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

Abstract

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 $p_1,\dots ,p_r\in C$. We study line bundles on $X$ and establish conditions for ampleness and $k$-very ampleness.

12 pages; some minor corrections and changes to exposition; to appear in J. Pure Appl. Algebra