Timeline for Proposition from Kollar's Rational Curves on Algebraic Varieties
Current License: CC BY-SA 4.0
7 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
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| Feb 5, 2020 at 21:16 | comment | added | Frank | Yes the assumption is made on p113 for the entire section. Probably a lot of statements hold more generally but I'd have to think a bit about it | |
| Feb 5, 2020 at 20:14 | comment | added | user267839 | @Frank:About the last point I'm not sure. You say "Both maps are linear maps of vector spaces over an algebraically closed field, so rank means the rank of the image". Did Kollar in his book somewhere assumed that base field $k$ is algebraically closed? Is it really neccessary to make this assumption? $H^0(C, T_{\mathbb{P}^1} \otimes I_B)$ has always structure of a vc over base field and the $k(p)$-vc $ T_{\mathbb{P}^1} \otimes k(p)$ obtains also $k$-space structure by transitivity. Where do we explicitly need alg. closeness? | |
| Feb 5, 2020 at 7:48 | comment | added | Frank | Q1. $B$ is a closed subscheme, so its ideal sheaf will be $I_B=\mathcal{O}_{\mathbb{P}^1}(-p-q)$ (where $p$ can be equal to $q$), so $|B|$ is meant to denote the degree of the closed subscheme, i.e., the numbers of points accounting for multiplicities. Since $T_{\mathbb{P}^1}\cong\mathcal{O}_{\mathbb{P}^1}(2)$, this line bundle will continue to be globally generated (implying the surjectivity you want) after we twist by up to 2 points, hence the assumption. Q2. Both maps are linear maps of vector spaces over an algebraically closed field, so rank means the rank of the image. | |
| Feb 5, 2020 at 6:00 | review | Close votes | |||
| Feb 10, 2020 at 3:05 | |||||
| Feb 5, 2020 at 3:43 | history | edited | LSpice | CC BY-SA 4.0 |
TeX fixes
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| Feb 5, 2020 at 3:11 | history | asked | user267839 | CC BY-SA 4.0 |