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Alexander polynomials of Hurwitz curves. (English. Russian original) Zbl 1112.14019
Izv. Math. 70, No. 1, 69-86 (2006); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 70, No. 1, 75-94 (2006).
The work under review is a sequel of the paper [G.-M.Greuel and V. S. Kulikov, Izv. Math. 69, No. 4, 667–701 (2005; Zbl 1093.14023)], where the Alexander polynomials of Hurwitz \(C\)-groups and the corresponding Hurwitz curves in \({\mathbb C}{\mathbb P}^2\) are studied.
In particular, the author proves that the necessary condition from [loc. cit.] for a polynomial \(P(t) \in \mathbb Z [t]\) to be the Alexander polynomial of an irreducible \(C\)-group is, in fact, also a sufficient one. This result leads to a complete description of the set of the Alexander polynomials of irreducible Hurwitz \(C\)-groups; it consists of all the polynomials \(P(t)\) such that \(P(1)=1\) and the roots of \(P(t)\) are roots of unity. The second theorem describes sufficient conditions on \(P(t)\) to be the Alexander polynomial of a reducible Hurwitz \(C\)-groups. The author also conjectures the necessity of these conditions.
As an application he proves that \(P(t)=(-1)^{n+k}(t-1)^{n}(t+ 1)^{k}\) is the Alexander polynomial of a Hurwitz \(C\)-group if and only if \( n\geq k.\) In conclusion two examples with explicitly given \(C\)-presentations are considered; they don’t correspond to a Hurwitz \(C\)-group since their Alexander polynomials \((t-1)^2\) and (\((1-t)(t+1)^2\) do not satisfy the necessary conditions.
MSC:
14F35 Homotopy theory and fundamental groups in algebraic geometry
14H30 Coverings of curves, fundamental group
14H50 Plane and space curves
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