Timeline for Is the set of all curves that have a Galois map to the projective line Zariski closed in M_g?
Current License: CC BY-SA 3.0
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| Sep 7, 2011 at 16:49 | answer | added | rita | timeline score: 10 | |
| Sep 6, 2011 at 0:54 | comment | added | Noam D. Elkies | @M.Duff: Not sure about either parts 1 or 2 of the argument. For a $g=2$ curve, the six unordered branch points of the 2:1 cover need not be definable over their field of moduli. $\phantom{ZZZZZ}$ For $g \geq 2$, yes, that's what I'm trying to say. There are finitely many possibilities for: $G$, the number $b$ of branch points, and the conjugacy classes in $G$ of the monodromy generator above each branch point; for each choice, the relevant part of the locus in ${\cal M}_g$ is identified with an unramified cover of the moduli space of $b$ unordered but distinct points in $S$, "etc." | |
| Sep 5, 2011 at 18:16 | comment | added | Makhalan Duff | You make a good point... Here was my argument (which I guess is false): 1. the field of moduli of the curve is equal to the field of moduli of the cover (considered without an isomorphism with group), 2. if a cover is Galois then its field of moduli is a field of definition, 3. therefore the curve is defined over its own field of moduli. About your $g\geq 2$ case: it is true that only finitely many group can arise. Are you saying that for each one we get a Zariski closed subset of $M_g$? | |
| Sep 5, 2011 at 18:10 | comment | added | Noam D. Elkies | How do you obtain that "if $C \rightarrow {\bf P}^2$ is Galois, this implies that $C$ is defined over its field of moduli (as a curve)"? Not all genus-2 curves are defined over their field of moduli, but as a Riemann surface every genus-2 curve is a 2:1 cover of the Riemann sphere $S$, and this cover is Galois. $$ $$ In general, if $g \leq 2$ then every genus-$g$ curve is a 2:1 cover of $S$, while if $g \geq 2$ the Hurwitz bound $84(g-1)$ on $\#({\rm Aut}(C))$ shows only finitely many groups can arise as ${\rm Gal}(C/S)$. In either case we get a Zariski-closed subset of ${\cal M}_g$. | |
| Sep 5, 2011 at 17:51 | history | edited | Makhalan Duff | CC BY-SA 3.0 |
Fixed grammar
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| Sep 5, 2011 at 17:26 | history | asked | Makhalan Duff | CC BY-SA 3.0 |