2 Had a statement ("3 critical values => unique") for which my argument in wrong -- removed it, and put some examples instead

Hurwitz's encoding counts the number of branched self-coverings of a sphere, with prescribed ramification degrees at the critical points, as numbers of factorizations of the identity in a symmetric group with given cycle lengths. My question is:

Is there a classification of all "Hurwitz data" (namely, degrees at critical points) for which the covering is determined uniquely?

For example, if the branch data are {d,d}, then the map has to be $z^d$, up to Möbius transformations. If there A few other cases I found out arethree critical points, :

• if the data are {d,m,(d+1-m)}, then the map has to be $\int z^{m-1}(1-z)^{d-m}$;
• if the data are {d,m+(d-m),2}, then the map has to be $z^m(1-z)^{d-m}$;
• if the data are {n+n,2+...+2,2+...+2}, then the map has to be $z^n/(1+z^n)^2$;
• if the data are {m+n,m+n,3}, the map is also uniquely determined. $z^m((m-n)z-(m+n))^n/((m+n)z+(m-n))^n$.

On the other hand, if all critical values are simple (so the data is {2,2,...,2}), then there are exponentially many branched coverings (something like a Catalan number).

I'm aware of various combinatorial tools to compute the number of coverings, including "integration on the Deligne-Mumford stack", but all the literature I was able to google up was concerned about cases where there are no branched coverings, or lots.

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# Unicity of branched covering of sphere, and Hurwitz numbers

Hurwitz's encoding counts the number of branched self-coverings of a sphere, with prescribed ramification degrees at the critical points, as numbers of factorizations of the identity in a symmetric group with given cycle lengths. My question is:

Is there a classification of all "Hurwitz data" (namely, degrees at critical points) for which the covering is determined uniquely?

For example, if the branch data are {d,d}, then the map has to be $z^d$, up to Möbius transformations. If there are three critical points, then the map is also uniquely determined. On the other hand, if all critical values are simple (so the data is {2,2,...,2}), then there are exponentially many branched coverings (something like a Catalan number).

I'm aware of various combinatorial tools to compute the number of coverings, including "integration on the Deligne-Mumford stack", but all the literature I was able to google up was concerned about cases where there are no branched coverings, or lots.