# Computing global monodromy of branched covering of Riemann sphere

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?