In mathematics, a line bundle expresses the concept of a 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. 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. 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 bundles. There are some celebrated ones, for example the Hopf fibrations of spheres 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 under the following conditions:
(I) If X {\displaystyle X} is a reduced and irreducible scheme, then every line bundle comes from a divisor. (II) If X {\displaystyle X} is a projective scheme then the same statement holds.
The tautological bundle on projective space
One of the most important line bundles in algebraic geometry is the tautological line bundle on projective space. The projectivization P ( V ) {\displaystyle \mathbf {P} (V)} of a vector space V {\displaystyle V} over a field k {\displaystyle k} is defined to be the quotient of V ∖ { 0 } {\displaystyle V\setminus \{0\}} by the action of the multiplicative group k × {\displaystyle k^{\times }} . Each point of P ( V ) {\displaystyle \mathbf {P} (V)} therefore corresponds to a copy of k × {\displaystyle k^{\times }} , and these copies of k × {\displaystyle k^{\times }} can be assembled into a k × {\displaystyle k^{\times }} -bundle over P ( V ) {\displaystyle \mathbf {P} (V)} . But k × {\displaystyle k^{\times }} differs from k {\displaystyle k} only by a single point, and by adjoining that point to each fiber, we get a line bundle on P ( V ) {\displaystyle \mathbf {P} (V)} . This line bundle is called the tautological line bundle. This line bundle is sometimes denoted O ( − 1 ) {\displaystyle {\mathcal {O}}(-1)} since it corresponds to the dual of the Serre twisting sheaf O ( 1 ) {\displaystyle {\mathcal {O}}(1)} .
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