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A [[Locally free sheaf|locally free sheaf]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i0525001.png" />-modules of rank 1 on a [[Ringed space|ringed space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i0525002.png" />. An equivalent definition is: A sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i0525003.png" />-modules that is locally isomorphic to the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i0525004.png" />. The invertible sheaves on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i0525005.png" />, considered up to isomorphism, form an Abelian group with respect to the operation of tensor multiplication over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i0525006.png" />. This group is called the [[Picard group|Picard group]] of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i0525007.png" />, and is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i0525008.png" />. The inverse of a sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i0525009.png" /> in this group is the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250010.png" /> dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250011.png" />. In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250012.png" /> is a [[Scheme|scheme]] (in particular, an algebraic variety) or an [[Analytic space|analytic space]], a sheaf of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250013.png" />-modules is invertible if and only if it is isomorphic to the sheaf of regular (respectively, analytic) sections of some algebraic (respectively, analytic) line bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250014.png" />.
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Invertible sheaves on schemes are closely connected with divisors (cf. [[Divisor|Divisor]]). With each Cartier divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250015.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250016.png" /> is associated an invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250017.png" />, thus defining an injective homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250019.png" /> is the group of classes of Cartier divisors on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250020.png" />. For integral schemes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250021.png" /> this homomorphism is an isomorphism.
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On a projective scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250022.png" /> Serre's twisted invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250023.png" /> can be defined. In fact, if an imbedding of the scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250024.png" /> in a projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250025.png" /> is given, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250026.png" /> corresponds to the class of a hyperplane section. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250027.png" /> is a projective space over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250028.png" />, then the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250029.png" /> is the direct image of the sheaf of linear functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250030.png" /> under the natural mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250031.png" />. The system of homogeneous coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250032.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250033.png" /> can be identified with a basis for the space of sections <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250034.png" />.
+
A [[Locally free sheaf|locally free sheaf]] of  $  {\mathcal O} _ {X} $-
 +
modules of rank 1 on a [[Ringed space|ringed space]]  $  ( X , {\mathcal O} _ {X} ) $.  
 +
An equivalent definition is: A sheaf of  $  {\mathcal O} _ {X} $-
 +
modules that is locally isomorphic to the sheaf  $  {\mathcal O} _ {X} $.  
 +
The invertible sheaves on  $  X $,  
 +
considered up to isomorphism, form an Abelian group with respect to the operation of tensor multiplication over  $  {\mathcal O} _ {X} $.  
 +
This group is called the [[Picard group|Picard group]] of the space  $  X $,
 +
and is denoted by  $  \mathop{\rm Pic}  X $.  
 +
The inverse of a sheaf  $  {\mathcal L} $
 +
in this group is the sheaf  $  {\mathcal L}  ^ {-} 1 = fs {Hom } ( {\mathcal L} , {\mathcal O} _ {X} ) $
 +
dual to  $  {\mathcal L} $.  
 +
In the case when  $  ( X , {\mathcal O} _ {X} ) $
 +
is a [[Scheme|scheme]] (in particular, an algebraic variety) or an [[Analytic space|analytic space]], a sheaf of  $  {\mathcal O} _ {X} $-
 +
modules is invertible if and only if it is isomorphic to the sheaf of regular (respectively, analytic) sections of some algebraic (respectively, analytic) line bundle over  $  X $.
  
