Intrinsicness of the Newton polygon for smooth curves on \({\mathbb {P}}^1\times {\mathbb {P}}^1\)

Abstract

Let C be a smooth projective curve in \({\mathbb {P}}^1 \times {\mathbb {P}}^1\) of genus \(g\ne 4\), and assume that it is birationally equivalent to a curve defined by a Laurent polynomial that is non-degenerate with respect to its Newton polygon \(\Delta \). Then we show that the convex hull \(\Delta ^{(1)}\) of the interior lattice points of \(\Delta \) is a standard rectangle, up to a unimodular transformation. Our main auxiliary result, which we believe to be interesting in its own right, is that the first scrollar Betti numbers of \(\Delta \)-non-degenerate curves are encoded in the combinatorics of \(\Delta ^{(1)}\), if \(\Delta \) satisfies some mild conditions.

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Acknowledgements

We would like to thank Christian Bopp, Marc Coppens and Jeroen Demeyer for some interesting discussions. This research was supported by Research Project G093913N of the Research Foundation—Flanders (FWO), by the European Community’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement 615722 MOTMELSUM, and by the Labex CEMPI (ANR-11-LABX-0007-01).

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Correspondence to Filip Cools.

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Castryck, W., Cools, F. Intrinsicness of the Newton polygon for smooth curves on \({\mathbb {P}}^1\times {\mathbb {P}}^1\) . Rev Mat Complut 30, 233–258 (2017). https://doi.org/10.1007/s13163-017-0224-7

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Keywords

  • Non-degenerate curve
  • Toric surface
  • Lattice polygon
  • Scrollar invariants

Mathematics Subject Classification

  • Primary 14H45
  • Secondary 14J25
  • 14M25