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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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##### References:
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