Timeline for Automorphisms of the affine quadric $z_1^2+\dots+z_n^2=1$
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Oct 24, 2013 at 2:43 | comment | added | Piojo | @NateBottman: Definitely you're right! Just modified my post. Thank you! | |
| Oct 23, 2013 at 16:00 | comment | added | Francesco Polizzi | @Piojo: oh right, Cartan B. Thanks! | |
| Oct 23, 2013 at 15:44 | comment | added | Piojo | @FrancescoPolizzi: See mathoverflow.net/questions/1878/… for instance. | |
| Oct 23, 2013 at 15:43 | comment | added | Piojo | @MarianoSuárez-Alvarez: Sorry for the vagueness, I simply mean an extension such that $g_1^2+\dots+g_n^2=z_1^2+\dots+z_n^2$. | |
| Oct 23, 2013 at 15:40 | history | edited | Piojo | CC BY-SA 3.0 |
added 114 characters in body
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| Oct 23, 2013 at 15:20 | comment | added | Francesco Polizzi | @Piojo: ok, inversion is an automorphism of the projective quadric. But, still, I do not understand why any automorphism of $V$ should extend to $\mathbb{C}^n$. Maybe this is trivial, but could you please explain this point? | |
| Oct 23, 2013 at 15:13 | comment | added | Mariano Suárez-Álvarez | What do you mean by good extension? | |
| Oct 23, 2013 at 14:35 | history | edited | Piojo | CC BY-SA 3.0 |
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| Oct 23, 2013 at 14:30 | comment | added | Piojo | To David: For n=1, 1/z=z is the extension you want. | |
| Oct 23, 2013 at 14:29 | comment | added | Piojo | To Polizzi: I guess your map is not defined at the points $(\frac{3}{4}+\frac{1}{4}i,\frac{3}{4}-\frac{1}{4}i)$ and $(\frac{3}{4}-\frac{1}{4}i,\frac{3}{4}+\frac{1}{4}i)$. | |
| Oct 23, 2013 at 14:16 | history | edited | Piojo | CC BY-SA 3.0 |
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| Oct 23, 2013 at 13:52 | comment | added | emperordali | I also do not understand. For n = 1 isn't 1/z holomorphic on V but not on $\mathbb{C}$? | |
| Oct 23, 2013 at 13:51 | comment | added | Francesco Polizzi | There is something I do not understand. Take $z_1^2+z_2^2-1=0$ in $\mathbb{C}^2$ and consider the inversion with respect to the point $(1,1)$. Then we obtain the automorphism of $V$ given by $$z_1 \to 1 + \frac{z_1-1}{3-2z_1-2z_2}, \quad z_2 \to 1 + \frac{z_2-1}{3-2z_1-2z_2}.$$ It seems to me that it does not extend to an automorphism of $\mathbb{C}^2$. Am I missing something? | |
| Oct 23, 2013 at 13:30 | comment | added | Nathaniel Bottman | I think this is not true. Set $g: V \to V$ to be the identity, and define $g_1, \ldots, g_n$ by setting $g_1(z) := z_1 + \left(z_1^2 + \cdots + z_n^2 - 1\right)$ and $g_i(z) := z_i$ for $i > 1$. Am I missing something? | |
| Oct 23, 2013 at 12:01 | review | First posts | |||
| Oct 23, 2013 at 12:12 | |||||
| Oct 23, 2013 at 11:42 | history | asked | Piojo | CC BY-SA 3.0 |