Critical k-Very Ampleness for Abelian Surfaces

Let $(S,L)$ be a polarized abelian surface of Picard rank one and let $\phi$ be the function which takes each ample line bundle $L'$ to the least integer $k$ such that $L'$ is $k$-very ample but not $(k+1)$-very ample. We use Bridgeland's stability conditions and Fourier-Mukai techniques to give a closed formula for $\phi(L^n)$ as a function of $n$ showing that it is linear in $n$ for $n>1$. As a byproduct, we calculate the walls in the Bridgeland stability space for certain Chern characters.