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The resolution of the universal Abel map via tropical geometry and applications. (English) Zbl 1469.14064

The universal Abel map \(\mathcal M_{g,n} \rightarrow \mathcal J_{g,n}\) extends in general only to a rational map \(\overline{ \mathcal M}_{g,n} \dashrightarrow \overline {\mathcal J}_{g,n}\) from the Deligne-Mumford compactification of the moduli space of smooth curves to a compactification of the universal Jacobian given by some choice of universal stability condition. Roughly speaking, the issue is that the line bundles one obtains as images of the Abel map need not be stable, and a stable representative depends on the choice of a one-parameter smoothing of the curve. There are two natural approaches to this issue: first, one can modify \(\overline{ \mathcal M}_{g,n}\) to resolve the indeterminancy as for example in [D. Holmes, J. Inst. Math. Jussieu 20, No. 1, 331–359 (2021; Zbl 1462.14031)] or in [S. Marcus and J. Wise, Proc. Lond. Math. Soc. (3) 121, No. 5, 1207–1250 (2020; Zbl 1455.14021)]; or second, one can tailor a stability condition to obtain a compactified Jacobian that avoids the issue for a given Abel map as in [J. L. Kass and N. Pagani, Trans. Am. Math. Soc. 372, No. 7, 4851–4887 (2019; Zbl 1423.14187)].
In this paper, the authors follow the first approach and describe a blow-up of \(\overline{ \mathcal M}_{g,n}\) that resolves the indeterminancy of the universal Abel map. This resolution is formulated in terms of tropical geometry. Namely, the tropical universal Abel map is not a morphism of generalized cone complexes, for the analogous reason as in the algebro-geometric setting: given a divisor on a tropical curve, there is a unique stable representative linearly equivalent to it, but this representative depends on the edge lengths of the underlying graph. Refining the cone structure of the moduli space of tropical curves turns the tropical universal Abel map into a morphism of cone complexes, which describes the desired blow-up of \(\overline{ \mathcal M}_{g,n}\) by a standard construction of toric geometry. Much of the tropical analysis is done in the authors’ previous work [A. Abreu and M. Pacini, Proc. Lond. Math. Soc. (3) 120, No. 3, 328–369 (2020; Zbl 1453.14082)], to which the current paper serves as an algebro-geometric counterpart. As an application, the authors give descriptions of the algebro-geometric and tropical double ramification cycles.

MSC:

14H10 Families, moduli of curves (algebraic)
14H40 Jacobians, Prym varieties
14T90 Applications of tropical geometry

Software:

CoCoA; SageMath
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Full Text: DOI arXiv

References:

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