Let /: X —> Y be a branched covering of compact complex surfaces, where the ramification set in X consists of smooth curves meeting with at most normal crossings and Y has ample cotangent bundle. We further assume that / is locally of form (u,υ) —> (un ,vm). We characterize ampleness of T*X . A class of examples of such X, which are branched covers of degree two, is provided. 1. Introduction. An interesting problem in surface theory is the construction and characterization of surfaces with ample cotangent bundle. They are necessarily algebraic surfaces of general type. Natural examples occur among the complete intersection surfaces of abelian varieties. More subtle examples are those constructed by Hirzebruch [6] using line-arrangements in the plane. The characterization of those of Hirzebruch's line-arrangement surfaces with ample cotangent bundle is due to Sommese [8]. In this article, we will give a characterization of ampleness of the cotangent bundle of a class of surfaces which branch cover another surface with ample cotangent bundle. We will also construct certain branched coverings of explicit line-arrangement surfaces; these constructions will again have ample cotangent bundle. For any vector bundle E over a base manifold M, the projectivization P(E) is a fiber bundle over M, with fiber Pq(E) over q e M given by T?g(E) « (2?*\0)/C*. There is a tautological linebundle ξβ over P(E) satisfying (i) ζE\F ^ « 0(1)P ^ V<? e M, and (ii) the projection pE: P(E) -» M gives /?^ (ξβ) ~ E. In the case that E = T*X we will denote p£ = Pτ*x simply by p. DEFINITION. The vector bundle E is ample if ζβ over ~P(E) is