Positivity of line bundles on general blow ups of P2P2

Let X   be the blow up of P2P2 at r   general points p1,…,pr∈P2p1,…,pr∈P2. We study line bundles on X given by plane curves of degree d   passing through pipi with multiplicity at least mimi. Motivated by results in [39], we establish conditions for ampleness, very ampleness and global generation of such line bundles.