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.
In mathematics, a projectiveplane has two possible definitions, one of them coming from linear algebra, and another (which is more general) coming from the combinatorics of block designs.
The most common projectiveplane is the real projectiveplane, which is a topological surface with surprising geometric properties; after that is the complexprojectiveplane of algebraic geometry, a topological four-dimensional manifold.
In the case of finite projectiveplanes, the only proof known of the purely geometric statement that Desargues theorem then implies Pappus' theorem (the converse being always true and provable geometrically) is through this algebraic route, using Wedderburn's theorem that finite division rings must be commutative.
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