The tropicalization of the moduli space of curves

Abstract

We show that the skeleton of the Deligne-Mumford-Knudsen moduli stack of stable curves is naturally identified with the moduli space of extended tropical curves, and that this is compatible with the "naive" set-theoretic tropicalization map. The proof passes through general structure results on the skeleton of a toroidal Deligne-Mumford stack. Furthermore, we construct tautological forgetful, clutching, and gluing maps between moduli spaces of extended tropical curves and show that they are compatible with the analogous tautological maps in the algebraic setting.

v2: 55 pages. Expanded Section 2 with improved treatment of the category of generalized cone complexes. Clarified the role of the coarse moduli space and its analytification in the construction of the skeleton for a toroidal DM stack

References in corpus (9)

• Tropical fans and the moduli spaces of tropical curves
• On the tropical Torelli map
• Moduli spaces of rational tropical curves
• Torelli theorem for graphs and tropical curves
• Non-archimedean tame topology and stably dominated types
• Analytification is the limit of all tropicalizations
• Logarithmic Gromov-Witten invariants
• Tropical geometry and correspondence theorems via toric stacks
• Tropical Teichmuller and Siegel spaces

Cited by in corpus (62)

• Tropical curves, graph complexes, and top weight cohomology of M_g
• Tropical Amplitudes
• Equations of tropical varieties
• Functorial tropicalization of logarithmic schemes: The case of constant coefficients
• Tropical Independence I: Shapes of Divisors and a Proof of the Gieseker-Petri Theorem
• A moduli stack of tropical curves
• Tropical independence II: The maximal rank conjecture for quadrics
• Moduli of Tropical Plane Curves
• Topology of moduli spaces of tropical curves with marked points
• Skeletons of stable maps I: Rational curves in toric varieties
• Decomposition of degenerate Gromov-Witten invariants
• Tropical refined curve counting via motivic integration
• Intersection Theory on Tropicalizations of Toroidal Embeddings
• Intrinsic Mirror Symmetry
• The tropicalization of the moduli space of curves II: Topology and applications
• The universal tropical Jacobian and the skeleton of the Esteves' universal Jacobian
• Moduli of stable maps in genus one and logarithmic geometry II
• Singular semipositive metrics in non-Archimedean geometry
• A theory of dormant opers on pointed stable curves -- a proof of Joshi's conjecture
• Tropical complexes
• Functorial factorization of birational maps for qe schemes in characteristic 0
• Realizability of tropical canonical divisors
• Topology of the tropical moduli spaces $M_{2,n}$
• Tropical moduli spaces as symmetric Delta-complexes
• A module-theoretic approach to matroids
• The Kodaira dimensions of ${\overline{\mathcal{M}}}_{22}$ and ${\overline{\mathcal{M}}}_{23}$
• Factorization for stacks and boundary complexes
• Curve counting in genus one: elliptic singularities & relative geometry
• Tropicalized quartics and canonical embeddings for tropical curves of genus 3
• A case study of intersections on blowups of the moduli of curves
• On the Severi problem in arbitrary characteristic
• Tropical geometry of genus two curves
• Towards a tropical Hodge bundle
• Enumerative geometry of elliptic curves on toric surfaces
• Tropical curves and covers and their moduli spaces
• Tropicalizing the space of admissible covers
• Log smoothness and polystability over valuation rings
• Rubber tori in the boundary of expanded stable maps
• Tropicalization of the universal Jacobian
• Smoothing of Limit Linear Series of Rank One on Saturated Metrized Complexes of Algebraic Curves
• Homology of tropical fans
• Splitting of Gromov-Witten Invariants with Toric Gluing Strata
• A universal tropical Jacobian over $M_g^{\trop}$
• Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry
• Degenerated Calabi-Yau varieties with infinite components, Moduli compactifications, and limit toroidal structures
• Realizability of tropical pluri-canonical divisors
• Logarithmically Regular Morphisms
• Tropical invariants for binary quintics and reduction types of Picard curves
• Moduli spaces of codimension-one subspaces in a linear variety and their tropicalization
• Toric graph associahedra and compactifications of $M_{0,n}$
• Tropical double ramification loci
• Tropical $\psi$ classes
• Stable models of Hecke operators
• The dual complex of ${\mathcal{M}}_{1,n}({\mathbb{P}}^r,d)$ via the geometry of the Vakil--Zinger moduli space
• Constructions of superabundant tropical curves in higher genus
• Tropical Igusa Invariants
• Automorphisms of tropical Hassett spaces
• Stacky fans and tropical moduli in polymake
• Tropical compactification in log-regular varieties
• Tropical Normal Functions -- Higher Abel-Jacobi Invariants of Tropical cycles
• The ${\mathbb{S}}_n$-equivariant Euler characteristic of the moduli space of graphs
• Tropical trigonal curves