It is well-known that a non-singular projective curve of genus at least $2$ has a finite number of rational points.
Does this still hold true if we drop the non-singular assumption? If not, can you provide a counter-example?
2026-04-12 19:43:22.1776023002
Does the Mordell conjecture fail for singular curves?
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Ok, so if we take the normalisation of a singular curve $C$, then the normalisation is smooth. The rational points whose fibres are non-singular clearly form a finite set. But as there are at most finitely many singular points, we cannot get infinitely many rational points from those. Hence, the Mordell conjecture still holds.