Editing Complex projective plane (section)

{{short description|2-dimensional complex projective space}}
{{refimprove|date=May 2010}}

In [[mathematics]], the '''complex projective plane''', usually denoted {{tmath|\mathbb{P}^2(\C)}} or {{tmath|\mathbb{CP}^2,}} is the two-dimensional [[complex projective space]]. It is a [[complex manifold]] of complex dimension 2, described by three complex coordinates

:<math>(Z_1,Z_2,Z_3) \in \C^3, \qquad (Z_1,Z_2,Z_3)\neq (0,0,0)</math>

where, however, the triples differing by an overall rescaling are identified:

:<math>(Z_1,Z_2,Z_3) \equiv (\lambda Z_1,\lambda Z_2, \lambda Z_3); \quad \lambda \in \C, \qquad \lambda \neq 0.</math>

That is, these are [[homogeneous coordinates]] in the traditional sense of [[projective geometry]].