Timeline for Do automorphisms actually prevent the formation of fine moduli spaces?
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8 events
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| Jun 21 at 9:01 | comment | added | Coherent Sheaf | @anon Appologies for the delayed reply, but I wanted to sleep on your comment once and be sure I understood it, and could generalise this to other moduli problems as well. Thank a lot for your comment. It makes complete sense and was really helpful! | |
| Jun 20 at 11:13 | comment | added | anon | What prevents the existence of fine moduli is having forms of $X$ over $k$, say, that are nonisomorphic over $k$ but become isomorphic over the algebraic closure of $k$. Such forms are classified by $H^1(k,Aut(X))$, which is generally nontrivial if $Aut(X)$ is nontrivial. | |
| Jun 20 at 7:09 | comment | added | Coherent Sheaf | @anon This comment is definitely helpful, but saying something as general as "automorphisms prevent fine moduli" would require more. For instance, if I restrict the problem to scheme over some $S$, where $\mathbb{Q}$-elliptic curves do not live, will automorphisms still create a problem (in the old style definition of moduli spaces)? All of this works out nicely in Fibered language. | |
| Jun 20 at 2:06 | comment | added | anon | Let $E$ be an elliptic curve over $\mathbb{Q}$ with $Aut(E)={\pm 1}$. Then the quadratic twists of $E$ are classified by $H^1(\mathbb{Q},Aut(E))$. | |
| Jun 19 at 21:05 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading and tidying
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| Jun 19 at 20:57 | history | edited | Coherent Sheaf | CC BY-SA 4.0 |
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| Jun 19 at 20:52 | history | edited | Coherent Sheaf | CC BY-SA 4.0 |
added 183 characters in body
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| Jun 19 at 20:47 | history | asked | Coherent Sheaf | CC BY-SA 4.0 |