Birational geometry
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In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension 2, the birational geometry of algebraic surfaces was largely worked out by the Italian school of algebraic geometry in the two decades on either side of the year 1900. From about 1970 advances have been made in higher dimensions, giving a good theory of birational geometry for dimension 3.
Birational geometry is largely a geometry of transformations; but it doesn't fit exactly with the Erlangen programme. One reason is that its nature is to deal with transformations that are only defined on an open, dense subset of an algebraic variety. Such transformations, given by rational functions in the co-ordinates, can be undefined not just at isolated points on curves, but on entire curves on a surface, and so on.
One of the first results in the subject is the birational isomorphism of the projective plane, and a non-singular quadric Q in projective 3-space. Already in this example one can see whole sets where the mappings are ill-defined: taking a point P on Q as origin, we can use lines through P, intersecting Q at one other point, to project to a plane - but this definition breaks down with all lines tangent to Q at P, which in a certain sense 'blow up' P into the intersection of the tangent plane with the plane to which we project.
That is, quite generally, we can expect birational mappings to act like relations, with graphs containing parts that are not functional. On an open dense set they do behave like functions; but the Zariski closure of their graphs are more complex correspondences on the product showing 'blowing up' and 'blowing down'. Quite detailed descriptions of those, in terms of projective spaces associated to tangent spaces can be given and justified by the theory.
An example is the Cremona group of birational automorphisms of the projective plane. In purely algebraic terms, for a given field K, this is the automorphism group over K of the field K(X,Y) of rational functions in two variables. Its structure has been analysed since the nineteenth century; but it is 'large' (while the corresponding group for the projective line consists only of Möbius transformations determined by three parameters). It is still the subject of research.