Linear system on an abelian surface

Question

On a K3 surface $S$, a linear system $|C|$ is said to be hyperelliptic if the corresponding map is of degree 2 and the image is of degree $g_a(C)-1$ in $\mathbb P^{g_a}$.

For $g_a(C) > 2$, if $|C|$ is without fixed components, then it is hyperelliptic if and only if in the following cases:

$i)$ there exists an irreducible curve $E$ with $g_a=1$ such that $C E=2$

$ii)$ there exists an irreducible curve $B$ with $g_a=2$ such that $|C|=|2B|$

Moreover, a generic member of an hyperelliptic linear system is a smooth irreducible hyperelliptic curve.

$\textbf{Is there a similar result in the case of an abelian surface A?}$

Answer 1

The situation is different for abelian surfaces: if $A$ contains no elliptic curves, a linear system $|C|$ on $A$ is very ample as soon as the genus of $C$ is $\geq 8$, see this paper of Ramanan, or §10 of Birkenhake-Lange.