Tropical Hurwitz Numbers

Cavalieri, Renzo; Johnson, Paul D.; Markwig, Hannah

doi:10.1017/cbo9781316569252.013

Cited by 24 publications

(65 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The zero cycles representing Hurwitz numbers are obtained as the degree of a naturally defined branch morphism on an appropriate compactification of the Hurwitz space, namely the stack of admissible covers. This recovers previously known results on double Hurwitz numbers [12]. The present framework also applies equally well to more general settings, such as higher genus targets, and arbitrary ramification profiles.…”

Section: Context and Motivationsupporting

confidence: 85%

“…6.2 by a computation on polyhedral domains in analytifications of formal tori. This provides a conceptual explanation for the determinantal formulas for tropical multiplicities obtained in [12], and related works. In particular, the result seems adaptable to more general moduli spaces of maps.…”

Section: Theorem 2 Let σ Be Any Fixed Top Dimensional Cone Of the Tromentioning

confidence: 54%

“…We show that tropicalization commutes with the natural tautological source and branch maps on these moduli spaces. Consequently, we recover the equality between tropical and classical Hurwitz numbers, originally proved via combinatorial and topological methods in [7,12].…”

Section: Introductionmentioning

confidence: 64%

“…For example, see [18,19,27]. Tropical Hurwitz theory was first introduced in [12], where the case of double Hurwitz numbers for genus 0 targets is investigated. Further steps in the theory of tropical Hurwitz covers have since been made by Bertrand et al [7], and by Buchholz and Markwig [9].…”

Section: Context and Motivationmentioning

confidence: 99%

“…In [12], the first and second authors, together with Paul Johnson, constructed and studied a tropical analogue of the Gromov-Witten moduli stack of stable maps to P 1 , relative to a two-point special branch divisor. The corresponding classical moduli space is a singular, non-equidimensional stack, and does not afford a direct application of the techniques of Abramovich, Caporaso, and Payne.…”

Section: Context and Motivationmentioning

confidence: 99%

See 4 more Smart Citations

Tropicalizing the space of admissible covers

2015

View full text Add to dashboard Cite

We study the relationship between tropical and classical Hurwitz moduli spaces. Following recent work of Abramovich, Caporaso and Payne, we outline a tropicalization for the moduli space of generalized Hurwitz covers of an arbitrary genus curve. Our approach is to appeal to the geometry of admissible covers, which compactify the Hurwitz scheme. We study the relationship between a combinatorial moduli space of tropical admissible covers and the skeleton of the Berkovich analytification of the classical space of admissible covers. We use techniques from non-archimedean geometry to show that the tropical and classical tautological maps are compatible via tropicalization, and that the degree of the classical branch map can be recovered from the tropical side. As a consequence, we obtain a proof, at the level of moduli spaces, of the equality of classical and tropical Hurwitz numbers.

show abstract

Section: Context and Motivationsupporting

confidence: 85%

Section: Theorem 2 Let σ Be Any Fixed Top Dimensional Cone Of the Tromentioning

confidence: 54%

Section: Introductionmentioning

confidence: 64%

Section: Context and Motivationmentioning

confidence: 99%

Section: Context and Motivationmentioning

confidence: 99%

See 3 more Smart Citations

Tropicalizing the space of admissible covers

2015

View full text Add to dashboard Cite

show abstract

Introduction to Modular Forms

Rose

2015

Calabi-Yau Varieties: Arithmetic, Geometry and Physics

View full text Add to dashboard Cite

show abstract

Dessins, Their Delta-Matroids and Partial Duals

Malić

2016

Symmetries in Graphs, Maps, and Polytopes

View full text Add to dashboard Cite

Given a map M on a connected and closed orientable surface, the deltamatroid of M is a combinatorial object associated to M which captures some topological information of the embedding. We explore how delta-matroids associated to dessins behave under the action of the absolute Galois group. Twists of deltamatroids are considered as well; they correspond to the recently introduced operation of partial duality of maps. Furthermore, we prove that every map has a partial dual defined over its field of moduli. A relationship between dessins, partial duals and tropical curves arising from the cartography groups of dessins is observed as well.

show abstract

Tropical Hurwitz Numbers

Cited by 24 publications

References 16 publications

Tropicalizing the space of admissible covers

Tropicalizing the space of admissible covers

Introduction to Modular Forms

Dessins, Their Delta-Matroids and Partial Duals

Contact Info

Product

Resources

About