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for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267026.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267027.png" /> [[#References|[3]]]. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267028.png" /> can be identified with the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267029.png" />-rational points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267031.png" /> if such exists.
 
for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267026.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267027.png" /> [[#References|[3]]]. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267028.png" /> can be identified with the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267029.png" />-rational points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267030.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267031.png" /> if such exists.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267032.png" /> is a projective morphism with geometrically-integral fibres, then the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267033.png" /> exists and is a locally finitely representable separable group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267034.png" />-scheme. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267035.png" />, then the connected component of the unit, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267036.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267037.png" /> is an algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267038.png" />-scheme, and the corresponding reduced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267039.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267040.png" /> is precisely the Picard variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267041.png" /> [[#References|[4]]]. The nilpotent elements in the local rings of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267042.png" /> give much additional information on the Picard scheme and enable one to explain various "pathologies" in algebraic geometry over a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267043.png" />. On the other hand, over a field of characteristic 0 the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267044.png" /> is always reduced [[#References|[6]]]. It is also known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267045.png" /> is a reduced scheme if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267046.png" /> is a smooth algebraic surface and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267047.png" /> [[#References|[5]]].
+
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267032.png" /> is a projective morphism with geometrically-integral fibres, then the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267033.png" /> exists and is a locally finitely representable separable group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267034.png" />-scheme. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267035.png" />, then the connected component of the unit, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267036.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267037.png" /> is an algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267038.png" />-scheme, and the corresponding reduced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267039.png" />-scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267040.png" /> is precisely the Picard variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267041.png" /> [[#References|[4]]]. The nilpotent elements in the local rings of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267042.png" /> give much additional information on the Picard scheme and enable one to explain various "pathologies" in algebraic geometry over a field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267043.png" />. On the other hand, over a field of characteristic 0 the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267044.png" /> is always reduced [[#References|[6]]]. It is also known that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267045.png" /> is a reduced scheme if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267046.png" /> is a smooth algebraic surface and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267047.png" /> [[#References|[5]]].
  
 
For any proper flat morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267048.png" /> (finitely representable if the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267049.png" /> is Noetherian) for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267050.png" />, the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267051.png" /> is an algebraic space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267052.png" /> for any base-change morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267053.png" /> [[#References|[1]]]. In particular, the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267054.png" /> is representable if the ground scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267055.png" /> is the spectrum of a local Artinian ring.
 
For any proper flat morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267048.png" /> (finitely representable if the base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267049.png" /> is Noetherian) for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267050.png" />, the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267051.png" /> is an algebraic space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267052.png" /> for any base-change morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267053.png" /> [[#References|[1]]]. In particular, the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267054.png" /> is representable if the ground scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072670/p07267055.png" /> is the spectrum of a local Artinian ring.
  
 
====References====
 
====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</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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Grothendieck,   "Eléments de géomètrie algébrique. I Le langage des schémas" ''Publ. Math. IHES'' : 4 (1960) pp. 1–228</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> D. Mumford,   "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966)</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</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</TD></TR></table>
+
<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 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 {{MR|1611170}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A. Grothendieck, "Eléments de géomètrie 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>
  
  
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====References====
 
====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)</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</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</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</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</TD></TR></table>
+
<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>

Revision as of 14:51, 24 March 2012

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
How to Cite This Entry:
Picard scheme. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Picard_scheme&oldid=15715
This article was adapted from an original article by V.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article