Editing Line bundle (section)

{{Short description|Vector bundle of rank 1}}
In [[mathematics]], a '''line bundle''' expresses the concept of a [[Line (geometry)|line]] that varies from point to point of a space. For example, a [[curve]] in the plane having a [[tangent]] line at each point determines a varying line: the ''[[tangent bundle]]'' is a way of organising these. More formally, in [[algebraic topology]] and [[differential topology]], a line bundle is defined as a ''[[vector bundle]]'' of rank 1.<ref>{{cite book|author=Hartshorne |title=Algebraic Geometry, Arcata 1974|year=1975|url={{Google books|plainurl=y|id=eICMfNiDdigC|page=7|text=line bundle}}|page=7}}</ref>

Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner.  In topological applications, this vector space is usually real or complex.  The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 [[invertible]] real matrices, which is [[homotopy]]-equivalent to a [[discrete two-point space]] by contracting the positive and negative reals each to a point; whereas removing the origin from the complex plane yields the 1×1 invertible complex matrices, which have the homotopy type of a circle.

From the perspective of [[homotopy theory]], a real line bundle therefore behaves much the same as a [[fiber bundle]] with a two-point fiber, that is, like a [[double cover (topology)|double cover]].  A special case of this is the [[orientable double cover]] of a [[differentiable manifold]], where the corresponding line bundle is the determinant bundle of the tangent bundle (see below).  The [[Möbius strip]] corresponds to a double cover of the circle (the θ → 2θ mapping) and by changing the fiber, can also be viewed as having a two-point fiber, the [[unit interval]] as a fiber, or the real line.

Complex line bundles are closely related to [[circle bundle]]s.  There are some celebrated ones, for example the [[Hopf fibration]]s of [[sphere]]s to spheres.

In [[algebraic geometry]], an [[invertible sheaf]] (i.e., [[locally free sheaf]] of rank one) is often called a '''line bundle'''.

Every line bundle arises from a [[Divisor (algebraic geometry)|divisor]] under the following conditions:

:(I) If <math>X</math> is a reduced and irreducible scheme, then every line bundle comes from a divisor.
:(II) If <math>X</math> is a [[projective scheme]] then the same statement holds.