  Let $(S,L_{S})$ be a polarized abelian surface, and let $M = c \cdot \pi^*L_S
- \alpha \cdot \sum_{i=1}^r E_i$ be a line bundle on ${\rm Bl}_{r}(S)$, where
$\pi:{\rm Bl}_{r}(S) \rightarrow S$ is the blow-up of $S$ at $r$ general points
with exceptional divisors $E_{1},\dots,E_{r}$. In this paper, we provide a
criterion for $k$-very ampleness of $M$. Also, we deal with the case when $S$
is an arbitrary surface of Picard number one with a numerically trivial
canonical divisor.
