Abelian tropical covers

Len, Yoav; Ulirsch, Martin; Zakharov, Dmitry

doi:10.48550/arxiv.1906.04215

Cited by 4 publications

(5 citation statements)

References 29 publications

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“…Jensen and Len [JL18] consider unramified Z/2Z-covers of arbitrary tropical curves, while Brandt and Helminck [BH17] study Galois covers of metric trees with arbitrary cyclic Galois group and study their locus in M trop g . In [LUZ19], Len and the authors generalize all these constructions and develop a theory of Galois covers of tropical curves with arbitrary abelian Galois group. The realizability problem is also central to tropical Hurwitz theory.…”

Section: Related Work 131 Hyperelliptic and D-gonal Tropical Curvesmentioning

confidence: 99%

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Tropical double ramification loci

Ulirsch¹,

Zakharov²

2019

Preprint

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Motivated by the realizability problem for principal tropical divisors with a fixed ramification profile, we explore the tropical geometry of the double ramification locus in M g,n .There are two ways to define a tropical analogue of the double ramification locus: one as a locus of principal divisors, the other as a locus of finite effective ramified covers of a tree. We show that both loci admit a structure of a generalized cone complex in M trop g,n , with the latter contained in the former. We prove that the locus of principal divisors has cones of codimension zero in M trop g,n , while the locus of ramified covers has the expected codimension g. This solves the deformation-theoretic part of the realizability problem for principal divisors, reducing it to the so-called Hurwitz existence problem for covers of a fixed ramification type.

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Section: Related Work 131 Hyperelliptic and D-gonal Tropical Curvesmentioning

confidence: 99%

“…Hence Proposition 5.19 follows as a natural generalization to double covers of tropical curves with legs. In [LUZ19], further generalize this construction to unramified harmonic covers ϕ : Γ → Γ with an action of a finite abelian group G, and Proposition 5.19 follows from [LUZ19, Theorem 4.1].…”

Section: The Hyperelliptic Casementioning

confidence: 99%

Tropical double ramification loci

Ulirsch¹,

Zakharov²

2019

Preprint

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“…Tropical covers and their inverse images under tropicalization are the topic of [3,4]. Tropical covers with a group action appear in [38,48]. Combinatorial classifications of tropical covers of trees are studied in [24].…”

Section: Introductionmentioning

confidence: 99%

Tropical Curves and Covers and Their Moduli Spaces

Markwig

2020

Jahresber. Dtsch. Math. Ver.

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Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and covers of tropical curves maps between metric graphs satisfying certain conditions. In this short survey, we offer an introduction to the combinatorial theory of abstract tropical curves and covers of curves, and their moduli spaces, and we showcase three results demonstrating how this theory can be applied in algebraic geometry.

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“…This first step in this direction is the paper [Zak20] by the second author. It would also be interesting to determine whether the Prym construction generalizes to other tropical abelian covers (see [LUZ19]).…”

Section: Introductionmentioning

confidence: 99%

Kirchhoff's theorem for Prym varieties

Len,

Zakharov

2020

Preprint

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We prove an analogue of Kirchhoff's matrix tree theorem for computing the order of the Prym group of a free double cover of graphs, and the volume of the tropical Prym variety in the metric setting. We interpret the latter result in terms of a semi-canonical decomposition of the tropical Prym variety, via a careful study of the tropical Abel-Prym map. In particular, we show that the map is harmonic, determine its degree at every cell of the decomposition, and prove that its global degree is 2 g−1 . As a counterpart, the appendix by Sebastian Casalaina-Martin shows that the degree of the algebraic Abel-Prym map is 2 g−1 as well.

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Abelian tropical covers

Cited by 4 publications

References 29 publications

Tropical double ramification loci

Tropical double ramification loci

Tropical Curves and Covers and Their Moduli Spaces

Kirchhoff's theorem for Prym varieties

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