Difference between revisions of "Invertible sheaf"
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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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> |
Revision as of 21:53, 30 March 2012
A locally free sheaf of -modules of rank 1 on a ringed space
. An equivalent definition is: A sheaf of
-modules that is locally isomorphic to the sheaf
. The invertible sheaves on
, considered up to isomorphism, form an Abelian group with respect to the operation of tensor multiplication over
. This group is called the Picard group of the space
, and is denoted by
. The inverse of a sheaf
in this group is the sheaf
dual to
. In the case when
is a scheme (in particular, an algebraic variety) or an analytic space, a sheaf of
-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
.
Invertible sheaves on schemes are closely connected with divisors (cf. Divisor). With each Cartier divisor on
is associated an invertible sheaf
, thus defining an injective homomorphism
, where
is the group of classes of Cartier divisors on
. For integral schemes
this homomorphism is an isomorphism.
On a projective scheme Serre's twisted invertible sheaf
can be defined. In fact, if an imbedding of the scheme
in a projective space
is given, then
corresponds to the class of a hyperplane section. In particular, if
is a projective space over a field
, then the sheaf
is the direct image of the sheaf of linear functions on
under the natural mapping
. The system of homogeneous coordinates
in
can be identified with a basis for the space of sections
.
An invertible sheaf on a scheme is related to rational mappings of
into projective spaces. Let
be an invertible sheaf on a scheme and let
be sections of
the values of which at any point
generate the stalk
over
. Then there exists a unique morphism
such that
and
, where
are homogeneous coordinates in
. An invertible sheaf on
is called very ample if there exists an imbedding
such that
. An invertible sheaf
on
is called ample if there exists a positive integer
for which
is very ample. On a Noetherian scheme
over
an invertible sheaf
is ample if and only if for each coherent sheaf
on
there exists an integer
such that the sheaf
is generated by its global sections for
.
If is an ample invertible sheaf on
corresponding to a divisor
, then
is called an ample divisor. A Cartier divisor
on a scheme
that is proper and smooth over an algebraically closed field
is ample if and only if for each closed integral subscheme
the intersection index
is positive, where
(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 on
which has
as its quotient. The direct image of the structure sheaf under the mapping
splits into a direct sum of invertible sheaves
,
, such that
acts on
via the character
.
Invertible sheaf. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Invertible_sheaf&oldid=12066