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Plane curve complements and curves on Hurwitz spaces. (English) Zbl 1074.14026

In this paper, a method of using Hurwitz moduli spaces of covers of \({\mathbb P}^1\) for the regular inverse Galois problem is refined and applied to several examples. One of the new ingredients of this paper is to introduce a very general class of rational curves on Hurwitz spaces arising from plane curves, where tools on plane curve singularities, dual curves, Puiseux expansions of algebraic functions etc. can play effective roles to analyze the resulting subfamily of Hurwitz spaces through the associated group theoretical data. This enables the author to find rigid tuples for a wide class of finite groups \(G\) that make \(G\) realized as Galois groups over the rational function field \(\mathbb Q(t)\). Among several illustrative examples to present this method, rigid tuples for the Mathieu groups \(M_{23}\), \(M_{11}\) are detected from the concrete plane curves \(x(y-27/4\cdot(x^3+x^2))=0\). Also, it is shown that the method combined with the theory of middle convolutions (or braid companion functors) yields a rigid tuple for SL\(_5(9)\) from the plane curve \(x(y+x^2+1)(y-(x-1)^2)(y-(x+1)^2)=0\).

MSC:

14H30 Coverings of curves, fundamental group
12F12 Inverse Galois theory
14J10 Families, moduli, classification: algebraic theory
14E20 Coverings in algebraic geometry

Software:

BRAID; GAP
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Full Text: DOI Link

References:

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