Tropicalization of the universal Jacobian

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

doi:10.46298/epiga.2022.8352

Cited by 3 publications

(2 citation statements)

References 41 publications

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“…The notions are well-known, but we include them to avoid excessive referencing and to fix notation. We will follow the definitions of graphs and their morphisms from [MMUV22], although we will substitute oriented edges in the place of half-edges in our presentation. Let Γ be a graph.…”

Section: Graphs Divisors and Flowsmentioning

confidence: 99%

“…The main technical tool that allows us to carry out this translation is the determination of a tropicalization of and the relevant auxiliary spaces. The tropicalization of is analogous to the tropicalization of the universal Jacobian, as discussed in [MMUV22], and is perhaps of independent interest to tropical geometers. The analogy with the tropicalization of the universal Jacobian can, in fact, be made precise by recognizing as the pullback of the universal Picard stack to via an Abel-Jacobi section, but we do not explore this direction here.…”

Section: Introductionmentioning

confidence: 99%

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Smooth Compactifications of the Abel-Jacobi Section

Molcho

2023

Forum of Mathematics, Sigma

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For $\theta $ a small generic universal stability condition of degree $0$ and A a vector of integers adding up to $-k(2g-2+n)$ , the spaces $\overline {\mathcal {M}}_{g,A}^\theta $ constructed in [AP21, HMP+22] are observed to lie inside the space $\textbf {Div}$ of [MW20], and their pullback under $\textbf {Rub} \to \textbf {Div}$ of loc. cit. to be smooth. This provides smooth and modular modifications $\widetilde {\mathcal {M}}_{g,A}^\theta $ of $\overline {\mathcal {M}}_{g,n}$ on which the logarithmic double ramification cycle can be calculated by several methods.

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Section: Graphs Divisors and Flowsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Smooth Compactifications of the Abel-Jacobi Section

Molcho

2023

Forum of Mathematics, Sigma

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Divisors and curves on logarithmic mapping spaces

Kennedy-Hunt,

Nabijou,

Shafi

et al. 2024

Sel. Math. New Ser.

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We determine the rational class and Picard groups of the moduli space of stable logarithmic maps in genus zero, with target projective space relative a hyperplane. For the class group we exhibit an explicit basis consisting of boundary divisors. For the Picard group we exhibit a spanning set indexed by piecewise-linear functions on the tropicalisation. In both cases a complete set of boundary relations is obtained by pulling back the WDVV relations from the space of stable curves. Our proofs hinge on a controlled technique for manufacturing test curves in logarithmic mapping spaces, opening up the topology of these spaces to further study.

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Stability conditions for line bundles on nodal curves

Pagani,

Tommasi

2024

Forum of Mathematics, Sigma

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We introduce the abstract notion of a smoothable fine compactified Jacobian of a nodal curve, and of a family of nodal curves whose general element is smooth. Then we introduce the combinatorial notion of a stability assignment for line bundles and their degenerations. We prove that smoothable fine compactified Jacobians are in bijection with these stability assignments. We then turn our attention to fine compactified universal Jacobians – that is, fine compactified Jacobians for the moduli space $\overline {\mathcal {M}}_g$ of stable curves (without marked points). We prove that every fine compactified universal Jacobian is isomorphic to the one first constructed by Caporaso, Pandharipande and Simpson in the nineties. In particular, without marked points, there exists no fine compactified universal Jacobian unless $\gcd (d+1-g, 2g-2)=1$ .

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Tropicalization of the universal Jacobian

Cited by 3 publications

References 41 publications

Smooth Compactifications of the Abel-Jacobi Section

Smooth Compactifications of the Abel-Jacobi Section

Divisors and curves on logarithmic mapping spaces

Stability conditions for line bundles on nodal curves

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