Chern Classes of Bundles over Rational Surfaces

Abstract

Consider the blow up $π: \widetilde{X} \to X$ of a rational surface $X$ at a point. Let $\widetilde{V}$ be a holomorphic bundle over $\widetilde{X}$ whose restriction to the exceptional divisor equals ${\cal{O}(j) \oplus {\cal O}(-j)$ and define $V =(π_*\widetilde{V})^{\vee \vee}.$ Friedman and Morgan gave the following bounds for the second Chern classes $j \leq c_2(\widetilde{V}) - c_2(V) \leq j^2.$ We show that these bounds are sharp.