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Results tagged with birational-geometry
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user 5659
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.
11
votes
Contracting divisors to a point
In the case of smooth surfaces, this is already a tough question :
If E is a rational curve, then E can be contracted to a point in a smooth variety if the self-intersection $E\cdotp E=-1$ (thm of C …
- 2,727
4
votes
Technique to prove basepoint-freeness
I don't think that your assertion is true; for example, Lazarsfeld gives an example (PAG, 2.3.3) of a big and nef divisor on a surface such that its graded algebra is not finitely generated, so that t …
- 2,727
16
votes
When does $\operatorname{Aut}(X)=\operatorname{Bir}(X)$ hold?
To complete the answer of Diverietti and the comment of Roy Smith, here is a statement which might interest you:
Theorem If $X$, $Y$ are varieties over a field $k$, assume $X$ is smooth and $Y$ proper …
- 2,727