Subspace configurations and low degree points on curves

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 ([9], [2]) that such curves, after a base change to $\bar{k}$, admit a map of degree at most d onto $P^1$ or an elliptic curve. For $d⩾4$ the analogous statement was shown to be false [3]. 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 from [3]. As an application of our methods, we obtain a classification of such curves over k for $d=2,3$, and a classification over $\overline{k}$ for $d=4,5$.