# Birational geometry as local algebraic geometry

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
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
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
The analogue in algebraic geometry of the "local differential geometry" is the étale topology. –  user18119 Oct 3 '12 at 8:14
@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. –  RghtHndSd Aug 8 '13 at 3:37

With a couple of exceptions, each birational equivalence class contains a minimal model, unique up to isomorphism, such that any member of the class admits a birational morphism to the minimal model (so a birational map that is actually defined everywhere). Furthermore, the minimal model has a concrete characterization (it contains no $-1$ curves). Finally, any birational map between two members of the birational equivalence class can be factored as the composition of blowings up and blowings down.