Functorial tropicalization of logarithmic schemes: the case of constant coefficients

Ulirsch, Martin

doi:10.1112/plms.12031

Cited by 45 publications

(58 citation statements)

References 52 publications

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“…We briefly describe the construction of the cone complex here. A detailed discussion of this retraction map in the setting of log structures may also be found in [31].…”

Section: Definition 24mentioning

confidence: 99%

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Tropicalizing the space of admissible covers

2015

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We study the relationship between tropical and classical Hurwitz moduli spaces. Following recent work of Abramovich, Caporaso and Payne, we outline a tropicalization for the moduli space of generalized Hurwitz covers of an arbitrary genus curve. Our approach is to appeal to the geometry of admissible covers, which compactify the Hurwitz scheme. We study the relationship between a combinatorial moduli space of tropical admissible covers and the skeleton of the Berkovich analytification of the classical space of admissible covers. We use techniques from non-archimedean geometry to show that the tropical and classical tautological maps are compatible via tropicalization, and that the degree of the classical branch map can be recovered from the tropical side. As a consequence, we obtain a proof, at the level of moduli spaces, of the equality of classical and tropical Hurwitz numbers.

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“…We briefly describe the construction of the cone complex here. A detailed discussion of this retraction map in the setting of log structures may also be found in [31].…”

Section: Definition 24mentioning

confidence: 99%

“…In the language of logarithmic geometry, sheafifying M + produces the characteristic monoid sheaf, and M produces the characteristic abelian sheaf. The connections between tropical geometry and log geometry have been explored by Gross and Siebert [21,Appendix B], and Ulirsch, see [31].…”

Section: Local Toric Modelsmentioning

confidence: 99%

Tropicalizing the space of admissible covers

2015

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“…The extended skeleton (X ) of X can now be constructed via a colimit of the skeleton for the diagram (R ⇒ U ) . This is carried out using only cosmetic modifications to the arguments already present in the literature, see [1,33,35,37]. The remaining details are left to an interested reader.…”

Section: Maps To a Xmentioning

confidence: 99%

Skeletons of stable maps II: superabundant geometries

Ranganathan

2017

Res Math Sci

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We implement new techniques involving Artin fans to study the realizability of tropical stable maps in superabundant combinatorial types. Our approach is to understand the skeleton of a fundamental object in logarithmic Gromov-Witten theory-the stack of prestable maps to the Artin fan. This is used to examine the structure of the locus of realizable tropical curves and derive three principal consequences. First, we prove a realizability theorem for limits of families of tropical stable maps. Second, we extend the sufficiency of Speyer's well-spacedness condition to the case of curves with good reduction. Finally, we demonstrate the existence of liftable genus 1 superabundant tropical curves that violate the well-spacedness condition. BackgroundCentral to the application of tropical techniques to questions in algebraic geometry are so-called lifting theorems. Given a "synthetic" tropical object, such as a weighted balanced polyhedral complex, one must understand whether this object is the tropicalization of an algebraic variety. We deal in this paper with the case of curves. The tropical lifting question in this setting asks, when does an embedded tropical curve in R n arise as the tropicalization of an algebraic curve in a torus over a nonarchimedean field? This question becomes highly nontrivial in the so-called superabundant case and has been the primary obstacle to the application of tropical curve counting techniques in high genus settings. A tropical curve in R n encodes the combinatorial data in a degenerate logarithmic stable map to a toric variety. If the tropical curve is superabundant, i.e., if the tropical deformation space is larger than expected, the obstruction group of this degenerate logarithmic map is nonzero. As a result, such a map may not deform and the tropical curve may fail to be realizable. See Sect. 2.2 for a precise definition of superabundance. The earliest realization theorems for superabundant curves are due to Speyer, who observed a subtle combinatorial condition guaranteeing the realizability of superabundant genus 1 tropical curves [31]. While there has been substantial additional work in the intervening years, the general question remains mysterious [7,15,19,[23][24][25]29,34].In this note, we use recent technical breakthroughs in nonarchimedean geometry and the theory of logarithmic maps to provide a conceptual framework in which the realizability question may be approached. To demonstrate the efficacy of this framework, we use it to give simple proofs of three new results for lifting tropical curves. The same frame- 0123456789().,-: vol

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“…This is a reformulation of the polyhedral cone complexes of [KKMSD73] which realizes them within the category of monoidal spaces. In [Uli13], one further generalizes the construction of the associated Kato fan to logarithmic structures without monodromy. As in [KKMSD73], Kato fans provide a satisfying theory encoding logarithmically smooth birational modifications in terms of subdivisions of Kato fans.…”

Section: Kato Fansmentioning

confidence: 99%

“…In this case, following [Uli13], one can use the theory of Kato fans in order to define a tropicalization map trop X : X → Σ X generalizing the one of toric varieties.…”

Section: Skeletons and Tropicalizationmentioning

confidence: 99%