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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References
Bopp, C., Hoff, M.: Resolutions of general canonical curves on rational normal scrolls. Arch. Math. 105, 239–249 (2015)
Casnati, G., Ekedahl, T.: Covers of algebraic varieties I. A general structure theorem for covers of degree 3, 4 and enriques surfaces. J. Algebraic Geom. 5, 439–460 (1996)
Castryck, Wouter., Cools, Filip.: Linear pencils encoded in the Newton polygon, to appear in International Mathematics Research Notices (arXiv:1402.4651)
Castryck, W., Cools, F.: A minimal set of generators for the canonical ideal of a non-degenerate curve. J. Aust. Math. Soc. 98, 311–323 (2015)
Castryck, W., Cools, F.: A combinatorial interpretation for Schreyer’s tetragonal invariants. Doc. Math. 20, 927–942 (2015)
Eisenbud, D., Harris, J.: On varieties of minimal degree (a centennial account). In: Proceedings in Symposia in Pure Mathematics, vol. 46, pp. 3–13 (1987)
Kawaguchi, R.: The gonality and the Clifford index of curves on a toric surface. J. Algebra 449, 660–686 (2016)
Khovanskii, A.G.: Newton polyhedra and toroidal varieties. Funct. Anal. Appl. 11, 289–296 (1977)
Schreyer, F.-O.: Syzygies of canonical curves and special linear series. Math. Ann. 275, 105–137 (1986)
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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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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DOI: https://doi.org/10.1007/s13163-017-0224-7