Mathematical group occurring in algebraic geometry and the theory of complex manifolds
In
mathematics , the Picard group of a
ringed space X , denoted by Pic(X ), is the group of
isomorphism classes of
invertible sheaves (or
line bundles ) on X , with the
group operation being
tensor product . This construction is a global version of the construction of the divisor class group, or
ideal class group , and is much used in
algebraic geometry and the theory of
complex manifolds .
Alternatively, the Picard group can be defined as the
sheaf cohomology group
H
1
(
X
,
O
X
∗
)
.
{\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*}).\,}
For integral
schemes the Picard group is isomorphic to the class group of
Cartier divisors . For complex manifolds the
exponential sheaf sequence gives basic information on the Picard group.
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
projective space P n (k ) for k a
field , are the
twisting
sheaves
O
(
m
)
,
{\displaystyle {\mathcal {O}}(m),\,}
so the Picard group of P n (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 f T * 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 X s (when the degree is defined for the Picard group of X s .)
See also
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
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