Finite isoresidual covers in strata of k-differentials

Consider the strata of primitive k-differentials on the Riemann sphere whose singularities, except for two, are poles of order divisible by k. The map that assigns to each k-differential the k-residues at these poles is a ramified cover of its image. Generalizing results known in the case of abelian differentials, we describe the ramification locus of this cover and provide a formula, involving the k-factorial function, for the cardinality of each fiber. We prove this formula using intersection calculations on the multi-scale compactification of the strata of k-differentials. In special cases, we also give alternative proofs using flat geometry. Finally, we present an application to cone spherical metrics with dihedral monodromy.