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.


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



Linear system on an abelian surface
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](http://www.cmi.ac.in/~ypandey/S.Ramanan/papers/85.pdf) of Ramanan, or §10 of Birkenhake-Lange.


