The logarithmic Picard group and its tropicalization

Molcho, Samouil; Wise, Jonathan

doi:10.48550/arxiv.1807.11364

Cited by 15 publications

(23 citation statements)

References 9 publications

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“…More recently, the polytopal decomposition of the tropical Jacobian associated to an Esteves compactified Jacobian is studied in [AP20], as a tropical analogue of the Oda-Seshadri compactified Jacobian. See also [MW18] for related constructions with logarithmic Picard groups.…”

Section: Introductionmentioning

confidence: 99%

Compactified Jacobians as Mumford models

Christ¹,

Payne²,

Shen³

2019

Preprint

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We show that relative compactified Jacobians of one-parameter smoothings of a nodal curve of genus g, as constructed by Caporaso, Esteves, and Simpson, are Mumford models of the generic fiber. Each such model is given by an admissible polytopal decomposition of the skeleton of the Jacobian. We describe the decompositions corresponding to compactified Jacobians explicitly in terms of the auxiliary stability data and find, in particular, that the Simpson compactified Jacobian in degree g is induced by the tropical break divisor decomposition.

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Section: Introductionmentioning

confidence: 99%

Compactified Jacobians as Mumford models

Christ¹,

Payne²,

Shen³

2019

Preprint

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“…A sequel on LogPic and TroPic. In [MW18] the authors have shown that in the category of logarithmic schemes (and stacks), there is a unique minimal model of the universal logarithmic Jacobian that is not representable by an algebraic stack. The study of this so-called universal logarithmic Picard variety LogPic g,n,d can be traced back to Illusie [Ill94] and has subsequently received attention in [Kaj93,Ols04,Bel16,FRTU19].…”

Section: Theorem Cmentioning

confidence: 99%

“…The study of this so-called universal logarithmic Picard variety LogPic g,n,d can be traced back to Illusie [Ill94] and has subsequently received attention in [Kaj93,Ols04,Bel16,FRTU19]. The authors of [MW18] also introduce the universal tropical Picard variety, which universally over M log g,n is denoted by T roPic g,n,d . It parametrizes tropical curves together with a torsor over the sheaf of harmonic or linear functions on Γ of degree d and it naturally arises as the tropicalization of LogPic g,n,d via a natural tropicalization morphism LogPic g,n,d → T roPic g,n,d .…”

Section: Theorem Cmentioning

confidence: 99%

Tropicalization of the universal Jacobian

Melo¹,

Molcho²,

Ulirsch³

et al. 2021

Preprint

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A. In this article we provide a stack-theoretic framework to study the universal tropical Jacobian over the moduli space of tropical curves. We develop two approaches to the process of tropicalization of the universal compactified Jacobian over the moduli space of curves -one from a logarithmic and the other from a non-Archimedean analytic point of view. The central result from both points of view is that the tropicalization of the universal compactified Jacobian is the universal tropical Jacobian and that the tropicalization maps in each of the two contexts are compatible with the tautological morphisms. In a sequel we will use the techniques developed here to provide explicit polyhedral models for the logarithmic Picard variety.

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“…Remark 2.1.2. The condition in Definition 2.1.1 says that f pulls back harmonic functions on Γ to harmonic functions on r Γ (see [MZ08,ABBR15,MW18] for more on this point of view).…”

Section: Harmonic Morphisms Let γ and R γ Be Tropical Curves A Contin...mentioning

confidence: 99%

Skeletons of Prym varieties and Brill–Noether theory

Len

Ulirsch

2021

Alg. Number Th.

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We show that the non-Archimedean skeleton of the Prym variety associated to an unramified double cover of an algebraic curve is naturally isomorphic (as a principally polarized tropical abelian variety) to the tropical Prym variety of the associated tropical double cover. This confirms a conjecture by Jensen and the first author. We prove a new upper bound on the dimension of the Prym-Brill-Noether locus for generic unramified double covers of curves with fixed even gonality on the base. Our methods also give a new proof of the classical Prym-Brill-Noether Theorem for generic unramified double covers that is originally due to Welters and Bertram.Prym-Brill-Noether theory. Let X be a smooth projective curve of genus g and π : r X → X an unramified double cover. Rather than working with the components of the kernel of the norm map Nm π , it is occasionally more convenient to consider the preimage Nm −1 π (ω X ) of the canonical line bundle in Pic 2g−2 ( r X). Since the components parameterizing line bundles of positive or negative parity are naturally a torsor over Pr(X, π), the above results transfer to this setting.This point of view paves the way to studying Brill-Noether loci in Prym varieties. Fix r ě −1. In [Wel85], Welters defines the Prym-Brill-Noether locus V r (X, π) to be the closed subsetin Pic 2g−2 ( r X). Bertram's existence theorem for Prym special divisors [Ber87, Theorem 1.4] shows that this locus is non-empty, as long asfor all curves X of genus g and all unramified double covers π : r X → X. Using these two facts, Welters' Prym-Gieseker-Petri Theorem [Wel85, Theorem 1.11] implies that, for a general unramified double cover, inequality (1) is an equality.Prym-Brill-Noether theory with gonality. In contrast, very little is known about special Prym curves. Using Theorem A, and expanding on the work of Pflueger [Pfl17a] (whose use of chains of loops builds of course on [CDPR12]), we find a previously unknown upper bound on the dimension of V r (X, π) for general unramified double covers of curves X whose gonality is either even or sufficiently large.Denote by R g the moduli space of unramified double covers, as e.g. introduced in [Bea77]. We refer to the locus of unramified double covers π : r X → X for which X has gonality k ě 2 as the k-gonal locus in R g . For r ě −1, we write

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The logarithmic Picard group and its tropicalization

Cited by 15 publications

References 9 publications

Compactified Jacobians as Mumford models

Compactified Jacobians as Mumford models

Tropicalization of the universal Jacobian

Skeletons of Prym varieties and Brill–Noether theory

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