On the Degree-1 Abel Map for Nodal Curves

Sercio, Frederico; Souza, Aldi Nestor de

doi:10.1007/s00574-018-00127-8

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“…This has been started in degree one in [9] and [8], in degree two in [10], for curves of compact type and any degree in [11], and for curves with two components and any degree in [2]. For the degree-1 map, see also the recent work [12]. The general problem is still open.…”

mentioning

confidence: 99%

Abel Maps for nodal curves via tropical geometry

Abreu¹,

Andria²,

Pacini³

2020

Preprint

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mentioning

confidence: 99%

Abel Maps for nodal curves via tropical geometry

Abreu¹,

Andria²,

Pacini³

2020

Preprint

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show abstract

Abel Maps for nodal curves via tropical geometry

Abreu¹,

Andria²,

Pacini³

2022

Math. Comp.

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We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem into an explicit combinatorial problem by means of tropical and toric geometry. We show that the solution of the combinatorial problem gives rise to an explicit resolution of the Abel map. We are able to use this technique to construct and study all the Abel maps of degree one. Finally, we write an algorithm, which we implemented in SageMath to compute explicitly the solution of the combinatorial problem which, provided the existence of certain subdivisions of a hypercube, give rise to the resolution of the geometric Abel map.

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On the Degree-1 Abel Map for Nodal Curves

Cited by 2 publications

References 18 publications

Abel Maps for nodal curves via tropical geometry

Abel Maps for nodal curves via tropical geometry

Abel Maps for nodal curves via tropical geometry

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