Positivity of line bundles on general blow ups of ${\mathbb{P}}^2$

Abstract

Let $X$ be the blow up of ${\mathbb{P}}^2$ at $r$ general points $p_1,\dots ,p_r\in {\mathbb{P}}^2$. We study line bundles on $X$ given by plane curves of degree $d$ passing through $p_i$ with multiplicity $m_i$. We establish conditions for ampleness, very ampleness and global generation of such line bundles.

arXiv comment20 pages; a few minor changes since the last version