Picard schemes of acyclic schemes

Abstract

In his work extending rational simple connectedness to schemes with higher Picard rank, Yi Zhu introduced hypotheses for schemes insuring that the relative Picard functor is representable and is \'{e}tale locally constant with finite free stalks. We give examples showing that one cannot eliminate any of the hypotheses and still have a representable Picard functor that is locally constant with finite free stalks. We also prove that the hypotheses are compatible with composition and with hyperplane sections.