A moduli stack of tropical curves

Abstract

We contribute to the foundations of tropical geometry with a view towards 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.

69 pages, 14 figures. Incorporated referee's remarks. To appear in Forum Math. Sigma. Comments still very welcome!

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• The Log Product Formula
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• Tangent curves to degenerating hypersurfaces
• A classification of modular compactifications of the space of pointed elliptic curves by Gorenstein curves
• On the Severi problem in arbitrary characteristic
• Modular compactifications of ${\mathcal{M}}_{2,n}$ with Gorenstein singularities
• Rubber tori in the boundary of expanded stable maps
• Hyperelliptic Gorenstein curves and logarithmic differentials
• Stability conditions for line bundles on nodal curves
• Logarithmically Regular Morphisms
• The dual complex of ${\mathcal{M}}_{1,n}({\mathbb{P}}^r,d)$ via the geometry of the Vakil--Zinger moduli space