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](http://www.ma.utexas.edu/users/voloch/Preprints/unbounded.pdf).


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 $n$.



What is the exact statement about uniform boundedness of rational points on curves of genus greater than one? Singular points can be unbounded
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](http://www.ams.org/journals/jams/1997-10-01/S0894-0347-97-00195-1/S0894-0347-97-00195-1.pdf) (1997)


The conjecture refers indeed to *geometric genus*, following Faltings early results on finiteness of rational points.


Also note ([here of example](http://arxiv.org/pdf/alg-geom/9601004v1.pdf)) that this can be proved conditionally on Lang's conjecture.


