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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 pointswith exceptional divisors $E_{1},\dots,E_{r}$. In this paper, we provide acriterion 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 trivialcanonical divisor.

Positivity of line bundles on general blow-ups of abelian surfaces