We construct a foliation of a moduli space. This gives access to a conjecture on the possible size of Hecke orbits in positive characteristic, connecting to a result of Chai. We discuss properties of variation of the structure of p-divisible groups (partly joint work with Th. Zink). Written notes for Wednesday's survey talk and the result with Zink are available from http://www.math.upenn.edu/~chai/papers_pdf/survey02.pdf http://www.math.upenn.edu/~chai/papers_pdf/oort-zink.pdf