TITLE: What is the exact statement about uniform boundedness of rational points on curves of genus greater than one? Singular points can be unbounded
QUESTION [7 upvotes]: According to several sources, it is conjectured (or at least believed)
that the rational points of curves over the rationals of genus $g > 1$
are uniformly bounded by $g$. E.g. here p. 1.
Assuming the curve is irreducible, the singular points (which are also
rational) can easily be made unbounded for fixed $g > 1$.
For natural $n$, define $j(n)=\prod_{i=1}^n(x-i)$.
Consider the curve $C_n : j(n)^2(x^5+13)=y^2$.
It birationally equivalent to $x'^5+13=y'^2$ which is genus $2$.
$C_n$ has the rational (singular) points $(1,0),(2,0),\ldots(n,0)$
which are unbounded, since $n$ is unbounded.
The same applies for $2$ replaced by integer $d>1$.
For $d > 2$ the curve need not be hyperelliptic.

Q. What is the exact statement about uniform boundedness?

According to both sage and magma, $C_n$ is irreducible for small
values of $n$.

REPLY [6 votes]: The precise statement of the conjecture is:

Uniformity Conjecture. Let $K$ be a number field and $g\geq2$ an integer. There exists a number $B(K,g)$ such that for any smooth curve
  $X$ of genus $g$ defined over $K$
$$|X(K)|\leq B(K,g)$$

and the original reference for it is:

Lucia Caporaso, Joe Harris, Barry Mazur, Uniformity of rational points (1997)

The conjecture refers indeed to geometric genus, following Faltings early results on finiteness of rational points.
Also note (here of example) that this can be proved conditionally on Lang's conjecture.