Timeline for Smoothness of fix point components of finite group action on smooth variety
Current License: CC BY-SA 3.0
8 events
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| Jun 11, 2014 at 19:35 | comment | added | user27920 | @Qfwfq: The case of finite groups with order invertible on the base is in Edixhoven's 1992 paper "Neron models and tame ramification" in Compositio 81 (see 3.1--3.4), and the general case is Prop. A.8.11 in the book "Pseudo-reductive groups" (which gives a proof in a more general setting over rings that generalizes Edixhoven's Prop. 3.4; various aspects simplify when working over a field). | |
| Jun 11, 2014 at 19:03 | comment | added | Qfwfq | @user52824: could you please also give a reference? | |
| Jun 11, 2014 at 12:27 | comment | added | user27920 | If $H$ is a linearly reductive affine group scheme over any field $k$ (e.g., any affine algebraic group with reductive identity component in char. 0, which includes any finite group, as well as any group of multiplicative type in any characteristic) and $Y$ is a separated $k$-scheme of equipped with an $H$-action then the functor $Y^H$ of $H$-fixed points is represented by a closed subscheme of $Y$ (that much doesn't use the linear reductivity) which moreover is smooth whenever $Y$ is smooth. In particular, irreducible components of $Y^H$ are always pairwise disjoint. | |
| Jun 10, 2014 at 19:17 | vote | accept | Qfwfq | ||
| Jun 10, 2014 at 19:10 | answer | added | abx | timeline score: 9 | |
| Jun 10, 2014 at 19:08 | comment | added | Francesco Polizzi | I think this is a consequence of Cartan Lemma. In a neighborhood of a fixed point $p$ the action can be linearized, so locally analytically we may assume $X=\mathbb{C}^n$, $p=0$ and $G \subset \rm{GL}(n, \, \mathbb{C})$. Then in a neighborhood of $0$ the fixed locus is given by a union of linear subspaces and the claim follows. | |
| Jun 10, 2014 at 18:55 | history | edited | Qfwfq | CC BY-SA 3.0 |
added 152 characters in body
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| Jun 10, 2014 at 18:49 | history | asked | Qfwfq | CC BY-SA 3.0 |