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?
Birational geometry as local algebraic geometry
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algebraic-geometry
differential-geometry
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3They are very different because the Zariski topology is much coarser than classical Euclidean topology. – 2012-10-03
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1Birational 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. – 2012-10-03
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5Isn't birational geometry the exact opposite? Describing the *generic* behavior of a variety, disregarding anything special that might happen locally? – 2012-10-03
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0@MichaelJoyce Sure, I hoped there are some similarities nevertheless. – 2012-10-03
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0@Hurkyl Technically birational geometry is a geometry where morphisms only have to be defined on (dense) open sets, which is exactly like local differential geometry. – 2012-10-03
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0@Alexei: I guess I was focusing more on facts that a curve is birationally isomorphic to itself minus a point, and also to the singular curve by identifying two of its points: thus, losing the "local" information. I wasn't aware of the existence of a subdiscipline "local differential geometry", so I guess I can't really say anything about how they compare. – 2012-10-03
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2The analogue in algebraic geometry of the "local differential geometry" is the étale topology. – 2012-10-03
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1@Alexei: Having the "(dense)" in your comment (as it should be!) really destroys any comparison to local differential geometry, no? This is what Michael was saying. – 2013-08-08