Monday, 3 July 2006

ag.algebraic geometry - Fundamental groups of the spaces of rational functions

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 XX be a smooth complete complex curve (=a compact Riemann surface) of genus gg and let Rat(X,d)Rat(X,d) be the space of all regular (=holomorphic) maps from XX to mathbfP1(mathbfC)mathbfP1(mathbfC) of degree dd. In this question I'm interested in the fundamental group of the open subset U(X,d)U(X,d) of Rat(X,d)Rat(X,d) formed by all ff such that all critical points of ff are simple and all critical values are distinct. (A critical point is a point at which the derivative of ff vanishes; a critical value is the image of a critical point.) To be more specific, let's say I'd like to



  1. find a "nice" system of generators of pi1(U(X,d))pi1(U(X,d));


  2. to describe, for each of these generators, its image under the map induced by the map GG from U(X,d)U(X,d) to the configuration space B(mathbfP1(mathbfC),k)B(mathbfP1(mathbfC),k) of unordered subsets of mathbfP1(mathbfC)mathbfP1(mathbfC) of cardinality k=2(d+g1)k=2(d+g1) that takes ff 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)Rat(X,d). By associating to every function its divisor of poles we get a map FF from Rat(X,d)Rat(X,d) to the dd-th symmetric power Sd(X)Sd(X) of XX.



Assume d>2g2d>2g2. By the Riemann-Roch theorem, for any degree dd divisor DD the linear space calL(D)=H0(X,calO(D))calL(D)=H0(X,calO(D)) (which is formed by all rational functions ff such that for any xinXxinX the order of the pole of ff at xx is at most the multiplicity of xx in DD) is dg+1dg+1. So FF is surjective and a fiber of FF is mathbfCdg+1mathbfCdg+1 minus some number of hyperplanes (these are given by the condition that order the pole of ff at a point xx of DD is less then the multiplicity of xx in DD).



The map FF is probably not a fibration. However, the fundamental group of Rat(X,d)Rat(X,d) is spanned by the loops in a general fiber of FF going around one of the hyperplanes, and lifts of the loops in Sd(X)Sd(X) (these are all of the form "one of the points moves along a loop in XX and the other stand still").



Second, recall that the Jacobian J(X)J(X) of XX is defined as follows. Integration along cycles gives an injective map H1(X,mathbfZ)tomathbfCg=Hom(H0(X,OmegaX),mathbfC)H1(X,mathbfZ)tomathbfCg=Hom(H0(X,OmegaX),mathbfC) and the Jacobian of XX is the quotient. Moreover, once we have chosen a base point xx in XX, we get a natural map j:XtoJ(X)j:XtoJ(X) defined as follows: for any xinX 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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