Chern Numbers of Ample Vector Bundles on Toric Surfaces

Let $\text{sE}$ be an ample rank $r$ bundle on a smooth toric projective surface, $S$, whose topological Euler characteristic is $e(S)$. In this article, we prove a number of surprisingly strong lower bounds for $c_1(\text{sE}{)}^2$ and $c_2(\text{sE})$. We also enumerate the exceptions to either the inequality $c_1(\text{sE}{)}^2\ge 4e(S)$ or the inequality $c_2(\text{sE})\ge e(S)$ holding.