TITLE: Which curves can be found on Abelian varieties?
QUESTION [20 upvotes]: We know that each genus 2 curve is embedded into its degree 1 Jacobian. 
Under which conditions on $C$, $A$, $g$ and $n$ is it possible for a genus $g$ smooth curve $C$ to be embedded in an Abelian variety $A$ of dimension $n$?
What can be said in the case $n=2$? 
And what if $A$ is a Jacobian (and e.g. $n=2$)?

REPLY [6 votes]: Gian Pietro Pirola in  [Curves on generic Kummer varieties, 
 Duke Math. J. Volume 59, Number 3 (1989), 701-708] proves a rigidity theorem for curves of  genus $g\le q-2$ in the Kummer variety of a $q$-dimensional abelian variety. As a consequence, he shows that a generic abelian variety of dimension $q\ge 3$ does not contain hyperelliptic curves of any genus.