Timeline for Can a rational family of genus-g curves have generic gonality? Can it be Brill-Noether general?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Feb 8, 2011 at 4:27 | vote | accept | JSE | ||
| Feb 7, 2011 at 4:52 | comment | added | Felipe Voloch | I can't prove offhand that modular curves are not Brill-Noether generic, but here are things one can try: There are maps to elliptic curves (which in turn are double covers of the projective line) which might give low degree maps. The j-map has degree linear with the genus (1/12 or 12?). Another candidate is the line bundle $\omega$, whose global sections are the modular forms of weight one and there might be just enough of them. | |
| Feb 7, 2011 at 3:14 | comment | added | JSE | The only high-genus curves over Q I feel truly acquainted with are the modular curves, or Shimura curves more generally. Do they have unexpected linear systems? (Or rather: do we know they do? The gonality is certainly not known to be generic, though by work of Zograf and Abramovich it at least grows linearly in genus.) | |
| Feb 6, 2011 at 23:59 | answer | added | Gavril Farkas | timeline score: 12 | |
| Feb 6, 2011 at 20:37 | comment | added | Felipe Voloch | Slightly different question in very much the same spirit which I have asked many people and never got an answer: Is there ONE Brill-Noether generic curve defined over $\mathbb{Q}$ for every (or for infinitely many) genus $g$? | |
| Feb 6, 2011 at 19:18 | comment | added | David Zureick-Brown | I believe there is a conjecture that this is possible for gonality d iff d<g/3. I'll email the mathematician who told me this and ask for a reference. | |
| Feb 6, 2011 at 18:37 | history | asked | JSE | CC BY-SA 2.5 |