2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/79474Let $\sE$ 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 $c_1(\sE)^2$ and $c_2(\sE)$. We also enumerate the exceptions to either the inequality $c_1(\sE)^2\ge 4e(S)$ or the inequality $c_2(\sE)\ge e(S)$ holding.Latex file, 13 pagesAlgebraic Geometry14J60; 14Mtext