For a ringed space there is its Picard group of invertible objects in the category of -modules. When is a projective integral scheme over the Picard group underlies a -scheme, this is the Picard 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.
Often one considers just the connected component of the neutral element in , and often (such as in the discussion below, beware) it is that connected component (only) which is referred to by “Picard scheme”. The difference between the two is measured by the quotient , which is called the Néron-Severi group of . Though at least for an algebraic curve, goes by a separate name: it is the Jacobian variety of .
The 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 vanishing first Chern class.
The Picard scheme is a scheme representing the relative Picard functor by . In this generality the Picard functor 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 Poincaré 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 the stack of invertible sheaves on , i.e. the fiber category? over is the groupoid of line bundles on (not just their isomorphism classes). (Hence it is the Picard groupoid equipped with geometric structure).
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.
|Calabi-Cau n-fold||line n-bundle||moduli of line n-bundles||moduli of flat/degree-0 n-bundles||Artin-Mazur formal group of deformation moduli of line n-bundles||complex oriented cohomology theory||modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory|
|unit in structure sheaf||multiplicative group/group of units||formal multiplicative group||complex K-theory|
|elliptic curve||line bundle||Picard group/Picard scheme||Jacobian||formal Picard group||elliptic cohomology||3d Chern-Simons theory/WZW model|
|K3 surface||line 2-bundle||Brauer group||intermediate Jacobian||formal Brauer group||K3 cohomology|
|Calabi-Yau 3-fold||line 3-bundle||intermediate Jacobian||CY3 cohomology||7d Chern-Simons theory/M5-brane|
wikipedia Picard group
Specifically on the Picard stack: