Picard group

In

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

${\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*}).\,}$

For integral

The name is in honour of Émile Picard's theories, in particular of divisors on algebraic surfaces.

Examples

• The Picard group of the spectrum of a Dedekind domain is its ideal class group.
• The invertible sheaves on
  twisting sheaves
  ${\displaystyle {\mathcal {O}}(m),\,}$ 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 ${\displaystyle n}$-dimensional complex affine space: ${\displaystyle \operatorname {Pic} (\mathbb {C} ^{n})=0}$, indeed the
  exponential sequence
  yields the following long exact sequence in cohomology

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

and since ${\displaystyle H^{k}(\mathbb {C} ^{n},{\underline {\mathbb {Z} }})\simeq H_{\scriptscriptstyle {\rm {sing}}}^{k}(\mathbb {C} ^{n};\mathbb {Z} )}$^[1] we have ${\displaystyle H^{1}(\mathbb {C} ^{n},{\underline {\mathbb {Z} }})\simeq H^{2}(\mathbb {C} ^{n},{\underline {\mathbb {Z} }})\simeq 0}$ 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 the Dolbeault isomorphism to calculate ${\displaystyle H^{1}(\mathbb {C} ^{n},{\mathcal {O}}_{\mathbb {C} ^{n}})\simeq H^{1}(\mathbb {C} ^{n},\Omega _{\mathbb {C} ^{n}}^{0})\simeq H_{\bar {\partial }}^{0,1}(\mathbb {C} ^{n})=0}$ by the Dolbeault-Grothendieck lemma.

Picard scheme

The construction of a scheme structure on (

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

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,

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

where ${\displaystyle f_{T}:X_{T}\to T}$ is the base change of f and f_T ^* is the pullback.

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

See also

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

Notes

References

• 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
• OCLC 13348052
• PMID 16589782
• MR 2223410
• OCLC 171541070
• OCLC 138290