In mathematics, an invertible sheaf is a sheaf on a ringed space which has an inverse with respect to tensor product of sheaves of modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their interactions with Cartier divisors, they play a central role in the study of algebraic varieties.
itself. Invertible sheaves are the invertible elements of this monoid. Specifically, if L is a sheaf of OX-modules, then L is called invertible if it satisfies...
called the twisting sheaf of Serre. It can be checked that O ( 1 ) {\displaystyle {\mathcal {O}}(1)} is in fact an invertiblesheaf. One reason for the...
ample invertible sheaves states that if X is a quasi-compact quasi-separated scheme and L {\displaystyle {\mathcal {L}}} is an invertiblesheaf on X,...
duality on V {\displaystyle V} . It may equally well be considered as an invertiblesheaf. The canonical class is the divisor class of a Cartier divisor K {\displaystyle...
geometry. In algebraic geometry, the tautological line bundle (as invertiblesheaf) is O P n ( − 1 ) , {\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(-1)...
along s has degree r as an invertiblesheaf over the fiber Xs (when the degree is defined for the Picard group of Xs.) Sheaf cohomology Chow variety Cartier...
E is a locally free sheaf of finite rank. In particular, if L is locally free of rank one (such L is called an invertiblesheaf or a line bundle), then...
fibrations of spheres to spheres. In algebraic geometry, an invertiblesheaf (i.e., locally free sheaf of rank one) is often called a line bundle. Every line...
O X ( n ) {\displaystyle {\mathcal {O}}_{X}(n)} is a line bundle (invertiblesheaf) on X {\displaystyle X} and O X ( n ) {\displaystyle {\mathcal {O}}_{X}(n)}...
equivariant sheaf to be an equivariant object in the category of, say, coherent sheaves. A structure of an equivariant sheaf on an invertiblesheaf or a line...
{\displaystyle \pi ^{-1}{\mathcal {I}}\cdot {\mathcal {O}}_{\tilde {X}}} is an invertiblesheaf, characterized by this universal property: for any morphism f: Y →...
measured by the vanishing of the higher sheaf cohomology groups of the associated line bundle (formally, invertiblesheaf). As the terminology reflects, this...
Positivity of the line bundle L translates into the corresponding invertiblesheaf being ample (i.e., some tensor power gives a projective embedding)...
sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology...
class as its dual. An explicit isogeny can be constructed by use of an invertiblesheaf L on A (i.e. in this case a holomorphic line bundle), when the subgroup...
theorem proved by Mumford in 1967 states that if L is a semi-ample invertiblesheaf with Iitaka dimension at least 2 on a complex projective manifold,...
the n-th tensor power of the Serre twist sheaf O ( 1 ) {\displaystyle {\mathcal {O}}(1)} , the invertiblesheaf or line bundle with associated Cartier divisor...
giving rise to the projective embedding of V, such a line bundle (invertiblesheaf) is said to be normally generated if V as embedded is projectively...
scheme is a scheme admitting an ample family of invertible sheaves. A scheme admitting an ample invertiblesheaf is a basic example. dominant A morphism f :...
In mathematics, the gluing axiom is introduced to define what a sheaf F {\displaystyle {\mathcal {F}}} on a topological space X {\displaystyle X} must...
nonzero rational section s of L.) A line bundle may also be called an invertiblesheaf. The term "nef" was introduced by Miles Reid as a replacement for the...
calculated by homological algebra means, from a minimal resolution of an invertiblesheaf of high degree. In many cases the gonality is two more than the Clifford...
Global Information