# riemann surfaces – Recovering a family of rational functions from branch points

Let $$Y$$ be a compact Riemann surface and $$B$$ a finite subset of $$Y$$. It is a standard fact that isomorphism classes of holomorphic ramified covers $$f:Xrightarrow Y$$ of degree $$d$$ with branch points in $$B$$ are in a correspondence with homomorphisms $$rho:pi_1(Y-B)rightarrow S_d$$ with transitive image modulo conjugation by elements of the permutation group $$S_d$$. Writing a formula for $$f$$ from the knowledge of $$Bsubset Y$$ and $$rho$$ is often hard, e.g. the task of recovering a Belyi map from its dessin where $$|B|=3$$. I am interested in the case of $$X=Y=Bbb{CP}^1$$, and some points from $$B$$ moving in the Riemann sphere. Here is an example:

• Consider rational functions $$f:Bbb{CP}^1rightarrowBbb{CP}^1$$ of degree $$3$$ with four simple critical points that have $$1,omega,bar{omega}$$ among their critical values $$left(omega={rm{e}}^{frac{2pi{rm{i}}}{3}}right)$$, thus $$B={1,omega,bar{omega},beta}$$ with $$beta$$ varying in a punctured sphere. To fix an element in the isomorphism class, we can pre-compose $$f$$ with a suitable Möbius transformation so that $$1$$, $$bar{omega}$$ and $$omega$$ are the critical points lying above $$1$$, $$omega$$ and $$bar{omega}$$ respectively: $$f(1)=1, f(bar{omega})=omega, f(omega)=bar{omega}$$. A normal form for such functions is
$$left{f_alpha(z):=frac{alpha z^3+3z^2+2alpha}{2z^3+3alpha z+1}right}_alpha.$$
A simple computation shows that the fourth critical point is $$alpha^2$$, and hence $$beta=beta_alpha=:f_alpha(alpha^2)=frac{alpha^4+2alpha}{2alpha^3+1}$$. Here is my question:

Why $$beta$$ is not a degree one function of $$alpha?$$ Shouldn’t the knowledge of the branch locus and the monodromy determine $$f_alpha(z)$$ in the normalized form above? I presume the monodromy does not change because there are only finitely many possibilities for it and this is a continuous family.

To monodromy of $$f_alpha$$ is a homomorphism
$$rho_alpha:pi_1left(Bbb{CP}^1-{1,omega,bar{omega},beta_alpha}right)rightarrow S_3$$
where small loops around $$1,omega,bar{omega},beta_alpha$$ generate the fundamental group, and are mapped to transpositions in $$S_3$$ whose product is identity and are not all distinct. So I guess my question is how can such a discrete object vary with $$alpha$$; and if it doesn’t, why the assignment $$alphamapstobeta(alpha)$$ is not injective. The degree of this assignment is four, and there are also four conjugacy classes of homomorphisms
$$rho:langlesigma_1,sigma_2,sigma_3,sigma_4midsigma_1sigma_2sigma_3sigma_4=mathbf{1}ranglerightarrow S_3$$ with $${rm{Im}}(rho)$$ being a transitive subgroup of $$S_3$$ generated by transpositions $$rho(sigma_i)$$:
$$sigma_1mapsto (1,2),sigma_2mapsto (1,2),sigma_3mapsto (1,3), sigma_4mapsto (1,3);\ sigma_1mapsto (1,2),sigma_2mapsto (1,3),sigma_3mapsto (1,2), sigma_4mapsto (2,3);\ sigma_1mapsto (1,2),sigma_2mapsto (1,3),sigma_3mapsto (1,3), sigma_4mapsto (1,2);\ sigma_1mapsto (1,2),sigma_2mapsto (1,3),sigma_3mapsto (2,3), sigma_4mapsto (1,3).$$
Is it accidental that the degree of $$alphamapstobeta(alpha)$$ is the same as the number of possibilities for the monodromy representations compatible with our ramification structure?