Subspace configurations and low degree points on curves

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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 k‾, admit a map of degree at most d onto P1 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 k¯ for d=4,5.