Linear Pencils Encoded in the Newton Polygon

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 well-understood exceptions, every base-point 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 semi-group of embedding dimension 2, and that if a non-hyperelliptic smooth projective curve C of genus g >= 2 can be embedded in the nth Hirzebruch surface H-n, then n is actually an invariant of C. This article comes along with three Magma files: basic_commands.m, gonal.m, neargonal.m