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 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. We then give an interpretation for the scrollar invariants associated to 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 $g \geq 2$ can be embedded in the $n$th Hirzebruch surface $\mathcal{H}_n$, then $n$ is actually an invariant of $C$.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.4651
 Bibcode:
 2014arXiv1402.4651C
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics
 EPrint:
 This covers and extends sections 1 to 3.4 of our previously posted article "On the intrinsicness of the Newton polygon" (arXiv:1304.4997), which will eventually become obsolete. arXiv admin note: text overlap with arXiv:1304.4997