Functorial tropicalization of logarithmic schemes: The case of constant coefficients

Abstract

The purpose of this article is to develop foundational techniques from logarithmic geometry in order to define a functorial tropicalization map for fine and saturated logarithmic schemes in the case of constant coefficients. Our approach crucially uses the theory of fans in the sense of K. Kato and generalizes Thuillier's retraction map onto the non-Archimedean skeleton in the toroidal case. For the convenience of the reader many examples as well as an introductory treatment of the theory of Kato fans are included.

v4: 33 pages. Restructured introduction, otherwise minor changes. To appear in the Proceedings of the LMS

References in corpus (8)

• The tropicalization of the moduli space of curves
• Equations of tropical varieties
• Tropical varieties for non-archimedean analytic spaces
• Logarithmic Geometry and Moduli
• Intersection Theory on Tropicalizations of Toroidal Embeddings
• Tropical geometry of moduli spaces of weighted stable curves
• The dual complex of singularities
• Some remarks on toroidal morphisms

Cited by in corpus (20)

• A moduli stack of tropical curves
• Skeletons of stable maps I: Rational curves in toric varieties
• Decomposition of degenerate Gromov-Witten invariants
• Intersection Theory on Tropicalizations of Toroidal Embeddings
• Tropical geometry of moduli spaces of weighted stable curves
• Moduli of stable maps in genus one and logarithmic geometry II
• The Essential Skeleton of a product of degenerations
• A smooth compactification of the space of genus two curves in projective space via logarithmic geometry and Gorenstein curves
• A case study of intersections on blowups of the moduli of curves
• Enumerative geometry of elliptic curves on toric surfaces
• Towards a tropical Hodge bundle
• Tropicalizing the space of admissible covers
• Log smoothness and polystability over valuation rings
• Splitting of Gromov-Witten Invariants with Toric Gluing Strata
• Tropical $\psi$ classes
• Tropical Graph Curves
• Realizability of tropical pluri-canonical divisors
• Tropical Geometric Compactification of Moduli, II - $A_g$ case and holomorphic limits -
• A study of compatible deformations in non-Archimedean geometry
• Torification of diagonalizable group actions on toroidal schemes