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Examples

• The Picard group of the spectrum of a Dedekind domain is its ideal class group.
• The invertible sheaves on projective space Pn(k) for k a field, are the twisting sheaves O ( m ) , {\displaystyle {\mathcal {O}}(m),\,} so the Picard group of Pn(k) is isomorphic to Z.
• The Picard group of the affine line with two origins over k is isomorphic to Z.
• The Picard group of the n {\displaystyle n} -dimensional complex affine space: Pic ⁡ ( C n ) = 0 {\displaystyle \operatorname {Pic} (\mathbb {C} ^{n})=0} , indeed the exponential sequence yields the following long exact sequence in cohomology ⋯ → H 1 ( C n , Z _ ) → H 1 ( C n , O C n ) → H 1 ( C n , O C n ⋆ ) → H 2 ( C n , Z _ ) → ⋯ {\displaystyle \dots \to H^{1}(\mathbb {C} ^{n},{\underline {\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},{\underline {\mathbb {Z} }})\to \cdots }

and since H k ( C n , Z _ ) ≃ H s i n g k ( C n ; Z ) {\displaystyle H^{k}(\mathbb {C} ^{n},{\underline {\mathbb {Z} }})\simeq H_{\scriptscriptstyle {\rm {sing}}}^{k}(\mathbb {C} ^{n};\mathbb {Z} )} ¹ we have H 1 ( C n , Z _ ) ≃ H 2 ( C n , Z _ ) ≃ 0 {\displaystyle H^{1}(\mathbb {C} ^{n},{\underline {\mathbb {Z} }})\simeq H^{2}(\mathbb {C} ^{n},{\underline {\mathbb {Z} }})\simeq 0} because C n {\displaystyle \mathbb {C} ^{n}} is contractible, then H 1 ( C n , O C n ) ≃ H 1 ( C n , O C n ⋆ ) {\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 the Dolbeault isomorphism to calculate H 1 ( C n , O C n ) ≃ H 1 ( C n , Ω C n 0 ) ≃ H ∂ ¯ 0 , 1 ( C n ) = 0 {\displaystyle H^{1}(\mathbb {C} ^{n},{\mathcal {O}}_{\mathbb {C} ^{n}})\simeq H^{1}(\mathbb {C} ^{n},\Omega _{\mathbb {C} ^{n}}^{0})\simeq H_{\bar {\partial }}^{0,1}(\mathbb {C} ^{n})=0} by the Dolbeault–Grothendieck lemma.

Picard scheme

The construction of a scheme structure on (representable functor version of) the Picard group, the Picard scheme, is an important step in algebraic geometry, in particular in the duality theory of abelian varieties. It was constructed by Grothendieck (1962), and also described by Mumford (1966) and Kleiman (2005).

In the cases of most importance to classical algebraic geometry, for a non-singular complete variety V over a field of characteristic zero, the connected component of the identity in the Picard scheme is an abelian variety called the Picard variety and denoted Pic0(V). The dual of the Picard variety is the Albanese variety, and in the particular case where V is a curve, the Picard variety is naturally isomorphic to the Jacobian variety of V. For fields of positive characteristic however, Igusa constructed an example of a smooth projective surface S with Pic0(S) non-reduced, and hence not an abelian variety.

The quotient Pic(V)/Pic0(V) is a finitely-generated abelian group denoted NS(V), the Néron–Severi group of V. In other words, the Picard group fits into an exact sequence

1 → P i c 0 ( V ) → P i c ( V ) → N S ( V ) → 1. {\displaystyle 1\to \mathrm {Pic} ^{0}(V)\to \mathrm {Pic} (V)\to \mathrm {NS} (V)\to 1.\,}

The fact that the rank of NS(V) is finite is Francesco Severi's theorem of the base; the rank is the Picard number of V, often denoted ρ(V). Geometrically NS(V) describes the algebraic equivalence classes of divisors on V; that is, using a stronger, non-linear equivalence relation in place of linear equivalence of divisors, the classification becomes amenable to discrete invariants. Algebraic equivalence is closely related to numerical equivalence, an essentially topological classification by intersection numbers.

Relative Picard scheme

Let f: X →S be a morphism of schemes. The relative Picard functor (or relative Picard scheme if it is a scheme) is given by:² for any S-scheme T,

Pic X / S ⁡ ( T ) = Pic ⁡ ( X T ) / f T ∗ ( Pic ⁡ ( T ) ) {\displaystyle \operatorname {Pic} _{X/S}(T)=\operatorname {Pic} (X_{T})/f_{T}^{*}(\operatorname {Pic} (T))}

where f T : X T → T {\displaystyle f_{T}:X_{T}\to T} is the base change of f and fT * is the pullback.

We say an L in Pic X / S ⁡ ( T ) {\displaystyle \operatorname {Pic} _{X/S}(T)} has degree r if for any geometric point s → T the pullback s ∗ L {\displaystyle s^{*}L} of L along s has degree r as an invertible sheaf over the fiber Xs (when the degree is defined for the Picard group of Xs.)

See also

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

Notes

• 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

References

1. Sheaf cohomology#Sheaf cohomology with constant coefficients /wiki/Sheaf_cohomology#Sheaf_cohomology_with_constant_coefficients ↩

2. Kleiman 2005, Definition 9.2.2. - 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 https://arxiv.org/abs/math/0504020 ↩