Brill–Noether theory of curves on toric surfaces

A Laurent polynomial f   in two variables naturally describes a projective curve C(f)C(f) on a toric surface. We show that if C(f)C(f) is a smooth curve of genus at least 7, then C(f)C(f) is not Brill–Noether general. To accomplish this, we classify all Newton polygons that admit such curves whose divisors all have nonnegative Brill–Noether number.