Lascar and Morley Ranks Differ in Differentially Closed Fields

We note here, in answer to a question of Poizat, that the Morley and Lascar ranks need not coincide in differentially closed fields. We will approach this through the (perhaps) more fundamental issue of the variation of Morley rank in families. We will be interested here only in sets of finite Morley rank. § 1 consists of some general lemmas relating the above issues. § 2 points out a family of sets of finite Morley rank, whose Morley rank exhibits discontinuous upward jumps. To make the base of the family itself have finite Morley rank, we use a theorem of Buium. We thank John Baldwin, Anand Pillay, and Wai Yan Pong for reading an earlier version of this note and suggesting improvements.