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Picard scheme

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Idea

For a ringed space (X,𝒪 X)(X, \mathcal{O}_X) there is its Picard group of invertible objects in the category of 𝒪 X\mathcal{O}_X-modules. When XX is a projective integral scheme over kk the Picard group underlies a kk-scheme, this is the Picard scheme Pic XPic_X. This scheme varies in a family as XX varies in a family. From this starting point one can naturally generalize to more general relative situations.

Often one considers just the connected component Pic X 0Pic_X^0 of the neutral element in Pic XPic_X, 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 Pic X/Pic X 0Pic_X/Pic_X^0, which is called the Néron-Severi group of XX. Though at least for XX an algebraic curve, Pic X 0Pic_X^0 goes by a separate name: it is the Jacobian variety of XX.

The completion of the Picard scheme at its neutral element (hence either of Pic XPic_X or Pic X 0Pic_X^0) is the formal Picard group.

Definition

The Picard variety of a complete smooth algebraic variety XX over an algebraically closed field parametrizes the Picard group of XX, 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 Pic X/S:(Sch/S) opSetPic_{X/S}: (Sch/S)^{op}\to Set by TPic(X T)/f *Pic(T)T\mapsto Pic(X_T)/f^*Pic(T). 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 f *f_* is Pic X/S(T)=H 0(T,R 1f T*𝒪 X T *)Pic_{X/S}(T)=H^0(T, R^1f_{T*}\mathcal{O}_{X_T}^*).

Note we must work with the relative functor because the global Picard functor Pic X(T)=Pic(X T)Pic_X(T)=Pic(X_T) has no hope of being representable as it is not even a sheaf. Consider any non-trivial invertible sheaf in Pic(X T)Pic(X_T). This becomes trivial on some cover {T iT}\{T_i\to T\}, so Pic(X T)Pic(X T i)Pic(X_T)\to \prod Pic(X_{T_i}) is not injective.

Representability

For this section suppose f:XSf:X\to S 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:

  • Suppose 𝒪 Sf *𝒪 X\mathcal{O}_S\to f_*\mathcal{O}_X is universally an isomorphism (stays an isomorphism after any base change), then we have a comparison of relative Picard functors Pic X/SPic X/S,zarPic X/S,etPic X/S,fppfPic_{X/S}\hookrightarrow Pic_{X/S, zar}\hookrightarrow Pic_{X/S, et}\hookrightarrow Pic_{X/S, fppf}. They are all isomorphisms if ff has a section.

  • If Pic X/SPic_{X/S} 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 Pic X/SPic_{X/S} is representable then a universal sheaf 𝒫\mathcal{P} on X×Pic X/SX\times Pic_{X/S} is called a Poincaré sheaf. It is universal in the following sense: if TST\to S and \mathcal{L} is invertible on X TX_T, then there is a unique h:TPic X/Sh:T\to Pic_{X/S} such that for some 𝒩\mathcal{N} invertible on TT we get (1×h) *𝒫f T *𝒩\mathcal{L}\simeq (1\times h)^*\mathcal{P}\otimes f_T^*\mathcal{N}.

  • If ff is (Zariski) projective, flat with integral geometric fibers then Pic X/S,etPic_{X/S, et} is representable by a separated and locally of finite type scheme over SS.

  • Grothendieck’s Generic Representability: If ff is proper and SS is integral, then there is a nonempty open VSV\subset S such that Pic X V/V,fppfPic_{X_V/V, fppf} is representable and is a disjoint union of open quasi-projective subschemes.

  • If ff is a flat, cohomologically flat in dimension 0, proper, finitely presented map of of algebraic spaces, then Pic X/SPic_{X/S} is representable by an algebraic space locally of finite presentation over SS.

Picard Stack

The Picard stack 𝒫𝒾𝒸 X/S\mathcal{Pic}_{X/S} is the stack of invertible sheaves on X/SX/S, i.e. the fiber category? over TXT\to X is the groupoid of line bundles on X TX_T (not just their isomorphism classes). (Hence it is the Picard groupoid equipped with geometric structure).

If XX is proper and flat, then 𝒫𝒾𝒸 X/S\mathcal{Pic}_{X/S} is an Artin stack since 𝒫𝒾𝒸 X/S=ℋℴ𝓂(X,B𝔾 m)\mathcal{Pic}_{X/S}=\mathcal{Hom}(X, B\mathbb{G}_m) is the Hom stack which is Artin.

Note the following “failure” of the relative Picard scheme: points on Pic X/SPic_{X/S} do not parametrize line bundles. The low degree terms of the Leray spectral sequence give the following exact sequence H 1(X T,𝔾 m)H 0(T,R 1f *𝔾 m)H 2(T,𝔾 m)H 2(X T,𝔾 m)H^1(X_T, \mathbb{G}_m)\to H^0(T, R^1f_*\mathbb{G}_m)\to H^2(T, \mathbb{G}_m)\to H^2(X_T, \mathbb{G}_m), but as noted above Pic X/S(T)=H 0(T,R 1f *𝔾 m)Pic_{X/S}(T)=H^0(T, R^1f_*\mathbb{G}_m), so we see exactly when a TT-point comes from a line bundle it is when that point maps to 00 in this sequence.

This gives us an obstruction theory lying in H 2(T,𝔾 m)H^2(T, \mathbb{G}_m) for a point corresponding to a line bundle. If Pic X/SPic_{X/S} is representable we could take T=Pic X/ST=Pic_{X/S} 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 XX. The Picard scheme is the 𝔾 m\mathbb{G}_m-rigidification of the Picard stack.

The natural map 𝒫𝒾𝒸 X/SPic X/S\mathcal{Pic}_{X/S}\to Pic_{X/S} is a 𝔾 m\mathbb{G}_m-gerbe. But isomorphism classes of 𝔾 m\mathbb{G}_m-gerbes over TT are in bijective correspondence with H 2(T,𝔾 m)H^2(T, \mathbb{G}_m) and so the above map could be thought of as a geometric realization of the universal obstruction class.

moduli spaces of line n-bundles with connection on nn-dimensional XX

nnCalabi-Cau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory
n=0n = 0unit in structure sheafmultiplicative group/group of unitsformal multiplicative groupcomplex K-theory
n=1n = 1elliptic curveline bundlePicard group/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory/WZW model
n=2n = 2K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology
n=3n = 3Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory/M5-brane
nnintermediate Jacobian

References

Specifically on the Picard stack:

Revised on June 13, 2014 04:42:46 by Urs Schreiber (145.116.130.115)