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