An invertible sheaf on a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250035.png" /> is related to rational mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250036.png" /> into projective spaces. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250037.png" /> be an invertible sheaf on a scheme and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250038.png" /> be sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250039.png" /> the values of which at any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250040.png" /> generate the stalk <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250041.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250042.png" />. Then there exists a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250043.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250045.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250046.png" /> are homogeneous coordinates in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250047.png" />. An invertible sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250048.png" /> is called very ample if there exists an imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250049.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250050.png" />. An invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250051.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250052.png" /> is called ample if there exists a positive integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250053.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250054.png" /> is very ample. On a [[Noetherian scheme|Noetherian scheme]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250055.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250056.png" /> an invertible sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250057.png" /> is ample if and only if for each [[Coherent sheaf|coherent sheaf]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250058.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250059.png" /> there exists an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250060.png" /> such that the sheaf <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250061.png" /> is generated by its global sections for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250062.png" />.
+
Invertible sheaves on schemes are closely connected with divisors (cf. [[Divisor|Divisor]]). With each Cartier divisor  $  D $
 +
on  $  X $
 +
is associated an invertible sheaf $  {\mathcal O} _ {X} ( D) $,
 +
thus defining an injective homomorphism  $  \mathop{\rm Cl}  X \rightarrow  \mathop{\rm Pic}  X $,
 +
where  $  \mathop{\rm Cl}  X $
 +
is the group of classes of Cartier divisors on $  X $.  
 +
For integral schemes  $  X $
 +
this homomorphism is an isomorphism.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250063.png" /> is an ample invertible sheaf on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250064.png" /> corresponding to a divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250065.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250066.png" /> is called an [[ample divisor]]. A Cartier divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250067.png" /> on a scheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250068.png" /> that is proper and smooth over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250069.png" /> is ample if and only if for each closed integral subscheme <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250070.png" /> the intersection index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250071.png" /> is positive, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250072.png" /> (cf. [[Intersection index (in algebraic geometry)|Intersection index (in algebraic geometry)]]). For other criteria of ampleness see [[#References|[5]]]. There is also a generalization of the concept of an ample divisor on subvarieties of large codimension [[#References|[2]]].
+
On a projective scheme  $  X $
 +
Serre's twisted invertible sheaf  $  {\mathcal O} _ {X} ( 1) = {\mathcal O} ( 1) $
 +
can be defined. In fact, if an imbedding of the scheme  $  X $
 +
in a projective space  $  P  ^ {N} $
 +
is given, then  $  {\mathcal O} _ {X} $
 +
corresponds to the class of a hyperplane section. In particular, if  $  X = P  ^ {N} ( k) $
 +
is a projective space over a field  $  k $,
 +
then the sheaf  $  {\mathcal O} ( 1) $
 +
is the direct image of the sheaf of linear functions on  $  k  ^ {N+} 1 $
 +
under the natural mapping  $  k  ^ {N+} 1 \setminus  \{ 0 \} \rightarrow P  ^ {N} ( k) $.  
 +
The system of homogeneous coordinates  $  x _ {0} \dots x _ {n} $
 +
in  $  P  ^ {N} ( k) $
 +
can be identified with a basis for the space of sections  $  \Gamma ( P  ^ {N} , {\mathcal O} ( 1) ) $.
 +
 
 +
An invertible sheaf on a scheme  $  X $
 +
is related to rational mappings of  $  X $
 +
into projective spaces. Let  $  {\mathcal L} $
 +
be an invertible sheaf on a scheme and let  $  s _ {0} \dots s _ {N} $
 +
be sections of  $  {\mathcal L} $
 +
the values of which at any point  $  x \in X $
 +
generate the stalk  $  {\mathcal L} _ {x} $
 +
over  $  {\mathcal O} _ {x} $.
 +
Then there exists a unique morphism  $  \phi :  X \rightarrow P  ^ {N} ( k) $
 +
such that  $  \phi  ^ {*} {\mathcal O} ( 1) \cong {\mathcal L} $
 +
and  $  \phi  ^ {*} x _ {i} = s _ {i} $,
 +
where  $  x _ {0} \dots x _ {N} $
 +
are homogeneous coordinates in  $  P  ^ {N} ( k) $.
 +
An invertible sheaf on  $  X $
 +
is called very ample if there exists an imbedding  $  \phi : X \rightarrow P  ^ {N} ( k) $
 +
such that  $  \phi  ^ {*} {\mathcal O} ( 1) \cong {\mathcal L} $.  
 +
An invertible sheaf  $  {\mathcal L} $
 +
on  $  X $
 +
is called ample if there exists a positive integer  $  n $
 +
for which  $  {\mathcal L}  ^ {n} $
 +
is very ample. On a [[Noetherian scheme|Noetherian scheme]]  $  X $
 +
over  $  k $
 +
an invertible sheaf  $  {\mathcal L} $
 +
is ample if and only if for each [[Coherent sheaf|coherent sheaf]]  $  {\mathcal F} $
 +
on  $  X $
 +
there exists an integer  $  n _ {0} > 0 $
 +
such that the sheaf  $  {\mathcal F} \otimes {\mathcal L}  ^ {n} $
 +
is generated by its global sections for  $  n \geq  n _ {0} $.
 +
 
 +
If  $  {\mathcal L} $
 +
is an ample invertible sheaf on  $  X $
 +
corresponding to a divisor $  D $,  
 +
then $  D $
 +
is called an [[ample divisor]]. A Cartier divisor $  D $
 +
on a scheme $  X $
 +
that is proper and smooth over an algebraically closed field $  k $
 +
is ample if and only if for each closed integral subscheme $  Y \subseteq X $
 +
the intersection index $  D  ^ {r} \cdot Y $
 +
is positive, where $  r = \mathop{\rm dim}  Y $(
 +
cf. [[Intersection index (in algebraic geometry)|Intersection index (in algebraic geometry)]]). For other criteria of ampleness see [[#References|[5]]]. There is also a generalization of the concept of an ample divisor on subvarieties of large codimension [[#References|[2]]].
  
