The influence of the ideal edge barrier on the entry of vortices and on the magnetization curve of a thin superconducting film is studied within Ginzburg-Landau theory. A modulation instability of the superconducting order parameter is predicted to occur at the film edges when the longitudinal component of the kinematic momentum reaches a critical value at the edges. Successive instabilities lead to a sequence of vortex-chain entry fields and to a serrated structure in the magnetization curve.