Interpolation of low degree points on curves

TL;DR

This paper investigates low degree points on algebraic curves within toric surfaces, demonstrating that most such points can be obtained through intersections with ambient curves, using toric geometry techniques.

Contribution

It extends previous results by showing that low degree points on certain curves are mostly intersections with ambient surface curves, even for singular and higher degree cases.

Findings

Most low degree points are intersections with ambient curves.

Toric geometry methods effectively produce interpolating curves.

Generalizes results to singular plane curves and higher degrees.

Abstract

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.