Timeline for Groups acting on Riemann Surfaces
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 24, 2011 at 21:05 | comment | added | jlk | Torsten Ekedahl has a nice write-up of this approach in his answer to this question mathoverflow.net/questions/54994/… | |
| Feb 24, 2011 at 18:36 | answer | added | Yuri Zarhin | timeline score: 5 | |
| Feb 24, 2011 at 16:07 | answer | added | JSE | timeline score: 3 | |
| Feb 24, 2011 at 15:45 | comment | added | S. Carnahan♦ | If a complex curve of genus $g>1$ has nontrivial automorphism group, one can show that the (complex) dimension of the deformation space is $3g-3$, and use another theorem of Hurwitz to compare that with the deformation space of the quotient curve equipped with the finite collection of branch points. If $g>2$, you find that there are directions you can deform the curve such that any symmetry is destroyed - that is, the locus of curves with nontrivial automorphisms has strictly smaller dimension than the full moduli space. | |
| Feb 24, 2011 at 11:20 | comment | added | Daniel Loughran | Sorry I should have also made clear that these closed subsets are also not open, i.e. not the empty set or the whole space! | |
| Feb 24, 2011 at 11:18 | comment | added | Daniel Loughran | @Martin: A high-brow (and perhaps slightly vague) way to think about it is as follows. Let $M_g$ be the moduli space of curves (=Riemann surfaces) of genus $g>2$ (i.e. the space which parametrises all algebraic curves of genus $g$). Then the collection of curves with non-trivial automorphism group forms a closed subset, as do the collection of hyperelliptic curves. When you construct this moduli space you need to take these automorphisms into account, and the automorphism groups of your objects better be finite or things get funny. These ideas are prevalent in Geometric invariant theory. | |
| Feb 24, 2011 at 10:37 | comment | added | Martin David | @Daniel: Can you explain first statement in your comments? | |
| Feb 24, 2011 at 10:29 | answer | added | Charles Matthews | timeline score: 3 | |
| Feb 24, 2011 at 10:27 | comment | added | Daniel Loughran | I would have thought that "almost all" compact Riemann surfaces of genus >2 have trivial automorphism group. Any hyperelliptic curve (in particular any Riemann surface of genus two) has a hyperelliptic involution (i.e. action by $C_2$), but these surfaces are quite special. | |
| Feb 24, 2011 at 10:00 | history | asked | Martin David | CC BY-SA 2.5 |