Unicity of branched covering of sphere, and Hurwitz numbers - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T21:39:18Z http://mathoverflow.net/feeds/question/44530 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/44530/unicity-of-branched-covering-of-sphere-and-hurwitz-numbers Unicity of branched covering of sphere, and Hurwitz numbers grok 2010-11-02T09:13:09Z 2010-11-02T15:34:38Z <p>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:</p> <p>Is there a classification of all "Hurwitz data" (namely, degrees at critical points) for which the covering is determined uniquely?</p> <p>For example, if the branch data are {d,d}, then the map has to be $z^d$, up to Möbius transformations. A few other cases I found out are:</p> <ul> <li>if the data are {d,m,(d+1-m)}, then the map has to be $\int z^{m-1}(1-z)^{d-m}$;</li> <li>if the data are {d,m+(d-m),2}, then the map has to be $z^m(1-z)^{d-m}$;</li> <li>if the data are {n+n,2+...+2,2+...+2}, then the map has to be $z^n/(1+z^n)^2$;</li> <li>if the data are {m+n,m+n,3}, the map is $z^m((m-n)z-(m+n))^n/((m+n)z+(m-n))^n$.</li> </ul> <p>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).</p> <p>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.</p>