I'm trying to find an algebraic curve that represents a specific Riemann surface and my question goes like this:
Given divisors $(\omega_1) = P_1 + 5 P_2 + 2 P_3,$ $(\omega_2) = 5 P_1 + P_2 + 2 P_3,$ $(\omega_3) = 2 P_1 + 2 P_2 + 4 P_3,$
is there a way one can come up with an algebraic equation $P(\omega_1, \omega_2, \omega_3) = 0$ with no further information?
What I'm trying to do here is to paste 8 sheets of 3-punctured spheres. At each point $P_i$(or slit), the $j$-th sheet gets glued to $(j+ a_i)(mod \, 8)$-th sheet given $(a_1, a_2, a_3) = (1, 5, 2).$ $\omega_i$s are the holomorphic 1-forms on this surface and assuming that $P_1$ is sent to $\infty,$ $P_2$ to 0, and $P_3$ to 1, I end up with $(\omega_1^2 - \omega_2^2) \omega_1 \omega_2 = \omega_3^4.$ Though I was wondering if there is a way to find this when only the divisors $(\omega_i)$ are given.