Ample Line Bundles on Blown Up Surfaces

Abstract

Given a smooth complex projective surface $S$ and an ample divisor $H$ on $S$, consider the blow up of $S$ along $k$ points in general position. Let $H^{\prime }$ be the pullback of $H$ and $E_1,...,E_k$ be the exceptional divisors. We show that $L=n{H}^{\prime }-E_1-...-E_k$ is ample if and only if $L^2$ is positive provided the integer $n$ is at least 3.

arXiv comment4 pages, AMS-TeX 2.1