Timeline for Examples of the moduli space of X giving facts about a certain X
Current License: CC BY-SA 4.0
7 events
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| Oct 22, 2022 at 15:35 | history | edited | LSpice | CC BY-SA 4.0 |
`\operatorname`, while this is on the front page
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| May 28, 2010 at 17:31 | comment | added | BCnrd | @Dmitri: Some not-so-linguistic examples come to mind when moduli space is etale. Deformation theory (vanishing of tangent space) shows moduli spaces of polarized CM ab. var. in char. 0 and Hom-schemes between abelian varieties are etale, so CM ab. var. are defined over number field and "geometric" Homs between ab. var. are defined over sep'ble ext'n of ground field. (Pfs can be more direct.) Likewise, finiteness of set of finite maps between two curves of genus $> 1$ over alg. closed field of any char.: can prove suitable Hilbert scheme has tang. spaces = 0, hence etale, hence finite. | |
| May 28, 2010 at 6:30 | comment | added | Dmitri Panov | BCnrd, this is just lingustics indeed. I was just afraid that in algebraic geometry there will not be a non-trivial understandable statement, that motivates introduction of moduli spaces from the point of view of the preson who asked the question. By the way, can you formulate someting in algebraic geometry the seems to be interesting (while it is only linguistics), and uses the same trick - that the moduli space is a point? | |
| May 28, 2010 at 3:58 | comment | added | BCnrd | @Henry: OK, so it seems one does give a direct proof. But then I don't understand why rephrasing this as saying "moduli space is a point" is anything deeper than linguistics. That is, I don't see how this is an example of a moduli space being used in an essential way, since what we are learning is actually proved directly, without recourse to any concept of moduli space. I am not saying that this reformulation is a bad thing, but just that I don't see how introducing the moduli space viewpoint provides insight in this case. | |
| May 28, 2010 at 2:27 | comment | added | HJRW | BCnrd, this question gives a summary of the proof: mathoverflow.net/questions/21986/… . | |
| May 28, 2010 at 1:08 | comment | added | BCnrd | I don't know much about hyperbolic geometry, so maybe this is a naive question: does the proof that it is a point involve something other than proving directly the conclusion that you mention? | |
| May 27, 2010 at 22:01 | history | answered | Dmitri Panov | CC BY-SA 2.5 |