Picard scheme

A natural generalization of the concept of the Picard variety $ \mathfrak P (X) $ for a smooth algebraic variety $ X $ within the framework of the theory of schemes. To define the Picard scheme for an arbitrary $ S $- scheme $ X $ one considers the relative Picard functor $ \mathop{\rm Pic}\nolimits _{X/S} $ in the category $ \mathop{\rm Sch}\nolimits /S $ of schemes over the scheme $ S $. The value of this functor on an $ S $- scheme $ S ^ \prime $ is the group $$ H ^{0} (S ^ \prime ,\ R _{fpqc} ^{1} f _{*} ^{ {\ } \prime} (G _{ {m,\ } X ^ \prime } )), $$ where $ f ^{ {\ } \prime} : \ X \times _{S} S ^ \prime \rightarrow S ^ \prime $ is the base-change morphism and $ R _{fpqc} ^{1} f _{*} ^{ {\ } \prime} (G _{ {m,\ } X ^ \prime } ) $ is the sheaf in the Grothendieck topology $ S _{fpqc} ^ \prime $ of strictly-flat quasi-compact morphisms associated with the pre-sheaf $$ T \rightarrow H ^{1} (T _{fpqc} ,\ G _{m} ) = H ^{1} (T _{ \textrm et} ,\ G _{m} ), $$ and $ G _{m} $ denotes the standard multiplicative group sheaf. If the Picard functor $ \mathop{\rm Pic}\nolimits _{X/S} $ is representable on $ \mathop{\rm Sch}\nolimits /S $, then the $ S $- scheme representing it is called the relative Picard scheme for the $ S $- scheme $ X $ and is denoted by $ \mathop{\rm Pic}\nolimits under _{X/S} $. If $ X $ is an algebraic scheme over a certain field $ k $ having a rational $ k $- point, then $$ \mathop{\rm Pic}\nolimits _{X/k} (S ^ \prime ) = \mathop{\rm Pic}\nolimits (X \times _{k} S ^ \prime )/ \mathop{\rm Pic}\nolimits (S ^ \prime ) $$ for any $ k $- scheme $ S ^ \prime $[[# References|[3]]]. In particular, $ \mathop{\rm Pic}\nolimits _{X/k} (k) = \mathop{\rm Pic}\nolimits (X) $ can be identified with the group of $ k $- rational points $ \mathop{\rm Pic}\nolimits _{X/k} (k) $ of $ \mathop{\rm Pic}\nolimits _{X/k} $ if such exists.

If $ f: \ X \rightarrow S $ is a projective morphism with geometrically-integral fibres, then the scheme $ \mathop{\rm Pic}\nolimits under _{X/S} $ exists and is a locally finitely representable separable group $ S $- scheme. If $ S = \mathop{\rm Spec}\nolimits (k) $, then the connected component of the unit, $ \mathop{\rm Pic}\nolimits under _{X/k} ^{0} $, of $ \mathop{\rm Pic}\nolimits under _{X/k} $ is an algebraic $ k $- scheme, and the corresponding reduced $ k $- scheme $ ( \mathop{\rm Pic}\nolimits _{X/k} ^{0} ) _{ {fnnme} red} $ is precisely the Picard variety $ \mathfrak P _{c} (X) $[[# References|[4]]]. The nilpotent elements in the local rings of the scheme $ \mathop{\rm Pic}\nolimits under _{X/k} ^{0} $ give much additional information on the Picard scheme and enable one to explain various "pathologies" in algebraic geometry over a field of characteristic $ p > 0 $. On the other hand, over a field of characteristic 0 the scheme $ \mathop{\rm Pic}\nolimits under _{K/k} ^{0} $ is always reduced [6]. It is also known that $ \mathop{\rm Pic}\nolimits _{F/k} $ is a reduced scheme if $ F $ is a smooth algebraic surface and $ H ^{2} (F,\ {\mathcal O} _{F} ) = 0 $[[# References|[5]]].

For any proper flat morphism $ f: \ X \rightarrow S $( finitely representable if the base $ S $ is Noetherian) for which $ f _{*} ^{ {\ } \prime} ( {\mathcal O} _{ {X} ^ \prime } ) = {\mathcal O} _{ {S} ^ \prime } $, the functor $ \mathop{\rm Pic}\nolimits _{X/S} $ is an algebraic space over $ S $ for any base-change morphism $ f ^{ {\ } \prime} : \ X ^ \prime = X \times _{S} S ^ \prime \rightarrow S $[[# References|[1]]]. In particular, the functor $ \mathop{\rm Pic}\nolimits _{X/S} $ is representable if the ground scheme $ S $ 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 $ X $ assigns to an affine open set $ U $ in $ X $ the group $ \Gamma ( U,\ {\mathcal O} _{X} ) ^{*} $ of units of $ \Gamma (U ,\ {\mathcal O} _{X} ) $.

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