The gonality and the Clifford index of curves on a toric surface

Abstract

We determine the gonality and the Clifford index for curves on a compact smooth toric surface. Moreover, it is shown that their gonality are computed by pencils on the ambient surface. From the geometrical view point, this means that the gonality can be read off from the lattice polygon associated to the curve.

The lattice size of a lattice polygon

“…(ii) its gonality c equals lw(∆(f ) (1) ) + 2, unless ∆(f ) ≃ 2Υ in which case the gonality equals 3; this is [8, Cor. 6.2], whose proof strongly builds on previous work of Kawaguchi [19];…”

“…But we stress that no circular reasoning is being made: no statements in [8] make use of any of the results of this section. Moreover, some of the results of [8] that we need appear (in more disguised terms) in an earlier article by Kawaguchi [19]. Finally, we emphasize that the primary aim of this section is to convince the reader that the bounds from Theorem 1.3 often give the correct values of s 2 (U f ) and s 1,1 (U f ), and to give some evidence in favor of Conjecture 1.4; we will not push the limits of our exposition.…”

Linear Pencils Encoded in the Newton Polygon

“…It is in the same vein as Kawaguchi's notion of relative minimality [29,Def. 3.9], and can be proven more directly, by noting that ∆ is obtained from ∆ max by clipping off a number of vertices, without affecting the interior.…”

“…We also express our gratitude to Ryo Kawaguchi for sending us a preprint of [29], which was the main source of inspiration for this research. The first author thanks the Massachusetts Institute of Technology for its hospitality.…”

“…This is again inspired by [29], but thanks to our coverage of the case of smooth projective plane curves (i.e., curves of Clifford dimension 2) we are able to fill in the missing spots. In particular, we obtain a purely combinatorial criterion for determining whether C is birationally equivalent to a smooth projective curve in P 2 or not.…”