Timeline for Lie Algebra of Automorphism Group of $\mathbb{P}_k^1$
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 8, 2019 at 23:07 | comment | added | user | Now $\operatorname{Lie}(\operatorname{PGL}(2,k))= sl_{2,k}$ is isomorphic to the group of automorphisms of the trivial deformation $\mathcal{X}$ of $\mathbb{P}_k^1$ over $k[\epsilon]$. Locally automorphism of $\mathcal{X}$ are given by $z \mapsto z + \epsilon(a_0 + a_1z + a_2 z^2)$. Unfortunately, there is no obvious global description of these automorphism. However, knowing that $\operatorname{Aut}(\mathcal{X}) \cong sl_{2,k}$, is there a way to determine $\operatorname{Aut}(\mathcal{X})$ from the given local description? | |
| Jun 2, 2019 at 15:05 | comment | added | Julian Rosen | The map $\mathbb{G}_m\times\mathbb{G}_a\to \operatorname{Aut}(\mathbb{A}^1)$ is a monomorphism which is an isomorphism on $k$ points, but this map is not an isomorphism because, for example, $x\mapsto x+\epsilon x^2$ is an automorphism of $\mathbb{A}^1\times\operatorname{Spec}\, k[\epsilon]/\epsilon^2$ that does not come from a $k[\epsilon]/\epsilon^2$ point of $\mathbb{G}_m\times\mathbb{G}_a$. | |
| Jun 1, 2019 at 22:19 | comment | added | user | Thanks again. Do you know why the equality $\operatorname{Lie}(G)=H^0(X, \mathcal{T}X)$ doesn't hold when $X= \mathbb{A}_k^1$ ? Here, global sections are of the form $f(z) \partial_z$ which is an infinite dimensional $k$ vector space. However, $G= \operatorname{Aut}(X)$ is only two-dimensional, I think isomorphic to $k^* \times k$. | |
| May 29, 2019 at 15:07 | comment | added | Julian Rosen | In general, I'm not sure you can get an automorphism of $X$ from a global vector field. Setting $\epsilon=1$ won't work because we need $\epsilon^2=0$. Exponentiation is a transcendental operation, and might not make sense algebraically (e.g. the exponential of $z\partial_z$ wants to be multiplication by $e$, but why should $e$ be an element of $k$?). | |
| May 29, 2019 at 4:54 | comment | added | user | Thank you for the answer. How do I now use this to write down the automorphisms of $X$? It seems like we set $t=\epsilon$ and then evaluate at $t=1$, i.e something akin to the exponential map. | |
| May 28, 2019 at 23:10 | vote | accept | user | ||
| May 28, 2019 at 18:59 | vote | accept | user | ||
| May 28, 2019 at 20:20 | |||||
| May 28, 2019 at 3:55 | history | answered | Julian Rosen | CC BY-SA 4.0 |