Linear pencils encoded in the Newton polygon
 Author
 Wouter Castryck (UGent) and Filip Cools
 Organization
 Abstract
 Let C be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon Delta. It is classical that the geometric genus of C equals the number of lattice points in the interior of Delta. In this paper we give similar combinatorial interpretations for the gonality, the Clifford index, and the Clifford dimension, by removing a technical assumption from a recent result of Kawaguchi. More generally, the method shows that apart from certain wellunderstood exceptions, every basepoint free pencil whose degree equals or slightly exceeds the gonality is combinatorial, in the sense that it corresponds to projecting C along a lattice direction. Along the way we prove various features of combinatorial pencils. For instance, we give an interpretation for the scrollar invariants associated with a combinatorial pencil, and show how one can tell whether the pencil is complete or not. Among the applications, we find that every smooth projective curve admits at most one Weierstrass semigroup of embedding dimension 2, and that if a nonhyperelliptic smooth projective curve C of genus g >= 2 can be embedded in the nth Hirzebruch surface Hn, then n is actually an invariant of C. This article comes along with three Magma files: basic_commands.m, gonal.m, neargonal.m
 Keywords
 CAB CURVES, LATTICE POLYGONS, CLIFFORD INDEX, SURFACES, GONALITY, THEOREM
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Citation
Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU7174297
 MLA
 Castryck, Wouter, and Filip Cools. “Linear Pencils Encoded in the Newton Polygon.” INTERNATIONAL MATHEMATICS RESEARCH NOTICES 10 (2017): 2998–3049. Print.
 APA
 Castryck, W., & Cools, F. (2017). Linear pencils encoded in the Newton polygon. INTERNATIONAL MATHEMATICS RESEARCH NOTICES, (10), 2998–3049.
 Chicago authordate
 Castryck, Wouter, and Filip Cools. 2017. “Linear Pencils Encoded in the Newton Polygon.” International Mathematics Research Notices (10): 2998–3049.
 Chicago authordate (all authors)
 Castryck, Wouter, and Filip Cools. 2017. “Linear Pencils Encoded in the Newton Polygon.” International Mathematics Research Notices (10): 2998–3049.
 Vancouver
 1.Castryck W, Cools F. Linear pencils encoded in the Newton polygon. INTERNATIONAL MATHEMATICS RESEARCH NOTICES. 2017;(10):2998–3049.
 IEEE
 [1]W. Castryck and F. Cools, “Linear pencils encoded in the Newton polygon,” INTERNATIONAL MATHEMATICS RESEARCH NOTICES, no. 10, pp. 2998–3049, 2017.
@article{7174297, abstract = {Let C be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon Delta. It is classical that the geometric genus of C equals the number of lattice points in the interior of Delta. In this paper we give similar combinatorial interpretations for the gonality, the Clifford index, and the Clifford dimension, by removing a technical assumption from a recent result of Kawaguchi. More generally, the method shows that apart from certain wellunderstood exceptions, every basepoint free pencil whose degree equals or slightly exceeds the gonality is combinatorial, in the sense that it corresponds to projecting C along a lattice direction. Along the way we prove various features of combinatorial pencils. For instance, we give an interpretation for the scrollar invariants associated with a combinatorial pencil, and show how one can tell whether the pencil is complete or not. Among the applications, we find that every smooth projective curve admits at most one Weierstrass semigroup of embedding dimension 2, and that if a nonhyperelliptic smooth projective curve C of genus g >= 2 can be embedded in the nth Hirzebruch surface Hn, then n is actually an invariant of C. This article comes along with three Magma files: basic_commands.m, gonal.m, neargonal.m}, author = {Castryck, Wouter and Cools, Filip}, issn = {10737928}, journal = {INTERNATIONAL MATHEMATICS RESEARCH NOTICES}, keywords = {CAB CURVES,LATTICE POLYGONS,CLIFFORD INDEX,SURFACES,GONALITY,THEOREM}, language = {eng}, number = {10}, pages = {29983049}, title = {Linear pencils encoded in the Newton polygon}, url = {http://dx.doi.org/10.1093/imrn/rnw082}, year = {2017}, }
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