Timeline for Choosing a group action to do GIT of hypersurfaces
Current License: CC BY-SA 3.0
11 events
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| Oct 1, 2015 at 14:39 | comment | added | Jason Starr | @John: Not every linear representation of $\textbf{SL}_{n+1}$ factors through $\textbf{PGL}_{n+1}$. Thus, for an action of $\textbf{PGL}_{n+1}$ on $X$, for the induced action of $\textbf{SL}_{n+1}$, a $\textbf{SL}_{n+1}$-linearized ample invertible sheaf $\mathcal{O}(1)$ may not factor through a $\textbf{PGL}_{n+1}$-linearization. Abstractly, we could just pass from $\mathcal{O}(1)$ to $\mathcal{O}(n+1)$. However, this is analogous to passing from the ideal of a projective variety to the ideal of its Veronese image, already nontrivial for projective space. | |
| Oct 1, 2015 at 13:55 | history | edited | John | CC BY-SA 3.0 |
Added question following comments.
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| Oct 1, 2015 at 13:52 | comment | added | John | @AriyanJavanpeykar I cannot notify two people in one comment, but the comment to Jason was for you too ;-) | |
| Oct 1, 2015 at 13:52 | comment | added | John | @JasonStarr Thanks for the comments. After reading Dolgachev I don't understand why many authors use $SL_{n+1}$ rather than $PGL_{n+1}$. e.g, Laza here also uses SL to compactify the moduli of cubic fourfolds. Dolgachev in "Lectures on Invariant Theory", Chapter 10, uses $SL_{n+1}$ to construct compactifications of the moduli of hypersurfaces. I understand SL gives a unique linearization. In particular, in the case of cubic surfaces he uses $SL_{4}$ and gets the same result as Mukai, who uses $GL_4$. Why both groups give the same compactification? | |
| Sep 25, 2015 at 13:18 | comment | added | Jason Starr | Ariyan is correct. Even in those cases where $\textbf{PGL}_{n+1}$ does not linearize to $\mathcal{O}(1)$ on $\mathbb{P}k[x_0,\dots,x_n]_d$, it does linearize some positive tensor power, e.g., $\mathcal{O}(n)$. So that is not, by itself, a reason one must use $\textbf{SL}_{n+1}$ rather than $\textbf{PGL}_{n+1}$. | |
| Sep 24, 2015 at 13:28 | comment | added | Ariyan Javanpeykar | The group $PGL_{N+1}$ is used in www-irma.u-strasbg.fr/~benoist/articles/Thesefinale.pdf and arxiv.org/abs/1505.02249 | |
| Sep 23, 2015 at 17:04 | comment | added | Jason Starr | In Chapter 7 of Dolgachev's "Lectures on Invariant Theory", he describes the obstruction to linearizing a given action. | |
| Sep 23, 2015 at 16:58 | comment | added | John | Thank you Jason, that explains Mukai's choice. Do you have a reference for that statement? (Or an idea of where to look for one) | |
| Sep 23, 2015 at 16:52 | comment | added | Jason Starr | Although there exists an action of $\textbf{PGL}_{n+1}$ on $\mathbb{P}k[x_0,\dots,x_n]_d$ for every $d$, this only lifts to a linear action on the vector space $k[x_0,\dots,x_n]$ if $n+1$ divides $d$. | |
| Sep 23, 2015 at 16:47 | review | First posts | |||
| Sep 23, 2015 at 17:46 | |||||
| Sep 23, 2015 at 16:47 | history | asked | John | CC BY-SA 3.0 |