Linear Pencils Encoded in the Newton Polygon

Abstract

Abstract. Let C be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon ∆. It is classical that the geometric genus of C equals the number of lattice points in the interior of ∆. 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 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 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 n th Hirzebruch surface H n , then n is actually an invariant of C.

The lattice size of a lattice polygon

“…Using the example ∆ from the end of the previous section, we see that ls (∆) = ls (∅) + 5 + 2 + 2 = 8. A Magma implementation can be found in the file basic_commands.m that accompanies [8]. For instance, the foregoing example can be treated as follows:…”

“…Let ∆ be a two-dimensional lattice polygon such that ∆ (1) ≃ dΣ for some d ≥ 1. Then there exists a unimodular transformation mapping ∆ inside (8). If not then at least one of the vertices of (d + 3)Σ is not contained in ∆.…”

Computing graded Betti tables of toric surfaces

“…These are well-defined, i.e. on each height j there is at least one lattice point in ∆, see for instance [10,Lem. 5.2].…”

Point counting on curves using a gonality preserving lift

“…Remark 4. The results from [10] are presented in characteristic zero only, but [10,Cor. 6.2] holds in finite characteristic too, as can be seen as follows.…”

“…Geometrically, what happens is that the points of C on are both mapped to (0 : 1 : 0) under projection from (0 : 0 : 0 : 1), creating a singularity there, which in terms of the 10 : else 11 : P := Random(C(F q 2 )); P := Conjugate(P ) 12 :…”