Interpolation of low degree points on curves

We study points of moderately low degree on a curve $C$ over a number field, which is embedded on a nice toric surface $S$. Recently, Smith and Vogt related the linear equivalence classes of such points to intersections of $C$ with curves in the ambient surface. We show that when $C$ is sufficiently effective and ample with simple singularities, all but finitely many sufficiently low degree points are obtained via intersections of $C$ with curves in the ambient surface $S$. Our methods make use of the toric geometry of the ambient surface $S$ to produce curves which interpolate the low degree points on $C$. Furthermore, we generalise a result by Debarre and Klassen to singular plane curves and higher degrees of points.