Picard scheme
A natural generalization of the concept of the Picard variety for a smooth algebraic variety
within the framework of the theory of schemes. To define the Picard scheme for an arbitrary
-scheme
one considers the relative Picard functor
in the category
of schemes over the scheme
. The value of this functor on an
-scheme
is the group
![]() |
where is the base-change morphism and
is the sheaf in the Grothendieck topology
of strictly-flat quasi-compact morphisms associated with the pre-sheaf
![]() |
and denotes the standard multiplicative group sheaf. If the Picard functor
is representable on
, then the
-scheme representing it is called the relative Picard scheme for the
-scheme
and is denoted by
. If
is an algebraic scheme over a certain field
having a rational
-point, then
![]() |
for any -scheme
[3]. In particular,
can be identified with the group of
-rational points
of
if such exists.
If is a projective morphism with geometrically-integral fibres, then the scheme
exists and is a locally finitely representable separable group
-scheme. If
, then the connected component of the unit,
, of
is an algebraic
-scheme, and the corresponding reduced
-scheme
is precisely the Picard variety
[4]. The nilpotent elements in the local rings of the scheme
give much additional information on the Picard scheme and enable one to explain various "pathologies" in algebraic geometry over a field of characteristic
. On the other hand, over a field of characteristic 0 the scheme
is always reduced [6]. It is also known that
is a reduced scheme if
is a smooth algebraic surface and
[5].
For any proper flat morphism (finitely representable if the base
is Noetherian) for which
, the functor
is an algebraic space over
for any base-change morphism
[1]. In particular, the functor
is representable if the ground scheme
is the spectrum of a local Artinian ring.
References
[1] | M. Artin, "Algebraization of formal moduli I" D.C. Spencer (ed.) S. Iyanaga (ed.) , Global analysis (papers in honor of K. Kodaira) , Univ. Tokyo Press (1969) pp. 21–72 MR0260746 Zbl 0205.50402 |
[2] | C. Chevalley, "Sur la théorie de la variété de Picard" Amer. J. Math. , 82 (1960) pp. 435–490 MR0118723 Zbl 0127.37701 |
[3] | A. Grothendieck, "Technique de déscente et théorèmes d'existence en géometrie algébrique. V. Les schémas de Picard. Théorèmes d'existence" Sém. Bourbaki , 14 (1962) pp. 232/01–232/19 MR1611170 |
[4] | A. Grothendieck, "Eléments de géomètrie algébrique. I Le langage des schémas" Publ. Math. IHES : 4 (1960) pp. 1–228 MR0217083 MR0163908 Zbl 0118.36206 |
[5] | D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701 |
[6] | F. Oort, "Algebraic group schemes in character zero are reduced" Invent. Math. , 2 : 1 (1966) pp. 79–80 MR206005 |
[7] | I.V Dolgachev, "Abstract algebraic geometry" J. Soviet Math. , 2 : 3 (1974) pp. 264–303 Itogi Nauk. i Tekhn. Algebra. Topol. Geom. , 10 (1972) pp. 47–112 Zbl 1068.14059 |
Comments
The standard multiplicative sheaf over a scheme assigns to an affine open set
in
the group
of units of
.
References
[a1] | A. Grothendieck, "Fondements de la géométrie algébrique" , Secr. Math. Univ. Paris (1961/62) (Extracts Sem. Bourbaki 1957–1962) MR1611235 MR1086880 MR0146040 Zbl 0239.14002 Zbl 0239.14001 |
[a2] | A. Altman, S. Kleiman, "Compactification of the Picard scheme I" Adv. in Math. , 35 (1980) pp. 50–112 |
[a3] | A. Altman, S. Kleiman, "Compactification of the Picard scheme II" Amer. J. Math. , 101 (1979) pp. 10–41 |
[a4] | J.P. Murre, "On contravariant functors from the category of preschemes over a field into the category of abelian groups (with an application to the Picard functor)" Publ. Math. IHES , 23 (1964) pp. 581–619 MR206011 Zbl 0142.18402 |
[a5] | F. Oort, "Sur le schéma de Picard" Bull. Soc. Math. France , 90 (1962) pp. 1–14 MR0138627 Zbl 0123.13901 |
Picard scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Picard_scheme&oldid=15715