# 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

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 $\Gamma (U ,\ {\mathcal O} _{X} )$.