A Moduli Stack of Tropical Curves

Cavalieri, Renzo; Chan, Melody; Ulirsch, Martin; Wise, Jonathan

doi:10.1017/fms.2020.16

Cited by 49 publications

(129 citation statements)

References 71 publications

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“…For more of the general theory of logarithmic structures, we refer the reader to Kato's original article [26]. A detailed study of the relationship between tropical and logarithmic geometry from a categorical point of view is undertaken in [13]. with connected and reduced geometric fibers of dimension 1.…”

Section: Sections Of the Groupification Mmentioning

confidence: 99%

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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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Section: Sections Of the Groupification Mmentioning

confidence: 99%

“…For more about logarithmic curves and their relationship to tropical curves, the reader may consult [13].…”

Section: Definition 242mentioning

confidence: 99%

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

2019

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“…For more details on tropical moduli spaces of curves, see e.g. [1,10,11,19,29]. Recall that a tropical curve is called hyperelliptic, if it admits a divisor of degree 2 and rank 1, [12,Definition 2.3].…”

Section: Preliminariesmentioning

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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“…In this way, the tropical moduli space M trop g,n (respectively, M trop g ) of n-marked (respectively, unmarked) curves of genus g inherits the structure of an abstract cone complex. For more details on tropical moduli spaces of curves, we refer to [1,16,18,23,44].…”

Section: Tropical Moduli Spacesmentioning

confidence: 99%

“…The induced poset of skeleta is depicted on the left of Figure 1 . It has the structure of a stacky fan with seven cones, each labeled by a type and isomorphic to an orthant of dimension equal to the number of edges on the skeleton [1,16,18]. We dicuss this space in more detail in Section 2.…”

Section: Introductionmentioning

confidence: 99%

Tropical geometry of genus two curves

Cueto

Markwig

2019

Journal of Algebra

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We exploit three classical characterizations of smooth genus two curves to study their tropical and analytic counterparts. First, we provide a combinatorial rule to determine the dual graph of each algebraic curve and the metric structure on its minimal Berkovich skeleton. Our main tool is the description of genus two curves via hyperelliptic covers of the projective line with six branch points. Given the valuations of these six points and their differences, our algorithm provides an explicit harmonic 2-to-1 map to a metric tree on six leaves. Second, we use tropical modifications to produce a faithful tropicalization in dimension three starting from a planar hyperelliptic embedding.Finally, we consider the moduli space of abstract genus two tropical curves and translate the classical Igusa invariants characterizing isomorphism classes of genus two algebraic curves into the tropical realm. While these tropical Igusa functions do not yield coordinates in the tropical moduli space, we propose an alternative set of invariants that provides new length data.

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A Moduli Stack of Tropical Curves

Cited by 49 publications

References 71 publications

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

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

Tropicalized Quartics and Canonical Embeddings for Tropical Curves of Genus 3

Tropical geometry of genus two curves

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