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On the Hurwitz scheme and its monodromy. (English) Zbl 0726.14022
The author works over the complex field \({\mathbb{C}}\). - Theorem 1: Let f,g: \(C\to {\mathbb{P}}^ 1\) be two coverings of \({\mathbb{P}}^ 1\) by a smooth irreducible curve C of genus \(\geq 1\). Assume that both have simple branching and that the branching occurs over the same set \(\Gamma \subset {\mathbb{P}}^ 1\). If \(\Gamma\) is sufficiently general, then f and g are isomorphic as coverings of \({\mathbb{P}}^ 1.\)
This theorem was previously known if \(d<(g-1)/2\) [E. Arbarello and M. Cornalba, Math. Ann. 256, 341-362 (1981; Zbl 0454.14023)]. - Examples showing that the conditions: genus\((C)\geq 1\) and \(\Gamma\) sufficiently general are necessary, are given. - The authors show that this theorem implies that the Tyrell conjecture (saying that the 40 elliptic curves tangent to 6 general concurrent lines are pairwise nonisomorphic) is true.
In the last section some geometric results on the monodromy of the covering \({\mathcal H}_{d,g}\to {\mathcal P}_ b\) (where \({\mathcal H}_{d,g}\) is the Hurwitz scheme of degree \(d\) branched covers of \({\mathbb{P}}^ 1\) by curves of genus \(g\geq 1\) and \({\mathcal P}_ b\) is the moduli space of b- pointed rational curves with \(b=2d-2+2g)\) are presented. Geometric interpretations of a Cohen result for \(d=3\) [D. B. Cohen, J. Algebra 32, 501-517 (1974; Zbl 0343.20002)] and a Maclachlan remark for \(d=4\) [C. Maclachlan, Mich. Math. J. 25, 235-244 (1978; Zbl 0366.20032), last paragraph] are given.

MSC:
14H30 Coverings of curves, fundamental group
14F45 Topological properties in algebraic geometry
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