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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)

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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