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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Tropical curves in abelian surfaces II: Enumeration of curves in linear systems
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by Thomas Blomme
Trans. Amer. Math. Soc. 376 (2023), 5641-5691
DOI: https://doi.org/10.1090/tran/8941
Published electronically: May 31, 2023

Abstract:

In this paper, second installment in a series of three, we give a correspondence theorem to relate the count of genus $g$ curves in a fixed linear system of an abelian surface passing through $g-2$ points to a tropical count. To do this, we relate the linear system defined by a complex curve to certain integrals of $1$-forms over cycles on the curve. We then give an expression for the tropical multiplicity provided by the correspondence theorem, and prove the invariance for the associated refined multiplicity, thus introducing refined invariants of Block-Göttsche type in abelian surfaces.
References
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Bibliographic Information
  • Thomas Blomme
  • Affiliation: Université de Genève, Rue du Conseil Général 5-7, 1205 Genève, Suisse
  • MR Author ID: 1497057
  • ORCID: 0000-0001-5881-4750
  • Email: thomas.blomme@unige.ch
  • Received by editor(s): March 24, 2022
  • Received by editor(s) in revised form: November 13, 2022, January 16, 2023, and January 16, 2023
  • Published electronically: May 31, 2023
  • Additional Notes: This research was supported in part by the SNSF grant 204125.
  • © Copyright 2023 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 5641-5691
  • MSC (2020): Primary 14N10, 14T90, 11G10, 14K05, 14H99
  • DOI: https://doi.org/10.1090/tran/8941
  • MathSciNet review: 4630756