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.
Categories: math.AG