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Results tagged with birational-geometry
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user 21724
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.
15
votes
Accepted
When is the map $H^0(X,mK_X) \times H^0(X,nK_X) \to H^0(X,(m+n)K_X)$ surjective?
I assume you mean $H^0(X, K_X)^{\otimes m}$ rather than $\oplus_{i=1}^m H^0(X, K_X)$. If $X$ is a smooth projective connected complex curve of genus $g \geq 2$, then the map
$$H^0(X, K_X)^{\otimes m} …
- 7,249
2
votes
Accepted
Blowing-up an ideal generated by squares
If $\alpha \in \mathbb{N}^{r}$ satisfies $|\alpha| = r+1$ then $\exists i, \alpha_i \geq 2$ so that $f^{\alpha} \in f_i^2 J^{r-1}$.
Thus $J^{r+1} = I J^{r-1}$. In particular $I$ becomes invertible on …
- 7,249