Birational Geometry
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High Quality Content by WIKIPEDIA articles! 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 two, the birational geometry of algebraic surfaces was largely worked out by the Italian school of algebraic geometry in the years 1890 1910. From about 1970 advances have been made in higher dimensions, giving a good theory of birational geometry for dimension three. Birational geometry is largely a geometry of transformations, but it doesn't…mehr

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High Quality Content by WIKIPEDIA articles! 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 two, the birational geometry of algebraic surfaces was largely worked out by the Italian school of algebraic geometry in the years 1890 1910. From about 1970 advances have been made in higher dimensions, giving a good theory of birational geometry for dimension three. 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.