Timeline for Construction of curves and morphisms
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 28, 2015 at 15:53 | comment | added | abx | You better see this homomorphism as an action of $\pi _1$ on a set $E$ with $d$ elements. The fiber at $p_i$ is naturally identified with the quotient of $E$ by the action of $c_i\in \pi _1$ (the class of a small loop around $p_i$). For instance, you have only one simple ramification point iff the image of $c_i$ in $\mathfrak{S}_d$ is a transposition. | |
| Apr 28, 2015 at 12:21 | comment | added | user39380 | Sorry, does $\pi_1\to S_d$ gives a degree $d!$ covering? If we choose a quotient of $S_d$ of order $d$, we get a $d$ sheet covering. How to read the ramification | |
| Apr 28, 2015 at 3:48 | comment | added | abx | No, it is much more complicated. Given such a choice, you must still choose a homomorphism $\pi _1\rightarrow \mathfrak{S}_d$ with certain properties, modulo conjugation. This is a difficult and interesting question -- google "Hurwitz numbers". | |
| Apr 28, 2015 at 1:56 | comment | added | user39380 | If we choose points $p_i$ on $C_g$ with prescribed ramification index $e_i$, such that $\sum (e_i-1)=2g-2-d(2g'-2)$. What will be the number of isomorphic classes of branched covering $C_g'\to C_g$? | |
| Apr 27, 2015 at 14:26 | vote | accept | CommunityBot | ||
| Apr 23, 2015 at 14:53 | history | answered | abx | CC BY-SA 3.0 |