Here is a question which I asked myself (and couldn't answer) while reading "The topology of spaces of rational functions" by G. Segal.
Let X be a smooth complete complex curve (=a compact Riemann surface) of genus g and let Rat(X,d) be the space of all regular (=holomorphic) maps from X to mathbfP1(mathbfC) of degree d. In this question I'm interested in the fundamental group of the open subset U(X,d) of Rat(X,d) formed by all f such that all critical points of f are simple and all critical values are distinct. (A critical point is a point at which the derivative of f vanishes; a critical value is the image of a critical point.) To be more specific, let's say I'd like to
find a "nice" system of generators of pi1(U(X,d));
to describe, for each of these generators, its image under the map induced by the map G from U(X,d) to the configuration space B(mathbfP1(mathbfC),k) of unordered subsets of mathbfP1(mathbfC) of cardinality k=2(d+g−1) that takes f to its branch divisor (i.e. the divisor of the critical points).
Here are some remarks that may be useful (or may not):
First, here is how one can think of the fundamental group of Rat(X,d). By associating to every function its divisor of poles we get a map F from Rat(X,d) to the d-th symmetric power Sd(X) of X.
Assume d>2g−2. By the Riemann-Roch theorem, for any degree d divisor D the linear space calL(D)=H0(X,calO(D)) (which is formed by all rational functions f such that for any xinX the order of the pole of f at x is at most the multiplicity of x in D) is d−g+1. So F is surjective and a fiber of F is mathbfCd−g+1 minus some number of hyperplanes (these are given by the condition that order the pole of f at a point x of D is less then the multiplicity of x in D).
The map F is probably not a fibration. However, the fundamental group of Rat(X,d) is spanned by the loops in a general fiber of F going around one of the hyperplanes, and lifts of the loops in Sd(X) (these are all of the form "one of the points moves along a loop in X and the other stand still").
Second, recall that the Jacobian J(X) of X is defined as follows. Integration along cycles gives an injective map H1(X,mathbfZ)tomathbfCg=Hom(H0(X,OmegaX),mathbfC) and the Jacobian of X is the quotient. Moreover, once we have chosen a base point x in X, we get a natural map j:XtoJ(X) defined as follows: for any x′inX take a path gamma from x to x′ and set j(x′) to be the image in J(X) of the "integration along gamma function". This is well defined map that can be extended by mathbfZ-linearity to Sd(X).
Abel's theorem says that two disjoint effective divisors are the divisors of the zeros and the poles of a rational function if and only if their images under j coincide. This may be useful in this problem, but I don't see how.
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