Timeline for automorphisms of an étale cover of a curve
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| S Jul 30, 2020 at 17:06 | history | suggested | RobPratt |
added a tag
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| Jul 30, 2020 at 16:19 | review | Close votes | |||
| Aug 5, 2020 at 3:06 | |||||
| Jul 30, 2020 at 16:13 | review | Suggested edits | |||
| S Jul 30, 2020 at 17:06 | |||||
| Jul 30, 2020 at 4:41 | comment | added | Will Chen | A $\pi_1(X)$-set is a set (finite in our case) equipped with an action by the group $\pi_1(X)$. By the Galois correspondence, if you fix a geometric point $x\in X$, taking fibers gives an equivalence of categories between finite etale covers of $X$ and finite $\pi_1(X,x)$-sets. So any question internal to the category of finite etale covers of $X$ (e.g., existence of automorphisms), can be translated into a combinatorial problem involving sets with group-action. | |
| Jul 30, 2020 at 4:38 | comment | added | user190964 | Could explain what is the corresponding $\pi_1(X)$-set? How can say that the automorphism is trivial in your example? | |
| Jul 30, 2020 at 4:37 | comment | added | user190964 | I just checked your additional comments and try to make the picture clear. $K$ means the base field, and my comments comes from the correspondences between 1. smooth algebraic curves and its function field $X\leftrightarrow K(X)$, 2. finite etale morphism $Y\to X$ and finite separable field extension $K(Y)/K(X)$, and 3. the automorphism groups $\mathrm{Aut}(Y/X)\leftrightarrow \mathrm{Aut}(K(Y)/K(X))$. | |
| Jul 30, 2020 at 4:30 | comment | added | Will Chen | What is $F(X)$? | |
| Jul 30, 2020 at 4:24 | comment | added | user190964 | I mean finite étale maps. I am confused that there may exist a nontrivial finite separable extension $F$ of $K(X)$ with trivial automorphism group $\mathrm{Aut}(F/K(X))$ and $F$ corresponds to a curve $Y$ by $Y=F(X)$. | |
| Jul 30, 2020 at 4:17 | history | edited | user190964 | CC BY-SA 4.0 |
edited title
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| Jul 30, 2020 at 4:17 | comment | added | Will Chen | By etale covering do you mean finite etale map? If so, this is just Galois theory. The map corresponds to a finite index index $d$ subgroup $H\le\pi_1(X)$, and the automorphism group is just the automorphism group of the corresponding $\pi_1(X)$-set. IIRC this should just be the quotient of the normalizer of $H$ by $H$. Sometimes it'll be trivial. For example you can fix a surjection $f : \pi_1(X)\rightarrow S_3$ (symmetric group on 3 things), and let $H$ be the preimage of an order 2 subgroup of $S_3$. The resulting cover will have trivial aut group for the same reason that $Q(\sqrt{3})$ does | |
| Jul 30, 2020 at 4:11 | history | asked | user190964 | CC BY-SA 4.0 |