Let $X$ be a scheme. I believe that Grothendieck was the first to put a scheme structure on the Picard group $\operatorname{Pic}(X)$, where he assumed that X is projective and reduced. Then Mumford constructed a Picard scheme in the non-reduced case, still assuming projectivity, and possibly with other assumptions.
I would like to know why the projective hypothesis is necessary, and what are the key differences between the reduced and non-reduced cases.
Many thanks!
