For a ringed space one can form the Picard group. When is a projective? integral scheme over the Picard group underlies a -scheme. This scheme varies in a family as varies in a family. From this starting point one can naturally generalize to more general relative situations.
Picard variety of a complete smooth algebraic variety over an algebraically closed field parametrizes the Picard group of , more precisely the set of classes of isomorphic invertible quasicoherent sheaves with zero Chern class.
Picard scheme is a scheme representing the relative Picard functor by . Picard functor in this generality has been introduced by Grothendieck in FGA, along with the proof of representability. An alternate form of this functor in terms of the derived functor of is .
Note we must work with the relative functor because the global picard functor has no hope of being representable as it is not even a sheaf. Consider any non-trivial invertible sheaf in . This becomes trivial on some cover , so is not injective.
For this section suppose is s separated map, finite type map of schemes. Many general forms of representability have been proven several of which are given in FGA Explained. Here we list several of the common forms:
If is representable by a scheme, then by descent theory for sheaves it is representable by the same scheme in all the topologies listed above. In general, representability gives representability in a finer topology (of the ones listed).
If is representable then a universal sheaf on is called a Poincare sheaf. It is universal in the following sense: if and is invertible on , then there is a unique such that for some invertible on we get .
The Picard stack is just stack of invertible sheaves on , i.e. the fiber category? over is the category of line bundles on (not isomorphism classes). If is proper and flat, then is an Artin stack since is the Hom stack? which is Artin.
Note the following “failure” of the relative Picard scheme: points on do not parametrize line bundles. The low degree terms of the Leray spectral sequence give the following exact sequence , but as noted above , so we see exactly when a -point comes from a line bundle it is when that point maps to in this sequence.
This gives us an obstruction theory lying in for a point corresponding to a line bundle. If is representable we could take to find a universal obstruction. Intuitively this is because the Picard stack is the right object to look at for the moduli problem of line bundles over . The Picard scheme is the -rigidification of the Picard stack.
The natural map is a -gerbe. But isomorphism classes of -gerbes over are in bijective correspondence with and so the above map could be thought of as a geometric realization of the universal obstruction class.