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Complex projective plane
where, however, the triples differing by an overall rescaling are identified:
The Betti numbers of the complex projective plane are
- 1, 0, 1, 0, 1, 0, 0, ... .
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 .
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