  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.
