Scott ranks of models of a theory

Our investigation of Scott spectra leads to the resolution (in ZFC) of a number of open problems about Scott ranks. We answer a question of Montalbán by showing, for each α<ω1, that there is a Π2in theory with no models of Scott rank less than α. We also answer a question of Knight and Calvert by showing that there are computable models of high Scott rank which are not computably approximable by models of low Scott rank. Finally, we answer a question of Sacks and Marker by showing that δ21 is the least ordinal α such that if the models of a computable theory T have Scott rank bounded below ω1, then their Scott ranks are bounded below α.