Complex projective plane
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In mathematics, the complex projective plane, usually denoted CP2, is the two-dimensional complex projective space. It is a complex manifold described by three complex coordinates
- <math>(z_1,z_2,z_3) \in \mathbb{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.
The Betti numbers of the complex projective plane are
- 1, 0, 1, 0, 1, 0, 0, ... .
The middle dimension 2 is accounted for by the homology class of the complex projective line, or Riemann sphere, lying in the plane.
In birational geometry, a complex rational surface is any algebraic surface birationally equivalent to the complex projective plane. It is known that any non-singular rational variety is obtained from the plane by a sequence of blowing up transformations and their inverses ('blowing down') of curves, which must be of a very particular type. As a special case, a non-singular complex quadric in P3 is obtained from the plane by blowing up a single point to a curve; the inverse of this transformation can be seen by taking a point P on the quadric Q and projecting onto a general plane in P3 by drawing lines through P.
The group of birational automorphisms of the complex projective plane is the Cremona group.
See also: del Pezzo surface, toric geometry.