Skeletons and Fans of Logarithmic Structures

Abramovich, Dan; Chen, Qile; Marcus, Steffen; Ulirsch, Martin; Wise, Jonathan

doi:10.1007/978-3-319-30945-3_9

Cited by 37 publications

(62 citation statements)

References 52 publications

(83 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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“…In contrast to a general monoidal space, a fan consists of a small (typically finite) number of points; indeed, there is a bijection between the affine open sets of a fan and its points (c.f. Lemma 4.6, [1]). Certain sufficiently nice logarithmic schemes (X, M X , O X ) (analogous to our manifolds with embedded boundary faces) are associated to a canonical fan F via a morphism (X, M ♯ X ) → F which essentially collapses various strata (analogous to our interiors of boundary faces) down to points.…”

Section: Commentarymentioning

confidence: 94%

“…That general manifolds (without embedded boundary faces) do not admit monoidal complexes can be compared to the fact that not all logarithmic schemes admit fans [1]. To define blow-up for manifolds in general, it should still be possible to explicitly patch together the local constructions in §3.1 for a suitable notion of refinement of the monoidal space (M, B ♯ M ).…”

Section: Commentarymentioning

confidence: 99%

Blow-up in Manifolds with Generalized Corners

Kottke

2017

Int Math Res Notices

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We construct a functor from the category of manifolds with generalized corners to the category of complexes of toric monoids, and for every 'refinement' of the complex associated to a manifold, we show there is a unique 'blow-up', i.e., a new manifold mapping to the original one, which satisfies a universal property and whose complex realizes the refinement. This was inspired in part by the work of Gillam and Molcho, though we work with manifolds with generalized corners, as developed by Joyce, which have embedded boundary faces, for which the appropriate objects (i.e., complexes of monoids) are simpler than they would be otherwise (i.e., monoidal spaces in the sense of Kato).

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“…If E is a Gorenstein curve with a genus 1 singularity then ω E is generated in a neighborhood of its singular point by a meromorphic form (4), with c ′ = 0, where the x i are local parameters for the branches of E at the singular point.…”

Section: Preliminariesmentioning

confidence: 99%

Moduli of stable maps in genus one and logarithmic geometry, II

2019

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This is the second in a pair of papers developing a framework to apply logarithmic methods in the study of stable maps and singular curves of genus 1. This volume focuses on logarithmic Gromov-Witten theory and tropical geometry. We construct a logarithmically nonsingular and proper moduli space of genus 1 curves mapping to any toric variety. The space is a birational modification of the principal component of the Abramovich-Chen-Gross-Siebert space of logarithmic stable maps and produces logarithmic analogues of Vakil and Zinger's genus one reduced Gromov-Witten theory. We describe the non-archimedean analytic skeleton of this moduli space and, as a consequence, obtain a full resolution to the tropical realizability problem in genus 1.

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Skeletons and Fans of Logarithmic Structures

Cited by 37 publications

References 52 publications

Functorial tropicalization of logarithmic schemes: the case of constant coefficients

Functorial tropicalization of logarithmic schemes: the case of constant coefficients

Blow-up in Manifolds with Generalized Corners

Moduli of stable maps in genus one and logarithmic geometry, II

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