Groups as Galois Groups

Völklein, Helmut

doi:10.1017/cbo9780511471117

Cited by 157 publications

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“…The equivalence of (i) and (ii) is classical and quite general: the basic idea is that the monodromy cycles of a cover depend not only on the cover, but also on a choice of local monodromy generators of the fundamental group of the base; all such choices of generators are related by braid operations, and each braid operation can be achieved as monodromy of the Hurwitz space by moving the marked points of the base around one another. For a slightly different exposition, see [20,Prop. 10.14 (a)]; note that the situation is slightly different because he considers Hurwitz spaces with unordered branch points and full braid orbits, but the argument is the same in our case of ordered branch points and pure braid orbits.…”

Section: Finally We Recallmentioning

confidence: 99%

The irreducibility of certain pure-cycle Hurwitz spaces

Liu¹,

Osserman²

2008

ajm

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Abstract. We study "pure-cycle" Hurwitz spaces, parametrizing covers of the projective line having only one ramified point over each branch point. We start with the case of genus-0 covers, using a combination of limit linear series theory and group theory to show that these spaces are always irreducible. In the case of four branch points, we also compute the associated Hurwitz numbers. Finally, we give a conditional result in the higher-genus case, requiring at least 3g simply branched points. These results have equivalent formulations in group theory, and in this setting complement results of Conway-Fried-Parker-Völklein. Introduction.In this paper, we use a combination of geometric and group-theoretic techniques to prove a result with equivalent statements in both fields. The geometric statement is that certain genus-0 Hurwitz spaces (the "purecycle" cases) are always irreducible, while the group-theoretic statement is that the corresponding factorizations into cycles always lie in a single pure braid group orbit. "Pure-cycle" refers to the hypothesis that for our covers, there is only a single ramified point over each branch point. The main significance for us of this condition is that it allows us to pass relatively freely between the point of view of branched covers, where one moves the branch points freely on the base curve, and linear series, where one moves the ramification points freely on the covering curve. This facilitates induction, as it is easier to stay within the pure-cycle case from the point of view of linear series.Our result is close to optimal in the sense that if one drops either of the pure-cycle or genus-0 hypotheses, one quickly runs into cases where the Hurwitz spaces have more than one component. However, we do prove a conditional generalization to higher-genus pure-cycle Hurwitz spaces having at least 3g simply branched points, depending on a positive answer to a different geometric question which is closely related to an old question of Zariski.Our immediate motivation for studying the pure-cycle situation is its relation to linear series: specifically, if one wishes to prove statements on branched covers via linear series arguments, the pure-cycle situation is the natural context to examine. A good understanding of the classical situation is therefore important

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Section: Finally We Recallmentioning

confidence: 99%

The irreducibility of certain pure-cycle Hurwitz spaces

Liu¹,

Osserman²

2008

ajm

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“…We will freely use the general theory of Hurwitz spaces (see [Fried and Völklein 1991] and [Völklein 1996], for instance), and only recall here the description of the fibers of ψ r,G and ψ r,G in terms of Nielsen classes Ni(C, G) and straight Nielsen classes SN(C, G). Recall that…”

Section: Preliminariesmentioning

confidence: 99%

Counting real Galois covers of the projective line

Cadoret¹

2005

Pacific J. Math.

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For Galois covers of ‫ސ‬ 1 of a given ramification type -essentially, a given monodromy group G and branch locus, assumed to be defined over ‫ޒ‬ -we ask: How many covers are defined over ‫ޒ‬ and how many are not? J.-P. Serre showed that the number of all Galois covers with given ramification type can be computed from the character table of G. We adapt Serre's method of calculation to the more refined situation of Galois covers defined over ‫,ޒ‬ for which there is a group-theoretic characterization due to P. Dèbes and M. Fried. We obtain explicit answers to our problem. As an application, we exhibit new families of covers not defined over their field of moduli, the monodromy group of which can be chosen arbitrarily large. We also give examples of Galois covers defined over the field ‫ޑ‬ tr of totally real algebraic numbers with ‫-ޑ‬rational branch locus.

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“…, P r in P 1 z \ z z z based at a point z 0 (not in z z z). (For details see [Fri77], [MM], [Ser92] or [Völ96].) The salient points are as follows.…”

Section: Then G O Is the Genus Of The Nonsingular Completionhmentioning

confidence: 99%

“…Our illustrations especially discuss tools for treating the value sets of genus 0 covers. This will show elementary use of Hurwitz families and braid group action as does [Deb99], [Fri90,Fri95b,Fri95a], [MM] and [Völ96]. [Ser92] has simpler examples not requiring Hurwitz families and braid groups.…”

Section: Introduction and The Basic Familiesmentioning

confidence: 99%

Integral specialization of families of rational functions

Dèbes¹,

Fried²

1999

Pacific J. Math.

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Abstract. Suppose C is an algebraic curve, f is a rational function on C defined over Q, and A is a fractional ideal of Q. If f is not equivalent to a polynomial, then Siegel's theorem gives a necessary condition for the set C(Q) ∩ f −1 (A) to be infinite: C is of genus 0 and the fiber f −1 (∞) consists of two conjugate quadratic real points. We consider a converse. Let P be a parameter space for a smooth family Φ : T → P × P 1 of (degree n) genus 0 curves over Q. That is, the fiber Tp p p of points of T over p p p × P 1 has genus 0 forover ∞ consisting of two conjugate quadratic real points. The family Φ is then a Siegel family. We ask when the conclusion of Siegel's theorem -Φp p p(Q) ∩ A is infinite -holds for a Zariski dense subset of p p p ∈ P(Q).We show how braid action on covers and Hurwitz spaces can tackle this. It refines a unirationality criterion for Hurwitz spaces. A particular family, 10 Φ , of degree 10 rational functions, illustrates this. It arises as the exceptional case for a general result on Hilbert's Irreducibility Theorem. We suspect this result generalizes, thus codifying arithmetic accidents occurring in 10 P . To illustrate, we've cast this paper as a collection of elementary group theory tools for extracting from a family of covers special cases with specific arithmetic properties. Examples of Siegel and Néron families show the efficiency of the tools, though each case leaves a diophantine mystery.

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Groups as Galois Groups

Cited by 157 publications

References 0 publications

The irreducibility of certain pure-cycle Hurwitz spaces

The irreducibility of certain pure-cycle Hurwitz spaces

Counting real Galois covers of the projective line

Integral specialization of families of rational functions

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