Timeline for Is dimension invariant under blow-ups?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| S May 27, 2018 at 10:45 | history | suggested | CommunityBot | CC BY-SA 4.0 |
corrected reference to Liu
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| May 27, 2018 at 8:43 | review | Suggested edits | |||
| S May 27, 2018 at 10:45 | |||||
| Apr 24, 2015 at 9:10 | history | edited | Alexander Voitovitch | CC BY-SA 3.0 |
added 1094 characters in body
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| Apr 23, 2015 at 6:53 | comment | added | Alexander Voitovitch | Thank you for your comments! I had a completely false perception of blow-ups in centers which contain a irreducible component. For $X$ locally Noetherian I should be able to to reduce my problem to the case that $X$ is a locally Noetherian integral scheme. | |
| Apr 22, 2015 at 14:47 | comment | added | Karl Schwede | If you blow up the zero ideal, you get the empty scheme. If you blowup an irreducible component, then that component goes away. But yes, outside of that situation everything is fine as the other comments said. | |
| Apr 22, 2015 at 11:56 | comment | added | diverietti | This is certainly true if $X$ is a locally Noetherian integral scheme and you blow-up a non-zero quasi-coherent sheaf of ideals. In this case, the blow-up morphism is proper and birational, and $X'$ integral. Finally, if $f\colon Z\to Y$ is any proper birational morphism, where $Y$ is a locally Noetherian integral scheme, then $\dim Z=\dim Y$. | |
| Apr 22, 2015 at 11:18 | comment | added | Francesco Polizzi | The dimension of an algebraic variety is a birational invariant (it equals the trascendental degree of the function field) so it is invariant under birational modifications, in particular invariant under blow-up. | |
| Apr 22, 2015 at 11:10 | review | First posts | |||
| Apr 22, 2015 at 11:12 | |||||
| Apr 22, 2015 at 11:10 | history | asked | Alexander Voitovitch | CC BY-SA 3.0 |