Line Bundle - The Tautological Bundle On Projective Space

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) of a vector space V over a field k is defined to be the quotient of by the action of the multiplicative group k×. Each point of P(V) therefore corresponds to a copy of k×, and these copies of k× can be assembled into a k×-bundle over P(V). k× differs from k only by a single point, and by adjoining that point to each fiber, we get a line bundle on P(V). This line bundle is called the tautological line bundle. This line bundle is sometimes denoted since it corresponds to the dual of the Serre twisting sheaf .