一个从代数曲线的模空间到热带曲线的模空间的平滑热带化态射，并且我们证明了这个态射与所有重言态射对易。