The three-point Gaudin model and branched coverings of the Riemann sphere

We study the three-point sl₂-Gaudin model. In this case the compactification of the parameter space is M0,4(C), which is the Riemann sphere. We analyze these sphere coverings by the joint spectrum of the Gaudin Hamiltonians treating them as algebraic curves. We write equations of these curves as determinants of tridiagonal matrices and observe that their branch points form remarkable patterns.