View source for Picard scheme

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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 [[scheme]]s. 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 [[#References|[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====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top"> 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 {{MR|0260746}} {{ZBL|0205.50402}} </TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top"> C. Chevalley, "Sur la théorie de la variété de Picard" ''Amer. J. Math.'' , '''82''' (1960) pp. 435–490 {{MR|0118723}} {{ZBL|0127.37701}} </TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top"> A. Grothendieck, "Technique de descente et théorèmes d'existence en géométrie algébrique. V. Les schémas de Picard. Théorèmes d'existence" ''Sém. Bourbaki'' , '''14''' (1962) pp. 232/01–232/19 {{MR|1611170}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Grothendieck, "Eléments de géométrie algébrique. I Le langage des schémas" ''Publ. Math. IHES'' : 4 (1960) pp. 1–228 {{MR|0217083}} {{MR|0163908}} {{ZBL|0118.36206}} </TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top"> D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) {{MR|0209285}} {{ZBL|0187.42701}} </TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top"> F. Oort, "Algebraic group schemes in character zero are reduced" ''Invent. Math.'' , '''2''' : 1 (1966) pp. 79–80 {{MR|206005}} {{ZBL|}} </TD></TR>
<TR><TD valign="top">[7]</TD> <TD valign="top"> 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 {{MR|}} {{ZBL|1068.14059}} </TD></TR>
</table>

====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====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Grothendieck, "Fondements de la géométrie algébrique" , Secr. Math. Univ. Paris (1961/62) (Extracts Sem. Bourbaki 1957–1962) {{MR|1611235}} {{MR|1086880}} {{MR|0146040}} {{ZBL|0239.14002}} {{ZBL|0239.14001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A. Altman, S. Kleiman, "Compactification of the Picard scheme I" ''Adv. in Math.'' , '''35''' (1980) pp. 50–112 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Altman, S. Kleiman, "Compactification of the Picard scheme II" ''Amer. J. Math.'' , '''101''' (1979) pp. 10–41 {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> 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 {{MR|206011}} {{ZBL|0142.18402}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> F. Oort, "Sur le schéma de Picard" ''Bull. Soc. Math. France'' , '''90''' (1962) pp. 1–14 {{MR|0138627}} {{ZBL|0123.13901}} </TD></TR></table>