Timeline for Curves which are not covers of P^1 with four branch points
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 5, 2011 at 16:17 | comment | added | JSE | Aha, gotcha! That's reassuring -- and indeed my memory is that this is just how Kodaira constructed compact families in M_g in the first place. | |
| Aug 5, 2011 at 15:49 | comment | added | Jason Starr | Edit -- $\OO$ should be $\mathcal{O}$. | |
| Aug 5, 2011 at 15:49 | comment | added | Jason Starr | You are correct that the abstract curve $C_b$ is not changing! But we can run the same argument with higher genus curves. Just let $f:E\to \mathbb{P}^1$ be a genus $g>1$ Belyi-curve. Let $g:E'\to E$ be an 'etale map of even degree. Let $B$ parameterize divisors $D_x$ together with a square root of $\OO_{E'}(D_x)$, etc. This will produce a complete family where the moduli of the curves $C_b$ actually does change. | |
| Aug 5, 2011 at 15:34 | comment | added | JSE | Let me ask a dumb question: is the cover h_b: C_b -> E' actually changing as b moves around? Any choice of D_x is isomorphic (via translation on E') to any other, right? | |
| Aug 5, 2011 at 13:02 | comment | added | Jason Starr | Start with an elliptic curve which is a Belyi-branched cover of $\mathbb{P}^1$, say $f:E\to \mathbb{P}^1$. Let $g:E'\to E$ be some isogeny of elliptic curves of even degree $d$. For every point $x$ in $E$, let $D_x$ be the preimage divisor in $E'$. Let $u:B \to E$ be the parameter space for a point $x=u(b)$ together with an invertible sheaf $L_b$ on $E'$ such that $L_b^2 = O_{E'}(D_x)$. Associated to $D_x$ and $L_b$, there is a degree 2 branched cover $h_b:C_b \to E'$ branched over only $D_x$. So the composition $f\circ g \circ h_b:C_b\to \mathbb{P}^1$ is branched over only $4$ points. | |
| Aug 5, 2011 at 13:00 | comment | added | JSE | That's an excellent point, Jason -- as you say, it's definitely the case that there are spaces of n-branched covers which miss the boundary. On the other hand, there's somehow more room for a curve to intersect the boundary in a "funny way" in M_{0,n} than in M_{0,4}.... Basically, I don't know! | |
| Aug 5, 2011 at 12:26 | comment | added | Jason Starr | I apologize, but I don't see anything in the argument above that could not be generalized to covers of $\mathbb{P}^1$ branched over $n$ points, where $n$ is any integer. We know that there are complete curves in $M_g$ parameterizing covers of $\mathbb{P}^1$ branched over $n$ points. Why is $n=4$ special? By the way, although I completely agree that connected components of Hurwitz spaces are complicated, there is quite a bit known about them, for instance, cf. Kluitman's article in the Braids volume (reporting on Deligne's results), Richard Hamilton's thesis, work of Gabai and Kazez, ... | |
| Aug 4, 2011 at 22:33 | comment | added | JSE | No, I'm saying that it's very believable that compact curves in M_g provide examples, but it might be hard to prove. | |
| Aug 4, 2011 at 21:49 | comment | added | Jason Starr | So are you saying that you believe Ben's argument above is rigorous? | |
| Aug 4, 2011 at 18:33 | history | answered | JSE | CC BY-SA 3.0 |