  Let $A$ be a simple abelian surface over an algebraically closed field $k$.
Let $S\subset A(k)$ be the set of torsion points $x$ of $A$ such that there
exists a genus $2$ curve $C$ and a map $f: C\to A$ such that $x$ is in the
image of $f$, and $f$ sends a Weierstrass point of $C$ to the origin of $A$.
The purpose of this note is to show that if $k$ has characteristic zero, then
$S$ is finite -- this is in contrast to the situation where $k$ is the
algebraic closure of a finite field, where $S=A(k)$, as shown by Bogomolov and
Tschinkel. We deduce that if $k=\bar{\mathbb{Q}}$, the Kummer surface
associated to $A$ has infinitely many $k$-points not contained in a rational
curve arising from a genus $2$ curve in $A$, again in contrast to the situation
over the algebraic closure of a finite field.
