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Birational geometry is a field of algebraic geometry the goal of which is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rational functions rather than polynomials; the map may fail to be defined where the rational functions have poles.

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What is $h^0(\mathcal O_F)$ where $F$ is a fiber of a normal surface over a smooth curve?

I believe that you misinterpreted the top of page 92 of Debarre's notes. At that stage Debarre works with a flat morphism $\pi: S\to \overline{T}$, where $S$ is a normal surface and the general fiber …
Remke Kloosterman's user avatar