Timeline for Is the affine closure of a quasi-affine variety again a variety?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Jun 23, 2015 at 8:27 | comment | added | Urs Hartl | Dear Colleagues, thank you for your counterexamples. Are there any results on the positive side, guaranteeing that $\Gamma(X,\mathcal{O}_X)$ is a finitely generated $k$-algebra? | |
| Jun 16, 2015 at 18:33 | answer | added | Jason Starr | timeline score: 4 | |
| Jun 16, 2015 at 13:06 | comment | added | Jason Starr | There are much more elementary examples than that one. Let $Y$ be the affine scheme $\text{Spec}(k[s,t,u]/\langle tu \rangle)$. Let $Z$ be the closed subset $\text{Zero}(s,t)$. Let $X$ be the quasi-affine complement $Y\setminus Z$. For every $n\in \mathbb{N}$, the element $s^{-n}u$ is an element in $\Gamma(X,\mathcal{O}_X)$. | |
| Jun 16, 2015 at 12:23 | comment | added | grghxy | A quasi-affine example with $A$ not of finite type is given by purely algebraic (rather than analytic) means in math.stanford.edu/~vakil/files/nonfg.pdf over any field $k$ such that there exists an elliptic curve $E$ over $k$ with $E(k)$ not a torsion group (so not for $k$ algebraic over a finite field). | |
| Jun 16, 2015 at 11:40 | review | First posts | |||
| Jun 16, 2015 at 11:56 | |||||
| Jun 16, 2015 at 11:38 | history | asked | Urs Hartl | CC BY-SA 3.0 |