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.
Categories: math.AG