 
The concepts of very ample and ample invertible sheaves can be carried over to the case of analytic spaces (for criteria for ampleness in this situation see [[Positive vector bundle|Positive vector bundle]]).
 
The concepts of very ample and ample invertible sheaves can be carried over to the case of analytic spaces (for criteria for ampleness in this situation see [[Positive vector bundle|Positive vector bundle]]).
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Hartshorne, "Ample subvarieties of algebraic varieties" , Springer (1970) {{MR|0282977}} {{ZBL|0208.48901}} </TD></TR><TR><TD valign="top">[3]</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">[4]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[5]</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>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> R. Hartshorne, "Ample subvarieties of algebraic varieties" , Springer (1970) {{MR|0282977}} {{ZBL|0208.48901}} </TD></TR><TR><TD valign="top">[3]</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">[4]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR><TR><TD valign="top">[5]</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====
 
====Comments====
The definition of Serre's twisted invertible sheaf is not precise enough. There is an action of the multiplicative group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250073.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250074.png" /> which has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250075.png" /> as its quotient. The direct image of the structure sheaf under the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250076.png" /> splits into a direct sum of invertible sheaves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250078.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250079.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250080.png" /> via the character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052500/i05250081.png" />.
+
The definition of Serre's twisted invertible sheaf is not precise enough. There is an action of the multiplicative group $  k  ^  \times  $
 +
on $  k  ^ {N+} 1 \setminus  \{ 0 \} $
 +
which has $  P  ^ {N} ( k) $
 +
as its quotient. The direct image of the structure sheaf under the mapping $  k  ^ {N+} 1 \setminus  \{ 0 \} \rightarrow P  ^ {N} ( k) $
 +
splits into a direct sum of invertible sheaves $  {\mathcal O} ( n) $,  
 +
$  n \in \mathbf Z $,  
 +
such that $  k  ^  \times  $
 +
acts on $  {\mathcal O} ( n) $
 +
via the character $  t \rightarrow t  ^ {n} $.

Revision as of 22:13, 5 June 2020


A locally free sheaf of $ {\mathcal O} _ {X} $- modules of rank 1 on a ringed space $ ( X , {\mathcal O} _ {X} ) $. An equivalent definition is: A sheaf of $ {\mathcal O} _ {X} $- modules that is locally isomorphic to the sheaf $ {\mathcal O} _ {X} $. The invertible sheaves on $ X $, considered up to isomorphism, form an Abelian group with respect to the operation of tensor multiplication over $ {\mathcal O} _ {X} $. This group is called the Picard group of the space $ X $, and is denoted by $ \mathop{\rm Pic} X $. The inverse of a sheaf $ {\mathcal L} $ in this group is the sheaf $ {\mathcal L} ^ {-} 1 = fs {Hom } ( {\mathcal L} , {\mathcal O} _ {X} ) $ dual to $ {\mathcal L} $. In the case when $ ( X , {\mathcal O} _ {X} ) $ is a scheme (in particular, an algebraic variety) or an analytic space, a sheaf of $ {\mathcal O} _ {X} $- modules is invertible if and only if it is isomorphic to the sheaf of regular (respectively, analytic) sections of some algebraic (respectively, analytic) line bundle over $ X $.

Invertible sheaves on schemes are closely connected with divisors (cf. Divisor). With each Cartier divisor $ D $ on $ X $ is associated an invertible sheaf $ {\mathcal O} _ {X} ( D) $, thus defining an injective homomorphism $ \mathop{\rm Cl} X \rightarrow \mathop{\rm Pic} X $, where $ \mathop{\rm Cl} X $ is the group of classes of Cartier divisors on $ X $. For integral schemes $ X $ this homomorphism is an isomorphism.

