Boundedness of the space of stable logarithmic maps

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

doi:10.4171/jems/728

Cited by 46 publications

(80 citation statements)

References 27 publications

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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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“…Modifications are compatible with toroidal morphisms in the sense that if

f : X \to Y

is toroidal, and

Σ^{'}

and

Δ^{'}

are subdivisions of

Σ (X)

and

Σ (Y)

, respectively, such that

Trop (f)

induces a morphisms

{normalΣ}^{'} \to {normalΔ}^{'}

, then

f

lifts to a toroidal morphism

f^{'} : X \times_{Σ (X)} {normalΣ}^{'} \to Y \times_{Σ (Y)} Δ

[, Lemma 1.11]. Remark The notation for toroidal modifications is due to Kazuya Kato and is not as abusive as it may seem, see [, Proposition 9.6.14; , Corollary 4.4.3].…”

Section: Cone Complexes and Toroidal Embeddingsmentioning

confidence: 99%

Intersection theory on tropicalizations of toroidal embeddings

Groß

2018

Proc. London Math. Soc.

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We show how to equip the cone complexes of toroidal embeddings with additional structure that allows to define a balancing condition for weighted subcomplexes. We then proceed to develop the foundations of an intersection theory on cone complexes including push-forwards, intersections with tropical divisors, and rational equivalence. These constructions are shown to have an algebraic interpretation: Ulirsch's tropicalizations of subvarieties of toroidal embeddings carry natural multiplicities making them tropical cycles, and the induced tropicalization map for cycles respects push-forwards, intersections with boundary divisors, and rational equivalence. As an application we prove a correspondence between the genus 0 tropical descendant Gromov-Witten invariants introduced by Markwig and Rau and the genus 0 logarithmic descendant Gromov-Witten invariants of toric varieties.

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“…When the non-proper variety admits a log smooth compactification, the recent developments on log stable maps provide us a powerful tool to study A 1 -connectedness. We refer to [Kat89] for the basics of logarithmic geometry, and to [GS13,Che14,AC14,ACMW14,Wis14] for the details of the theory of stable log maps.…”

Section: Introductionmentioning

confidence: 99%

$$\mathbb {A}^1$$-connected varieties of rank one over nonclosed fields

Chen¹,

Zhu²

2015

Math. Ann.

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Abstract. In this paper, we proved two results regarding the arithmetics of separably A 1 -connected varieties of rank one. First we proved over a large field, there is an A 1 -curve through any rational point of the boundary, if the boundary divisor is smooth and separably rationally connected. Secondly, we generalize a theorem of Hassett-Tschinkel for the Zariski density of integral points over function fields of curves.

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Boundedness of the space of stable logarithmic maps

Cited by 46 publications

References 27 publications

Functorial tropicalization of logarithmic schemes: the case of constant coefficients

Functorial tropicalization of logarithmic schemes: the case of constant coefficients

Intersection theory on tropicalizations of toroidal embeddings

$$\mathbb {A}^1$$-connected varieties of rank one over nonclosed fields

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