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

(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 two points to curves, and then blowing down the line through these two points; the inverse of this transformation can be seen by taking a point P on the quadric Q, blowing it up, 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.

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