Timeline for Finding an algebraic equation given divisors
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 5, 2014 at 9:26 | comment | added | Roberto Pignatelli | Dami Lee is indeed describing the smooth quartic curve $XY(X^2-Y^2)=Z^4$ pointed by abx. The $\omega_i$ have a nontrivial common divisors $\eta_3=P_1+P_2+2P_3$ and can be written $\omega_i=\eta_i +\eta_3$. Then $\eta_1, \eta_2, \eta_3$ are a basis of the canonical system, cut by the hyplerplane sections $X, Y, Z$. | |
| Aug 5, 2014 at 4:22 | comment | added | abx | If your surface has genus 3, the divisor of a nonzero holomorphic 1-form has degree 4. | |
| Aug 4, 2014 at 23:12 | comment | added | Dami Lee | I'm not familiar with Etale covers yet, but thank you for the reference. Let me show you how I got to this equation. First of all, I think this surface is of genus 3 (using Hurwitz formula). Secondly, let me define $f:= \omega_1/\omega_3, g:= \omega_2/\omega_3.$ Then sending $P_i$s to $\infty, 0 ,1$ resp. and normalizing the quotients give me $\left((f/g)^2 - 1\right) f g^3 = 1.$ Please feel free to point out anything that I'm missing. | |
| Aug 4, 2014 at 18:58 | history | answered | abx | CC BY-SA 3.0 |