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

Abstract

Let $(S,L_S)$ be a polarized abelian surface, and let $M=c\cdot {\pi }^{\ast }{L}_S-\alpha \cdot {\sum }_{i=1}^r{E}_i$ be a line bundle on ${\mathrm{Bl}}_r(S)$, where $\pi :{\mathrm{Bl}}_r(S)\to S$ 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.

17 pages, comments are welcome, minor changes on Question 4.4 and Theorem 4.5