# Difference between revisions of "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

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 $t \rightarrow t ^ {n}$.