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Revista Matemática Complutense

, Volume 30, Issue 2, pp 233–258 | Cite as

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

  • Wouter Castryck
  • Filip CoolsEmail author
Article

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.

Keywords

Non-degenerate curve Toric surface Lattice polygon Scrollar invariants 

Mathematics Subject Classification

Primary 14H45 Secondary 14J25 14M25 

Notes

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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Copyright information

© Universidad Complutense de Madrid 2017

Authors and Affiliations

  1. 1.Laboratoire Paul PainlevéUniversité de Lille-1Villeneuve d’Ascq CedexFrance
  2. 2.Departement ElektrotechniekKU Leuven and imecLeuvenBelgium
  3. 3.Departement WiskundeKU LeuvenLeuvenBelgium

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