Dual Projective Space
When the construction above is applied to the dual space V* rather than V, one obtains the dual projective space, which can be canonically identified with the space of hyperplanes through the origin of V. That is, if V is n dimensional, then P(V*) is the Grassmannian of n−1 planes in V.
In algebraic geometry, this construction allows for greater flexibility in the construction of projective bundles. One would like to be able associate a projective space to every quasi-coherent sheaf E over a scheme Y, not just the locally free ones. See EGAII, Chap. II, par. 4 for more details.
Read more about this topic: Projective Space
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