2022/08/01 by Borys Kadets, Isabel Vogt, Kadets, Borys +1 · 3 citations
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2208.01067
openalex publication_date 2022/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper is devoted to understanding curves X over a number field k that possess infinitely many solutions in extensions of k of degree at most d; such solutions are the titular low degree points. For d=2,3 it is known (by the work of Harris-Silverman and Abramovich-Harris) that such curves, after a base change to k, admit a map of degree at most d onto ℙ1 or an elliptic curve. For d \geqslant 4 the analogous statement was shown to be false by Debarre and Fahlaoui. We prove that once the genus of X is high enough, the low degree points still have geometric origin: they can be obtained as pullbacks of low degree points from a lower genus curve. We introduce a discrete-geometric invariant attached to such curves: a family of subspace configurations, with many interesting properties. This structure gives a natural alternative construction of curves with many low degree points, that were first discovered by Debarre and Fahlaoui. As an application of our methods, we obtain a classification of such curves over k for d=2,3, and a classification over k for d=4,5.