A study of the polymer structure in the vicinity of an interface with a rigid, curved wall is presented. The study is performed by means of lattice Monte Carlo simulations in melts of low and high density. Both purely entropic and energetic systems are considered. The configurational entropy of the chains as well as the cohesive interactions in the bulk polymer lead to the formation of a low-density layer in the close neighborhood of the wall. This layer is about one monomer thick in the purely entropic case, and has a thickness of about three radii of gyration in presence of energetic interactions. Increasing the wall curvature leads to an increase in density in the depleted layer, the effect being more pronounced in the energetic system. Chain end segregation at the wall is observed in all cases. This effect increases with increasing chain length and decreases with increasing wall curvature. The bonds are preferentially oriented in the direction tangential to the wall. The degree of orientation decreases with increasing wall curvature and is independent of chain length. Finally, the evolution of the density, of the segregation effect and of the bond preferential orientation with temperature is investigated. The density at the wall decreases with decreasing temperature in the melt state, while the segregation becomes more pronounced. The volume of polymer in which the structure is affected by the presence of the wall increases with decreasing temperature.