Birational invariance in logarithmic Gromov–Witten theory

Abramovich, Dan; Wise, Jonathan

doi:10.1112/s0010437x17007667

Cited by 70 publications

(162 citation statements)

References 29 publications

(68 reference statements)

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“…In and (also see and ), the authors develop the theory of Artin fans , an incarnation of the theory of Kato fans in the category of logarithmic algebraic stacks that is more suitable to deal with logarithmic structures that have monodromy. In particuar, for every logarithmic scheme there is an Artin fan

A_{X}

and an essentially unique strict morphism

X \to {scriptA}_{X}

that is a lift of the characteristic morphism to this category.…”

Section: Overview and Statement Of The Main Resultsmentioning

confidence: 99%

Functorial tropicalization of logarithmic schemes: the case of constant coefficients

Ulirsch

2017

Proc. London Math. Soc.

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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.

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A_{X}

and an essentially unique strict morphism

X \to {scriptA}_{X}

that is a lift of the characteristic morphism to this category.…”

Section: Overview and Statement Of The Main Resultsmentioning

confidence: 99%

Functorial tropicalization of logarithmic schemes: the case of constant coefficients

Ulirsch

2017

Proc. London Math. Soc.

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“…The natural logarithmic structure in this case is coherent, but not fine. In this case, one may approximate R by sub-DVR's and pass to a limit, see [5,Appendix A.1]. Alternatively, in Theorem A, one can pass to a subsequence of [0, 1) converging to 1, parameterizing tropical maps with rational edge lengths.…”

Section: Theorem 22 the Moduli Space L (X) Of Minimal Logarithmic Stmentioning

confidence: 99%

“…In [5], Abramovich and Wise introduce a stack of prestable logarithmic morphisms to the Artin fan A X itself. Fixing the discrete data as before, we will use the following result of theirs.…”

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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“…Under mild assumptions on X there is an initial factorization of this map through a strict, representable,étale map X → Log. Following [AW13] we call X the Artin fan of X.…”

Section: Theorem 112 ([Wis14])mentioning

confidence: 99%

“…Our strategy is to use the "virtual birational invariance" of the moduli spaces, proven in [AW13] when X is logarithmically smooth. Specifically, we construct a proper and logarithmicallyétale morphism Y → X such that the characteristic sheaf M Y is globally generated (Proposition 1.3.1).…”

Section: Theorem 112 ([Wis14])mentioning

confidence: 99%