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 .
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