Given an irreducible polynomial $F(X,Y)\in\mathbb{C}[X,Y]$, there is an associated Riemann surface $S$ obtained from the zero set of $F$. Projection onto the second coordinate gives a map $\pi:S\to \widehat{\mathbb{C}}$ from $S$ to the Riemann sphere which is a branched covering, say of degree $d$. This map is ramified at points $P=(x,y)$ for which $y$ is a repeated root of $F(x,Y)$. As discussed in Computing the monodromy for a cover of the Riemann sphere (and Puiseux expansions), it is possible to compute the local monodromy of the covering about such a point using Puiseux series. Specifically, we can determine the cycle type of the monodromy around $P$; and this is a complete description of the monodromy locally, since labelling of the sheets in a neighborhood of $P$ is arbitrary.

However, to compute a global representation of the monodromy, we need to keep better track of the sheets. If $P_1,...,P_k$ are the ramification points, and $\sigma_1,...\sigma_k$ the monodromy permutations about each, it's not enough to know the cycle types of each $\sigma_i$ considered as an element of the symmetric group $S_d$. The two questions determine the interactions between the $\sigma_i$ for specific coverings The monodromy representation of the projection map from a Fermat Curve, Calculating monodromy, but answers to both use particular features of the covering. Is there a general technique?


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