Tropical geometry and correspondence theorems via toric stacks

Tyomkin, Ilya

doi:10.1007/s00208-011-0702-z

Cited by 85 publications

(95 citation statements)

References 14 publications

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“…We shall partially compactify this family to include all the large radius limit points of X Σ with Σ ∈ Fan(S). We construct partial compactifications of (C × ) S and L ⊗ C × as possibly singular toric DM stacks in the sense of Tyomkin [105]. According to Tyomkin [105, §4.1], a singular toric DM stack can be described by toric stacky data (L, Ξ, ) such that:…”

Section: Orbifold Cohomologymentioning

confidence: 99%

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Global Mirrors and Discrepant Transformations for Toric Deligne-Mumford Stacks

Iritani

2020

SIGMA

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We introduce a global Landau-Ginzburg model which is mirror to several toric Deligne-Mumford stacks and describe the change of the Gromov-Witten theories under discrepant transformations. We prove a formal decomposition of the quantum cohomology D-modules (and of the all-genus Gromov-Witten potentials) under a discrepant toric wallcrossing. In the case of weighted blowups of weak-Fano compact toric stacks along toric centres, we show that an analytic lift of the formal decomposition corresponds, via the Γintegral structure, to an Orlov-type semiorthogonal decomposition of topological K-groups. We state a conjectural functoriality of Gromov-Witten theories under discrepant transformations in terms of a Riemann-Hilbert problem.

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Section: Orbifold Cohomologymentioning

confidence: 99%

“…where Y, M are possibly singular toric DM stacks (in the sense of Tyomkin [105]). The base M of this LG model corresponds to the (extended) Kähler moduli space of X Σ (i.e.…”

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confidence: 99%

Global Mirrors and Discrepant Transformations for Toric Deligne-Mumford Stacks

Iritani

2020

SIGMA

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“…In [33, §2.1.1], an abstract tropical curve Γ C was associated to C in a canonical way by considering the dual graph of the stable reduction equipped with a natural metric coming from the geometry of the model. In terms of Berkovich spaces [7], the construction in [33] is equivalent to considering the minimal skeleton of Berkovich analytification C an , i.e. the minimal Γ ⊂ C an such that C an \ Γ is a disjoint union of discs.…”

Section: 4mentioning

confidence: 99%

Tropicalized Quartics and Canonical Embeddings for Tropical Curves of Genus 3

Hahn

Markwig

Ren

et al. 2019

International Mathematics Research Notices

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In [8], it was shown that not all abstract non-hyperelliptic tropical curves of genus 3 can be realized as a tropicalization of a quartic in R 2 . In this paper, we focus on the interior of the maximal cones in the moduli space and classify all curves which can be realized as a faithful tropicalization in a tropical plane. Reflecting the algebro-geometric world, we show that these are all curves but the tropicalizations of realizably hyperelliptic algebraic curves.Our approach is constructive: For a curve which is not the tropicalization of a hyperelliptic algebraic curve, we explicitly construct a realizable model of the tropical plane in R n , and a faithfully tropicalized quartic in it. These constructions rely on modifications resp. tropical refinements. Conversely, we prove that the tropicalizations of hyperelliptic algebraic curves cannot be embedded in such a fashion. For that, we rely on the theory of tropical divisors and embeddings from linear systems [3,21], and recent advances in the realizability of sections of the tropical canonical divisor [30].

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“…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].…”

Section: Introductionmentioning

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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Tropical geometry and correspondence theorems via toric stacks

Cited by 85 publications

References 14 publications

Global Mirrors and Discrepant Transformations for Toric Deligne-Mumford Stacks

Global Mirrors and Discrepant Transformations for Toric Deligne-Mumford Stacks

Tropicalized Quartics and Canonical Embeddings for Tropical Curves of Genus 3

Skeletons of stable maps II: superabundant geometries

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