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Curves that change genus can have arbitrarily many rational points

Jose' Felipe Voloch
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Abstract

A singular curve over a non-perfect field K may not have a smooth model over K. Those are said to "change genus". If K is a global field of positive characteristic and C/K a curve that change genus, then C(K) is known to be finite. The purpose of this note is to give examples of curves with fixed relative genus, defined over K for which #C(K) is arbitrarily large. The motivation for considering this problem comes from the work of Caporaso et al. [CHM], where they show that a conjecture of Lang implies that, for a number field K, #C(K) can be bounded in terms of g and K only for all curves C/K of genus g > 1.

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Jose' Felipe Voloch. 1995-06-06. Curves that change genus can have arbitrarily many rational points. https://arxiv.org/abs/alg-geom/9506009

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