We study the extent to which the widely used Boston Mecha- nism (BM) fosters ability and socioeconomic segregation across public schools. Our model encompasses an endogenous component of school quality -determined by the peer group- and an exogenous one, so that there is at least one bad school ex-ante. Even with no residential priorities, BM generates ability sorting between a priori equally good public schools: an elitist public school emerges. A richer model with some preference for closer schools and flexible residential choice does not eliminate this effect. It rather worsens the peer quality of the nonelitist good school. The existence of private schools makes the best public school more elitist, while the bad school loses peer quality. Their presence may also engender socioeconomic segregation. The main alternative assignment mechanism, Deferred Acceptance, is resilient to such sorting effects.