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Catalan numbers and branched coverings by the Riemann sphere. (English) Zbl 0732.14013
Let $$Rat_ d$$ be the space of degree d rational maps modulo post composition with Moebius transformations, and $$\rho(d):=d^{- 1}\binom{2d-2}{d-1}$$ the Catalan number. - The author proves that the number of classes in $$Rat_ d$$ with a given branch set is generically equal to the Catalan number $$\rho(d)$$.
Let $$Poly_ d$$ denote the space of polynomials whose degree is $$\leq d$$, and $$G_ 2(Poly_ d)$$ the Grassmann manifold of two dimensional subspaces of $$Poly_ d$$. There is an embedding of $$Rat_ d$$ into $$G_ 2(Poly_ d):$$ $$Rat_ d\ni [R]\to X_ R\in G_ 2(Poly_ d),$$ where $$R=P/Q$$ and $$X_ R$$ is generated by P and Q. There is a complex analytic map $$\Phi_ d: G_ 2(Poly_ d)\to {\mathbb{P}}^{2d-2}$$ which connects a subspace generated by P and Q to the polynomial $$PQ'-P'Q$$ whose roots are the critical points of P/Q. The author computes the degree of $$\Phi_ d$$ by transforming the problem to one of enumerative geometry and by using the Schubert calculus.
Reviewer: R.Horiuchi (Kyoto)

##### MSC:
 14H30 Coverings of curves, fundamental group 14M15 Grassmannians, Schubert varieties, flag manifolds 30C10 Polynomials and rational functions of one complex variable 14H55 Riemann surfaces; Weierstrass points; gap sequences 14N10 Enumerative problems (combinatorial problems) in algebraic geometry
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