Editing Projective space (section)

== Dual projective space ==
When the construction above is applied to the [[dual space]] {{math|''V''<sup>∗</sup>}} rather than {{math|''V''}}, one obtains the dual projective space, which can be canonically identified with the space of hyperplanes through the origin of {{math|''V''}}.  That is, if {{math|''V''}} is {{math|''n''}}-dimensional, then {{math|'''P'''(''V''<sup>∗</sup>)}} is the [[Grassmannian]] of {{math|''n'' − 1}} planes in {{math|''V''}}.

In algebraic geometry, this construction allows for greater flexibility in the construction of projective bundles.  One would like to be able to associate a projective space to ''every'' quasi-coherent sheaf {{math|''E''}} over a scheme {{math|''Y''}}, not just the locally free ones.{{clarify|And how does the dual space help?|date=January 2011}} See [[Éléments de géométrie algébrique|EGA]]<sub>II</sub>, Chap. II, par. 4 for more details.