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) eq (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 lambdain C,qquad lambda eq 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 P 3 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 P 3 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.