Invertible sheaf

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} $.