Critical k-very ampleness for abelian surfaces

Abstract

Let $(S,L)$ be a polarized abelian surface of Picard rank $1$, and let $\varphi$ be the function which takes each ample line bundle $L\text{'}$ to the least integer $k$ such that $L\text{'}$ 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 $\varphi \left(L^n\right)$ as a function of $n$, showing that it is linear in $n$ for $n>1$. As a by-product, we calculate the walls in the Bridgeland stability space for certain Chern characters.