Tropicalization of the universal Jacobian

Melo, Margarida; Molcho, Samouil; Ulirsch, Martin; Viviani, Filippo

doi:10.48550/arxiv.2108.04711

Cited by 3 publications

(7 citation statements)

References 30 publications

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“…In the proof of [2] that ( 31) is a subdivision, nothing about the specific θ is used, except nondegeneracy. Their proof essentially produces a subdivision (31) for any stability condition, as discussed in [46,Theorem 5.5]. By pulling-back the Abel-Jacobi section, we obtain the same subdivision Σ θ Mg,n that we construct in Lemma 26 for the case S = M g,n .…”

Section: Proof Of Theorem 23mentioning

confidence: 89%

“…The definitions related to stability of line bundles here are adapted from [46]. Let π : C → S be a logarithmic curve.…”

Section: Stability Conditions 41 Definitionsmentioning

confidence: 99%

“…The relative inertia of P θ π over S is given by G m . The associated rigidification over S, denoted by P θ π , is a proper algebraic space over S. Both P θ π and P θ π are log smooth over S, and their tropicalizations parametrize θ-semistable tropical divisors on the tropicalization Σ C → Σ S (see [46] for details).…”

Section: Stability Conditions 41 Definitionsmentioning

confidence: 99%

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Logarithmic intersections of double ramification cycles

Holmes¹,

Schwarz²

2022

Alg. Geom.

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Let A = (a 1 , . . . , a n ) be a vector of integers which sum to k(2g − 2 + n). The double ramification cycle DR g,A ∈ CH g (M g,n ) on the moduli space of curves is the virtual class of an Abel-Jacobi locus of pointed curves (C, x 1 , . . . , x n ) satisfyingThe Abel-Jacobi construction requires log blow-ups of M g,n to resolve the indeterminacies of the Abel-Jacobi map. Holmes [29] has shown that DR g,A admits a canonical lift logDR g,A ∈ logCH g (M g,n ) to the logarithmic Chow ring, which is the limit of the intersection theories of all such blow-ups.The main result of the paper is an explicit formula for logDR g,A which lifts Pixton's formula for DR g,A . The central idea is to study the universal Jacobian over the moduli space of curves (following Caporaso [12], Kass-Pagani [37], and Abreu-Pacini [3]) for certain stability conditions. Using the criterion of Holmes-Schwarz [34], the universal double ramification theory of Bae-Holmes-Pandharipande-Schmitt-Schwarz [5] applied to the universal line bundle determines the logarithmic double ramification cycle. The resulting formula, written in the language of piecewise polynomials, depends upon the stability condition (and admits a wall-crossing study). Several examples are computed.

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Section: Proof Of Theorem 23mentioning

confidence: 89%

“…The definitions related to stability of line bundles here are adapted from [46]. Let π : C → S be a logarithmic curve.…”

Section: Stability Conditions 41 Definitionsmentioning

confidence: 99%

Section: Stability Conditions 41 Definitionsmentioning

confidence: 99%

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Logarithmic intersections of double ramification cycles

Holmes¹,

Schwarz²

2022

Alg. Geom.

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“…The notion of semistability is designed so that the right hand side is a finite union. This stratification is essentially in the paper of Caporaso [Cap94] or Oda-Seshadri [OS79], see [CC19] for in-depth examples and [MMUV21] for a modern overview.…”

Section: Comparison To the Original Hitchin Fibersmentioning

confidence: 99%

Compactifying the rank two Hitchin system via spectral data on semistable curves

Horn¹,

Möller²

2022

Preprint

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We study resolutions of the rational map to the moduli space of stable curves that associates with a point in the Hitchin base the spectral curve. In the rank two case the answer is given in terms of the space of quadratic multi-scale differentials introduced in [BCGGM3]. This space defines a compactification (of the projectivization) of the regular locus of the GL(2, C)-Hitchin base and provides a compactification of the Hitchin system by compactified Jacobians of pointed stable curves.We show how the classical GL(2, C)-and SL(2, C)-spectral correspondence extend to the compactified Hitchin system by a correspondence along an admissible cover between torsion-free rank 1 sheaves and (multi-scale) Higgs pairs of rank 2. Contents 1. Introduction 1 2. The SL(2, C)-and the GL(2, C)-Hitchin system 6 3. The compactification of strata of quadratic differentials 9 4. Resolutions of the spectral map 12 5. The modified Hitchin base 15 6. Universal compactified Jacobians 16 7. The spectral correspondence 20 8. Proof of the spectral correspondence 26 9. Comparison to the original Hitchin fibers 34 10. The fixed determinant case 38 References 43

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“…The relation between algebraic geometry and tropical geometry is made more concrete with the work of [CCUW20] that constructs Artin stacks from tropical moduli spaces. Several such tropical analogues have been constructed: the moduli space of tropical curves in [Mik07] and [BMV11], the moduli space of tropical admissible covers in [CHMR16], the universal tropical Jacobian in [AP20] and [MMUV21], and the moduli space of tropical spin curves in [CMP20].…”

Section: Introductionmentioning

confidence: 99%

On moduli spaces of roots in algebraic and tropical geometry

Abreu¹,

Pacini²,

Secco³

2022

Preprint

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In this paper we construct a tropical moduli space parametrizing roots of divisors on tropical curves. We study the relation between this space and the skeleton of Jarvis moduli space of nets of limit roots on stable curves. We show that the combinatorics of the moduli space of tropical roots is governed by the poset of flows, a poset parametrizing certain flows on graphs. We show many properties of connectedness of this poset and generalize some results on linkage of graphs. We compare our results with recent results about the tropicalization of the space of admissible covers.

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Tropicalization of the universal Jacobian

Cited by 3 publications

References 30 publications

Logarithmic intersections of double ramification cycles

Logarithmic intersections of double ramification cycles

Compactifying the rank two Hitchin system via spectral data on semistable curves

On moduli spaces of roots in algebraic and tropical geometry

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