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Torsion divisors of plane curves with maximal flexes and Zariski pairs

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  • Enrique Artal Bartolo
  • Shinzo Bannai
  • Taketo Shirane
  • Hiro‐o Tokunaga

Abstract

There is a close relationship between the embedded topology of complex plane curves and the (group‐theoretic) arithmetic of elliptic curves. In a recent paper, we studied the topology of some arrangements of curves that include a special smooth component, via the torsion properties induced by the divisors in the special curve associated to the remaining components, which is an arithmetic property. When this special curve has maximal flexes, there is a natural isomorphism between its Jacobian variety and the degree zero part of its Picard group. In this paper, we consider curve arrangements that contain a special smooth component with a maximal flex and exploit these properties to obtain Zariski tuples, which show the interplay between topology, geometry, and arithmetic.

Suggested Citation

  • Enrique Artal Bartolo & Shinzo Bannai & Taketo Shirane & Hiro‐o Tokunaga, 2023. "Torsion divisors of plane curves with maximal flexes and Zariski pairs," Mathematische Nachrichten, Wiley Blackwell, vol. 296(6), pages 2214-2235, June.
    • Handle: RePEc:bla:mathna:v:296:y:2023:i:6:p:2214-2235
    DOI: 10.1002/mana.202000319
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