Complex projective plane

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

$(z_1,z_2,z_3) \in \mathbb{C}^3,\qquad (z_1,z_2,z_3)\neq (0,0,0)$

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

$(z_1,z_2,z_3) \equiv (\lambda z_1,\lambda z_2, \lambda z_3);\quad \lambda\in C,\qquad \lambda \neq 0.$

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.