Timeline for Picard group of $(SL(n)\times SL(m))$-orbits
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 18, 2021 at 7:59 | answer | added | Mikhail Borovoi | timeline score: 2 | |
| Jan 17, 2021 at 21:32 | comment | added | user114666 | It is the subgroup made of pair of matrices $(\left(\begin{array}{cc} A & B \\ 0 & C \end{array} \right), \left(\begin{array}{cc} A' & B' \\ 0 & C' \end{array} \right))$ such that $AA'^{T} = \lambda I_k$. | |
| Jan 17, 2021 at 20:31 | comment | added | Mikhail Borovoi | Write $G={\rm SL}(n)\times{\rm SL}(n)$, and let $H$ denote the stabilizer of $J_k$. Then ${\rm Pic }\,X_k$ is canonically isomorphic to the character group of $H$. What is $H$ in your case? | |
| Jan 17, 2021 at 20:12 | comment | added | user114666 | Thank you. Following the argument you used here mathoverflow.net/questions/379171/… it seems that $\text{Pic}(X_n)\cong \mathbb{Z}/n\mathbb{Z}$ while $\text{Pic}(X_k)\cong \mathbb{Z}\oplus\mathbb{Z}$ for $k< n$. The parity of $k$ does not seem to play any role here. Is this correct? | |
| Jan 17, 2021 at 18:55 | comment | added | Mikhail Borovoi | See also Proposition 6.10 in Sansuc's paper Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres | |
| Jan 17, 2021 at 18:47 | comment | added | Mikhail Borovoi | Have a look at this answer. | |
| Jan 17, 2021 at 14:33 | history | asked | user114666 | CC BY-SA 4.0 |