Timeline for What, precisely, is the relationship between "fields of moduli" and "moduli spaces"?
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| Jan 24, 2010 at 20:08 | comment | added | Felipe Voloch | Sorry, I don't think I can answer your more precise question about the intersection of fields of definition. If I really needed the answers, I'd do as Pete suggested and ask Harbater :-) | |
| Jan 24, 2010 at 19:29 | comment | added | H. Hasson | Sorry, I meant "field of moduli", not "moduli space". | |
| Jan 24, 2010 at 19:21 | comment | added | H. Hasson | I see. So your definition of "moduli space" would be closely related to the second (function field) definition. Can you see why it would be equal to the intersection of all finite extension of L where it's still Galois? | |
| Jan 24, 2010 at 19:16 | comment | added | H. Hasson | My familiarity with Hurwitz schemes is mostly from Fulton's "Hurwitz Schemes and Irreducibility of Moduli of Algebraic Curves" - but there he shows that if you fix the genus of the coverings, and the degree of the covers, AND you only look at simple covers (heuristically meaning that over each branch point there's only one ramified point, and it has ramification degree=2) then you have a FINE Hurwitz scheme. But I'm not sure that if you look at all covers you get a coarse moduli space. | |
| Jan 24, 2010 at 19:05 | history | answered | Felipe Voloch | CC BY-SA 2.5 |