On a projective scheme $ X $ Serre's twisted invertible sheaf $ {\mathcal O} _ {X} ( 1) = {\mathcal O} ( 1) $ can be defined. In fact, if an imbedding of the scheme $ X $ in a projective space $ P ^ {N} $ is given, then $ {\mathcal O} _ {X} $ corresponds to the class of a hyperplane section. In particular, if $ X = P ^ {N} ( k) $ is a projective space over a field $ k $, then the sheaf $ {\mathcal O} ( 1) $ is the direct image of the sheaf of linear functions on $ k ^ {N+} 1 $ under the natural mapping $ k ^ {N+} 1 \setminus \{ 0 \} \rightarrow P ^ {N} ( k) $. The system of homogeneous coordinates $ x _ {0} \dots x _ {n} $ in $ P ^ {N} ( k) $ can be identified with a basis for the space of sections $ \Gamma ( P ^ {N} , {\mathcal O} ( 1) ) $.

An invertible sheaf on a scheme $ X $ is related to rational mappings of $ X $ into projective spaces. Let $ {\mathcal L} $ be an invertible sheaf on a scheme and let $ s _ {0} \dots s _ {N} $ be sections of $ {\mathcal L} $ the values of which at any point $ x \in X $ generate the stalk $ {\mathcal L} _ {x} $ over $ {\mathcal O} _ {x} $. Then there exists a unique morphism $ \phi : X \rightarrow P ^ {N} ( k) $ such that $ \phi ^ {*} {\mathcal O} ( 1) \cong {\mathcal L} $ and $ \phi ^ {*} x _ {i} = s _ {i} $, where $ x _ {0} \dots x _ {N} $ are homogeneous coordinates in $ P ^ {N} ( k) $. An invertible sheaf on $ X $ is called very ample if there exists an imbedding $ \phi : X \rightarrow P ^ {N} ( k) $ such that $ \phi ^ {*} {\mathcal O} ( 1) \cong {\mathcal L} $. An invertible sheaf $ {\mathcal L} $ on $ X $ is called ample if there exists a positive integer $ n $ for which $ {\mathcal L} ^ {n} $ is very ample. On a Noetherian scheme $ X $ over $ k $ an invertible sheaf $ {\mathcal L} $ is ample if and only if for each coherent sheaf $ {\mathcal F} $ on $ X $ there exists an integer $ n _ {0} > 0 $ such that the sheaf $ {\mathcal F} \otimes {\mathcal L} ^ {n} $ is generated by its global sections for $ n \geq n _ {0} $.

If $ {\mathcal L} $ is an ample invertible sheaf on $ X $ corresponding to a divisor $ D $, then $ D $ is called an ample divisor. A Cartier divisor $ D $ on a scheme $ X $ that is proper and smooth over an algebraically closed field $ k $ is ample if and only if for each closed integral subscheme $ Y \subseteq X $ the intersection index $ D ^ {r} \cdot Y $ is positive, where $ r = \mathop{\rm dim} Y $( cf. Intersection index (in algebraic geometry)). For other criteria of ampleness see [5]. There is also a generalization of the concept of an ample divisor on subvarieties of large codimension [2].

The concepts of very ample and ample invertible sheaves can be carried over to the case of analytic spaces (for criteria for ampleness in this situation see Positive vector bundle).

References

[1] R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001
[2] R. Hartshorne, "Ample subvarieties of algebraic varieties" , Springer (1970) MR0282977 Zbl 0208.48901
[3] D. Mumford, "Lectures on curves on an algebraic surface" , Princeton Univ. Press (1966) MR0209285 Zbl 0187.42701
[4] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
[5] 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 definition of Serre's twisted invertible sheaf is not precise enough. There is an action of the multiplicative group $ k ^ \times $ on $ k ^ {N+} 1 \setminus \{ 0 \} $ which has $ P ^ {N} ( k) $ as its quotient. The direct image of the structure sheaf under the mapping $ k ^ {N+} 1 \setminus \{ 0 \} \rightarrow P ^ {N} ( k) $ splits into a direct sum of invertible sheaves $ {\mathcal O} ( n) $, $ n \in \mathbf Z $, such that $ k ^ \times $ acts on $ {\mathcal O} ( n) $ via the character $ t \rightarrow t ^ {n} $.

How to Cite This Entry:
Invertible sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Invertible_sheaf&oldid=47427
This article was adapted from an original article by V.A. Iskovskikh (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article