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

TL;DR

This paper computes the gonality and Clifford index of curves on smooth toric surfaces, showing these invariants can be derived from the associated lattice polygon, thus linking algebraic geometry with combinatorial data.

Contribution

It provides explicit formulas for gonality and Clifford index of curves on toric surfaces and demonstrates they are determined by the ambient surface's lattice polygon.

Findings

Gonality and Clifford index are explicitly computed for curves on toric surfaces.

Gonality is shown to be determined by pencils on the ambient surface.

Gonality can be read directly from the lattice polygon associated with the curve.

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.