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Technically birational geometry is local geometry of algebraic varieties, yet it feels completely different from local differential geometry, which is more or less trivial. Is there some subtle similarity between them?

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They are very different because the Zariski topology is much coarser than classical Euclidean topology. – Michael Joyce Oct 3 '12 at 0:49
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Birational geometry should not be thought of as analogous to local differential geometry but to local complex geometry (if that); morphisms of algebraic varieties behave much more like holomorphic maps than like smooth ones. – Qiaochu Yuan Oct 3 '12 at 1:08
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Isn't birational geometry the exact opposite? Describing the generic behavior of a variety, disregarding anything special that might happen locally? – Hurkyl Oct 3 '12 at 1:08
@MichaelJoyce Sure, I hoped there are some similarities nevertheless. – Alexei Averchenko Oct 3 '12 at 1:26
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The analogue in algebraic geometry of the "local differential geometry" is the étale topology. – QiL'8 Oct 3 '12 at 8:14
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