From Curves to Tropical Jacobians and Back

Bolognese, Barbara; Brandt, Madeline; Chua, Lynn

doi:10.1007/978-1-4939-7486-3_2

Cited by 14 publications

(22 citation statements)

References 27 publications

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“…Curves, their Jacobians, and the Schottky locus have natural counterparts in the combinatorial setting of tropical geometry. We review the basics from [1,2,3,19]. The role of a curve is played by a connected metric graph Γ = (V, E, l, w).…”

Section: The Tropical Schottky Problemmentioning

confidence: 99%

Schottky algorithms: Classical meets tropical

2018

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We present a new perspective on the Schottky problem that links numerical computing with tropical geometry. The task is to decide whether a symmetric matrix defines a Jacobian, and, if so, to compute the curve and its canonical embedding. We offer solutions and their implementations in genus four, both classically and tropically. The locus of cographic matroids arises from tropicalizing the Schottky-Igusa modular form.arXiv:1707.08520v2 [math.AG] 30 Sep 2018 the latter, τ already passed that test, and we compute a curve whose Jacobian is given by τ . The recovery problem also makes sense for g = 3, both classically [4] and tropically [1, §7].This paper is organized as follows. In Section 2 we tackle the classical Schottky problem as a task in numerical algebraic geometry [6,14,24]. For g = 4, we utilize the software abelfunctions [25] to test whether the Schottky-Igusa modular form vanishes. In the affirmative case, we use a numerical version of Kempf's method [17] to compute a canonical embedding into P 3 . Our main results in Section 3 are Algorithms 3.3 and 3.5. Based on the work in [7,8,27,28], these furnish a computational solution to the tropical Schottky problem. Key ingredients are cographic matroids and the f-vectors of Voronoi polytopes.Section 4 links the classical and tropical Schottky scenarios. Theorem 4.2 expresses the edge lengths of a metric graph in terms of tropical theta constants, and Theorem 4.9 explains what happens to the Schottky-Igusa modular form in the tropical limit. We found it especially gratifying to discover how the cographic locus is encoded in the classical theory.The software we describe in this paper is made available at the supplementary website http://eecs.berkeley.edu/~chualynn/schottky (2) This contains several pieces of code for the tropical Schottky problem, as well as a more coherent Sage program for the classical Schottky problem that makes calls to abelfunctions.

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Section: The Tropical Schottky Problemmentioning

confidence: 99%

Schottky algorithms: Classical meets tropical

2018

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“…minimal Berkovich skeleton, associated to a nonarchimedean curve is an interesting one. See [9] for more on the status of this problem as well as its relationship with computing tropical Jacobians.…”

Section: Proof (Proof Of Theorem 32)mentioning

confidence: 99%

Theta Characteristics of Tropical K 4-Curves

Chan

Jiradilok

2017

Fields Institute Communications

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A K 4 -curve is a smooth, proper curve X of genus 3 over a nonarchimedean field whose Berkovich skeleton Γ is a complete graph on 4 vertices. The curve X has 28 effective theta characteristics, i.e. the 28 bitangents to a canonical embedding, while Γ has exactly seven effective tropical theta characteristics, as shown by Zharkov. We prove that the 28 effective theta characteristics of a K 4 -curve specialize to the theta characteristics of its minimal skeleton in seven groups of four.

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“…This is the dehomogenization of (2) with respect to z. Using the partial derivatives q y (x, y) = x 3 + 3xy 2 + x and q x (x, y) = 3x 2 y + y 3 + y, we compute the differential forms in (5) and (6). We choose to evaluate the integrals in (7) over the lines y = 0 and x = 0.…”

Section: Symbolic Computations For Special Quarticsmentioning

confidence: 99%

“…The dual graphs play a key role in tropical geometry, namely in the tropicalization of curves and their Jacobians. We follow the combinatorial construction in [5,Section 5]. Fix one of the graphs in Fig.…”

Section: Degenerations Of Theta Functionsmentioning

confidence: 99%

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Theta Surfaces

et al. 2020

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A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that any analytic surface that is the Minkowski sum of two space curves in two different ways is a theta surface. The four space curves that generate such a double translation structure are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.

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From Curves to Tropical Jacobians and Back

Cited by 14 publications

References 27 publications

Schottky algorithms: Classical meets tropical

Schottky algorithms: Classical meets tropical

Theta Characteristics of Tropical K 4-Curves

Theta Surfaces

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