We know tha 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. $n=2$)?

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. $n=2$)?