Timeline for Relative compactification without resolutions of singularities
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jun 2, 2023 at 8:01 | comment | added | user197402 | Thank you for the very nice clarification :) | |
| May 31, 2023 at 22:16 | comment | added | R. van Dobben de Bruyn | There is an alternative argument for projective varieties $Y$ for which every morphism $\mathbf P^1\to Y$ is constant (this includes subvarieties of abelian varieties). Indeed, consider the closure $\bar\Gamma$ of the graph $\Gamma_\phi\subseteq X\times Y$, which maps birationally onto $X$ via the first projection $p\colon\bar\Gamma\to X$. When $X$ is smooth, such a map has linearly connected fibres by a result of Murre, which (by assumption on $Y$) are contracted by the second projection $q\colon\bar\Gamma\to Y$. Then $q$ factors through $p$, i.e. $\phi$ is everywhere defined. | |
| May 31, 2023 at 13:51 | comment | added | user197402 | Ok I see, this was definitely too optimistic. Thank you for the clarification! | |
| May 31, 2023 at 13:45 | comment | added | Jason Starr | There is almost certainly no general argument that does not use Hironaka. Let $Z$ be an arbitrary (singular) projective variety with an embedding in $X=\mathbb{P}^n$. Let $X'$ be the blowing up of $X$ along $Z$, and let $X'\hookrightarrow \mathbb{P}^r$ be a closed immersion. For this rational transformation, $X\dashrightarrow Y$, a sequence of blowups with smooth centers that regularizes the transformation also implies a resolution of singularities of $Z$. | |
| May 31, 2023 at 13:45 | comment | added | user197402 | Oh I see, yes I know the result on abelian varieties but I did not think about using it for the result, as abelian varieties are a bit too special. Thank you both very much! | |
| May 31, 2023 at 13:42 | comment | added | Jason Starr | It also extends to N'eron models of Abelian varieties (it is roughly the defining property of a N'eron model). Sometimes it is called "Weil extension." | |
| May 31, 2023 at 13:13 | comment | added | abx | This is usually stated with $Y$ an abelian variety. Then this implies the result for any subvariety of an abelian variety, for instance a curve of genus $\geq 1$. | |
| May 31, 2023 at 12:09 | comment | added | user197402 | Really? I have never seen a result like this, can you explain it to me? | |
| May 31, 2023 at 11:04 | history | edited | user197402 | CC BY-SA 4.0 |
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| May 31, 2023 at 9:46 | comment | added | abx | If $Y$ is a curve of genus $\geq 1$, $\phi$ extends to a morphism $X\rightarrow Y$... | |
| May 31, 2023 at 8:20 | history | asked | user197402 | CC BY-SA 4.0 |