Snyder did research on configurations of ruled surfaces and birational transformations.
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Besides their importance in describing birational transformations, blowups are also an important way of constructing new spaces.
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It starts with a plane curve, and repeatedly applies birational transformations to the plane to improve the curve.
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On an algebraic surface, blowing down a curve lying on the surface is a typical effect of a birational transformation.
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Ramanujam was a scribe for Igor Shafarevich's course of lectures in 1965 on minimal models and birational transformation of two dimensional schemes.
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The birational transformations are more complicated than the quadratic transformations used in Noether's method, but produce the better result that the only singularities are ordinary double points.
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He also applied his theory to algebraic surfaces ( with a presentation of this research in his book " Algebraische Fl�chen " ) and worked on birational transformations in the plane ( Cremona transformations ).
8.
The equation of a hyperbola; since inversion is a birational transformation and the hyperbola is a rational curve, this shows the lemniscate is also a rational curve, which is to say a curve of genus zero.
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