Subspace configurations and low degree points on curves.(English)Zbl 07966462
Summary: 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 \(\overline{k}\), admit a map of degree at most \(d\) onto \(\mathbb{P}^1\) or an elliptic curve. For \(d \geqslant 4\) 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 \(d = 4, 5\).
MSC:
| 11G30 | Curves of arbitrary genus or genus \(\ne 1\) over global fields |
| 14H25 | Arithmetic ground fields for curves |
| 14H51 | Special divisors on curves (gonality, Brill-Noether theory) |
Cite
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