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\section*{Picard scheme}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{complex_geometry}{}\paragraph*{{Complex geometry}}\label{complex_geometry}

[[!include complex geometry - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{representability}{Representability}\dotfill \pageref*{representability} \linebreak
\noindent\hyperlink{PicardStack}{Picard Stack}\dotfill \pageref*{PicardStack} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For a [[ringed space]] $(X, \mathcal{O}_X)$ there is its [[Picard group]] of [[invertible objects]] in the category of $\mathcal{O}_X$-[[modules]]. When $X$ is a [[projective morphism|projective]] [[integral scheme]] over $k$ the [[Picard group]] underlies a $k$-scheme, this is the \emph{Picard scheme} $Pic_X$. 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.

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

The [[completion]] of the Picard scheme at its neutral element (hence either of $Pic_X$ or $Pic_X^0$) is the \emph{[[formal Picard group]]}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

The \textbf{Picard variety} of a complete [[smooth scheme|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 vanishing [[first Chern class]].

The \textbf{Picard scheme} is a scheme [[representable functor|representing]] the relative Picard functor $Pic_{X/S}: (Sch/S)^{op}\to Set$ by $T\mapsto Pic(X_T)/f^*Pic(T)$. 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 (with respect to the Zariski topology) in terms of the [[derived functor]] of $f_*$ is $Pic_{X/S}(T)=H^0(T, R^1f_{T*}\mathcal{O}_{X_T}^*)$.

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

\hypertarget{representability}{}\subsection*{{Representability}}\label{representability}

For this section suppose $f:X\to S$ is s [[separated morphism of schemes|separated]] map, [[finite type]] map of schemes. Many general forms of [[representable functor|representability]] have been proven several of which are given in \emph{[[FGA explained]]}. Here we list several of the common forms:

\begin{itemize}%
\item Suppose $\mathcal{O}_S\to f_*\mathcal{O}_X$ is universally an isomorphism (stays an [[isomorphism]] after any [[base change]]), then we have a comparison of relative Picard functors $Pic_{X/S}\hookrightarrow Pic_{X/S, zar}\hookrightarrow Pic_{X/S, et}\hookrightarrow Pic_{X/S, fppf}$. They are all isomorphisms if $f$ has a section.


\item If $Pic_{X/S}$ 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).


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


\item If $f$ is (Zariski) [[projective morphism|projective]], [[flat morphism|flat]] with [[integral scheme|integral]] geometric fibers then $Pic_{X/S, et}$ is representable by a [[separated scheme|separated]] and [[morphism of finite type|locally of finite type]] scheme over $S$.


\item Grothendieck's Generic Representability: If $f$ is [[proper morphism|proper]] and $S$ is [[integral scheme|integral]], then there is a nonempty open $V\subset S$ such that $Pic_{X_V/V, fppf}$ is representable and is a disjoint union of open [[quasi-projective scheme| quasi-projective]] subschemes.


\item If $f$ is a flat, cohomologically flat in dimension 0, proper, [[finitely presented]] map of of [[algebraic space]]s, then $Pic_{X/S}$ is representable by an algebraic space locally of finite presentation over $S$.



\end{itemize}
\hypertarget{PicardStack}{}\subsection*{{Picard Stack}}\label{PicardStack}

The \emph{[[Picard stack]]} $\mathcal{Pic}_{X/S}$ is the [[stack]] of invertible sheaves on $X/S$, i.e. the [[fiber category]] over $T\to X$ is the [[groupoid]] of [[line bundles]] on $X_T$ (not just their [[isomorphism classes]]). (Hence it is the [[Picard groupoid of a monoidal category|Picard groupoid]] equipped with geometric structure).

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

Note the following ``failure'' of the relative Picard scheme: points on $Pic_{X/S}$ do not parametrize line bundles. The low degree terms of the Leray [[spectral sequence]] give the following exact sequence $H^1(X_T, \mathbb{G}_m)\to H^0(T, R^1f_*\mathbb{G}_m)\to H^2(T, \mathbb{G}_m)\to H^2(X_T, \mathbb{G}_m)$, but as noted above $Pic_{X/S}(T)=H^0(T, R^1f_*\mathbb{G}_m)$, 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 $H^2(T, \mathbb{G}_m)$ for a point corresponding to a line bundle. If $Pic_{X/S}$ is representable we could take $T=Pic_{X/S}$ 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 $\mathbb{G}_m$-[[rigidification]] of the Picard stack.

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

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[dual abelian group scheme]]


\item [[Poincaré line bundle]]


\item [[Albanese variety]]


\item [[Prym variety]]



\end{itemize}
[[!include moduli of higher lines -- table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Springer eom: \href{http://eom.springer.de/p/p072690.htm}{Picard variety}, \href{http://eom.springer.de/p/p072670.htm}{Picard scheme}


\item wikipedia \href{http://en.wikipedia.org/wiki/Picard_group}{Picard group}


\item [[Steven Kleiman|Steven L. Kleiman]], \emph{The Picard scheme}, pp. 235--321 in [[FGA explained]], MR2223410 (draft \href{http://cdsagenda5.ictp.it//askArchive.php?categ=a0255&id=a0255s6t3&ifd=15022&down=1&type=lecture_notes}{pdf}), \href{http://arxiv.org/abs/math/0504020}{arxiv}


\item [[Akhil Mathew]], \emph{\href{http://amathew.wordpress.com/2013/03/19/the-picard-scheme-i/}{The Picard Scheme I}}, \emph{\href{http://amathew.wordpress.com/2013/03/19/the-picard-scheme-ii-deformation-theory/}{The Picard Scheme II: deformation theory}}



\end{itemize}
Specifically on the [[Picard stack]]:

\begin{itemize}%
\item [[The Stacks Project]], \emph{\href{http://stacks.math.columbia.edu/tag/0372}{The Picard stack}}

\end{itemize}
category: algebraic geometry

[[!redirects Picard schemes]]

[[!redirects Picard variety]] [[!redirects Picard varieties]]



\end{document}