| Complex Projective Plane |
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| CATEGORIES ABOUT COMPLEX PROJECTIVE PLANE | |
| algebraic surfaces | |
| complex manifolds | |
| projective geometry | |
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: 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 Number s 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 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 ''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 . |
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