Complex projective plane

In mathematics, the complex projective plane, usually denoted P²(C), is the two-dimensional complex projective space. It is a complex manifold described by three complex coordinates

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

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

The Betti numbers of the complex projective plane are

The middle dimension 2 is accounted for by the homology class of the complex projective line, or Riemann sphere, lying in the plane. The nontrivial homotopy groups of the complex projective plane are ${\displaystyle \pi _{2}=\pi _{5}=\mathbb {Z} }$. The fundamental group is trivial and all other higher homotopy groups are those of the 5-sphere, i.e. torsion.

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.

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