Inmathematics, thePicard group of aringed spaceX, denoted by Pic(X), is the group ofisomorphism classes ofinvertible sheaves (orline bundles) onX, with thegroup operation beingtensor product. This construction is a global version of the construction of the divisor class group, orideal class group, and is much used inalgebraic geometry and the theory ofcomplex manifolds.

Alternatively, the Picard group can be defined as thesheaf cohomology group

${\displaystyle H^1(X,{\mathcal{O}}_X^{\ast }).}$

For integralschemes the Picard group is isomorphic to the class group ofCartier divisors. For complex manifolds theexponential sheaf sequence gives basic information on the Picard group.

The name is in honour ofÉmile Picard's theories, in particular of divisors onalgebraic surfaces.

• The Picard group of thespectrum of aDedekind domain is itsideal class group.
• The invertible sheaves onprojective spacePⁿ(k) fork afield, are thetwistingsheaves${\displaystyle \mathcal{O}(m),}$ so the Picard group ofPⁿ(k) is isomorphic toZ.
• The Picard group of the affine line with two origins overk is isomorphic toZ.
• The Picard group of the${\displaystyle n}$-dimensionalcomplex affine space:${\displaystyle \mathrm{Pic}({\mathbb{C}}^n)=0}$, indeed theexponential sequence yields the following long exact sequence in cohomology

  ${\displaystyle \cdots \to H^1({\mathbb{C}}^n,\underset{\_}{\mathbb{Z}})\to H^1({\mathbb{C}}^n,{\mathcal{O}}_{{\mathbb{C}}^n})\to H^1({\mathbb{C}}^n,{\mathcal{O}}_{{\mathbb{C}}^n}^{\star })\to H^2({\mathbb{C}}^n,\underset{\_}{\mathbb{Z}})\to \cdots }$

and since${\displaystyle H^k({\mathbb{C}}^n,\underset{\_}{\mathbb{Z}})\simeq H_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^k({\mathbb{C}}^n;\mathbb{Z})}$^[1] we have${\displaystyle H^1({\mathbb{C}}^n,\underset{\_}{\mathbb{Z}})\simeq H^2({\mathbb{C}}^n,\underset{\_}{\mathbb{Z}})\simeq 0}$ because${\displaystyle {\mathbb{C}}^n}$ is contractible, then${\displaystyle H^1({\mathbb{C}}^n,{\mathcal{O}}_{{\mathbb{C}}^n})\simeq H^1({\mathbb{C}}^n,{\mathcal{O}}_{{\mathbb{C}}^n}^{\star })}$ and we can apply theDolbeault isomorphism to calculate${\displaystyle H^1({\mathbb{C}}^n,{\mathcal{O}}_{{\mathbb{C}}^n})\simeq H^1({\mathbb{C}}^n,{\mathrm{\Omega }}_{{\mathbb{C}}^n}^0)\simeq H_{\overline{\mathrm{\partial }}}^{0,1}({\mathbb{C}}^n)=0}$ by theDolbeault–Grothendieck lemma.

The construction of a scheme structure on (therepresentable functor version of) the Picard group, thePicard scheme, is an important step in algebraic geometry, in particular in theduality theory of abelian varieties. It was constructed byGrothendieck (1962), and also described byMumford (1966) andKleiman (2005).

In the cases of most importance to classical algebraic geometry, for anon-singularcomplete varietyV over afield ofcharacteristic zero, theconnected component of the identity in the Picard scheme is anabelian variety called thePicard variety and denoted Pic⁰(V). The dual of the Picard variety is theAlbanese variety, and in the particular case whereV is a curve, the Picard variety isnaturally isomorphic to theJacobian variety ofV. For fields of positive characteristic however,Igusa constructed an example of a smooth projective surfaceS with Pic⁰(S) non-reduced, and hence not anabelian variety.

The quotient Pic(V)/Pic⁰(V) is afinitely-generated abelian group denoted NS(V), theNéron–Severi group ofV. In other words, the Picard group fits into anexact sequence

${\displaystyle 1\to {\mathrm{P}\mathrm{i}\mathrm{c}}^0(V)\to \mathrm{P}\mathrm{i}\mathrm{c}(V)\to \mathrm{N}\mathrm{S}(V)\to 1.}$

The fact that the rank of NS(V) is finite isFrancesco Severi'stheorem of the base; the rank is thePicard number ofV, often denoted ρ(V). Geometrically NS(V) describes thealgebraic equivalence classes ofdivisors onV; that is, using a stronger, non-linearequivalence relation in place oflinear equivalence of divisors, the classification becomes amenable to discrete invariants. Algebraic equivalence is closely related tonumerical equivalence, an essentially topological classification byintersection numbers.

Letf:X →S be amorphism of schemes. Therelative Picard functor (orrelative Picard scheme if it is a scheme) is given by:^[2] for anyS-schemeT,

${\displaystyle {\mathrm{Pic}}_{X/S}(T)=\mathrm{Pic}(X_T)/f_T^{\ast }(\mathrm{Pic}(T))}$

where${\displaystyle f_T:X_T\to T}$ is the base change off andf_T^* is the pullback.

We say anL in${\displaystyle {\mathrm{Pic}}_{X/S}(T)}$ has degreer if for any geometric points →T the pullback${\displaystyle s^{\ast }L}$ ofL alongs has degreer as an invertible sheaf over the fiberX_s (when the degree is defined for the Picard group ofX_s.)

• Sheaf cohomology
• Chow variety
• Cartier divisor
• Holomorphic line bundle
• Ideal class group
• Arakelov class group
• Group-stack
• Picard category

• Grothendieck, A. (1962),V. Les schémas de Picard. Théorèmes d'existence, Séminaire Bourbaki, t. 14: année 1961/62, exposés 223-240, no. 7, Talk no. 232, pp. 143–161
• Grothendieck, A. (1962),VI. Les schémas de Picard. Propriétés générales, Séminaire Bourbaki, t. 14: année 1961/62, exposés 223-240, no. 7, Talk no. 236, pp. 221–243
• Hartshorne, Robin (1977),Algebraic Geometry, Berlin, New York:Springer-Verlag,ISBN 978-0-387-90244-9,MR 0463157,OCLC 13348052
• Igusa, Jun-Ichi (1955), "On some problems in abstract algebraic geometry",Proc. Natl. Acad. Sci. U.S.A.,41 (11):964–967,Bibcode:1955PNAS...41..964I,doi:10.1073/pnas.41.11.964,PMC 534315,PMID 16589782
• Kleiman, Steven L. (2005), "The Picard scheme",Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Providence, R.I.:American Mathematical Society, pp. 235–321,arXiv:math/0504020,Bibcode:2005math......4020K,MR 2223410
• Mumford, David (1966),Lectures on Curves on an Algebraic Surface, Annals of Mathematics Studies, vol. 59,Princeton University Press,ISBN 978-0-691-07993-6,MR 0209285,OCLC 171541070
• Mumford, David (1970),Abelian varieties, Oxford:Oxford University Press,ISBN 978-0-19-560528-0,OCLC 138290

• Geometry of divisors
• Scheme theory
• Abelian varieties

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