  We study the three-point quantum $\mathfrak{sl_2}$-Gaudin model. In this case
the compactification of the parameter space is
$\overline{M_{0,4}(\mathbb{C})}$, which is the Riemann sphere. We analyze
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 deduce some consequences regarding the geometric
structure of the Gaudin coverings.
