  Let $C \subset \mathbb{P}^2$ be an irreducible and reduced curve of degree $e
> 0$. Let $X$ be the blow up of $\mathbb{P}^2$ at $r$ distinct smooth points
$p_1,\ldots,p_r \in C$. We study line bundles on $X$ and establish conditions
for ampleness and $k$-very ampleness.
