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Picard scheme

Idea

For a ringed space $X$ one can form the Picard group. When $X$ is a projective? integral scheme over $k$ the Picard group underlies a $k$-scheme. This scheme varies in a family as $X$ varies in a family. From this starting point one can naturally generalize to more general relative situations.

Definition

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

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

Representability

For this section suppose $f:X\to S$ 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 ${𝒪}_S\to f_{*}{𝒪}_X$ is universally an isomorphism (stays an isomorphism after any base change), then we have a comparison of relative Picard functors ${\mathrm{Pic}}_{X/S}\hookrightarrow {\mathrm{Pic}}_{X/S,\mathrm{zar}}\hookrightarrow {\mathrm{Pic}}_{X/S,\mathrm{et}}\hookrightarrow {\mathrm{Pic}}_{X/S,\mathrm{fppf}}$. They are all isomorphisms if $f$ has a section.

• If ${\mathrm{Pic}}_{X/S}$ 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 ${\mathrm{Pic}}_{X/S}$ is representable then a universal sheaf $𝒫$ on $X\times {\mathrm{Pic}}_{X/S}$ is called a Poincare sheaf. It is universal in the following sense: if $T\to S$ and $ℒ$ is invertible on $X_T$, then there is a unique $h:T\to {\mathrm{Pic}}_{X/S}$ such that for some $𝒩$ invertible on $T$ we get $ℒ\simeq (1\times h{)}^{*}𝒫\otimes f_T^{*}𝒩$.

• If $f$ is (Zariski) projective?, flat with integral geometric fibers then ${\mathrm{Pic}}_{X/S,\mathrm{et}}$ is representable by a separated and locally of finite type scheme over $S$.

• Grothendieck’s Generic Representability: If $f$ is proper and $S$ is integral, then there is a nonempty open $V\subset S$ such that ${\mathrm{Pic}}_{X_V/V,\mathrm{fppf}}$ is representable and is a disjoint union of open quasi-projective? subschemes.

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

Picard Stack

The Picard stack ${\mathrm{𝒫𝒾𝒸}}_{X/S}$ is just stack of invertible sheaves on $X/S$, i.e. the fiber category? over $T\to X$ is the category of line bundles on $X_T$ (not isomorphism classes). If $X$ is proper and flat, then ${\mathrm{𝒫𝒾𝒸}}_{X/S}$ is an Artin stack since ${\mathrm{𝒫𝒾𝒸}}_{X/S}=\mathrm{\mathscr{H}ℴ𝓂}(X,B{𝔾}_m)$ is the Hom stack? which is Artin.

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

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

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

References

• Springer eom: Picard variety, Picard scheme
• wikipedia Picard group
• Steven L. Kleiman, The Picard scheme, pp. 235–321 in FGA explained, MR2223410 (draft pdf), arxiv

category: algebraic geometry