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Mordell-Weil lattices and toric decompositions of plane curves


We extend results of Cogolludo-Agustin and Libgober relating the Alexander polynomial of a plane curve C with the Mordell–Weil rank of certain isotrivial families of jacobians over \(\mathbf {P}^2\) of discriminant C. In the second part we introduce a height pairing on the (2, 3, 6) quasi-toric decompositions of a plane curve. We use this pairing and the results in the first part of the paper to construct a pair of degree 12 curves with 30 cusps and Alexander polynomial \(t^2-t+1\), but with distinct height pairing. We use the height pairing to show that these curves from a Zariski pair.

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Corresponding author

Correspondence to Remke Kloosterman.

Additional information

Remke Kloosterman would like to thank Orsola Tommasi for several comments on a previous version of this paper. Remke Kloosterman would like to thank the referee for suggesting various improvements on the presentation.

Communicated by Ngaiming Mok.

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Kloosterman, R. Mordell-Weil lattices and toric decompositions of plane curves. Math. Ann. 367, 755–783 (2017).

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Mathematics Subject Classification

  • 14H50
  • 14H20
  • 14H30
  • 14H40
  • 14J27
  • 14J30
  • 14J70