In my previous talk, I briefly mentioned a result of Serre implies the 
existence of smooth projective surfaces whose Picard scheme is 
non-reduced.  Using some ideas of Godeaux, Serre was able to produce 
smooth projective surfaces that exhibit a range of weird behavior 
(non-liftability, p-torsion in cohomology, ect).  In my talk, I will 
discuss Serre's result as well as some generalizations due to Raynaud.