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Irreducible components of Hurwitz spaces parameterizing Galois coverings of curves of positive genus. (English) Zbl 1338.14031
Given a complex smooth projective curve \(Y\) of genus \(\geq 1\), a finite group \(G\) and a positive integer \(n\geq 1\), let \(H_n^G(Y)\) be the Hurwitz space classifying the \(G\)-equivalence classes of \(G\)-covers over \(Y\) branched over \(n\) points on \(Y\). Then, \(H_n^G(Y)\) is a finite etale cover over the symmetric configuration space \(Y^{(n)}\setminus\Delta\) parametrizing the sets of points on \(Y\) with cardinality \(n\).
The author describes monodromy actions of the explicit generators of \(\pi_1(Y^{(n)}\setminus\Delta)\) due to J. S. Birman [Commun. Pure Appl. Math. 22, 41–72 (1969; Zbl 0157.30904)] and G. P. Scott [Proc. Camb. Philos. Soc. 68, 605–617 (1970; Zbl 0203.56302)] on the Hurwitz systems \((t_1,\dots,t_n,\lambda_1,\mu_1,\dots,\lambda_g,\mu_g)\in G^{2g+n}\) which are by definition those tuples whose entries generate \(G\) and satisfy \(t_i\neq 1\) \((i=1,\dots,n)\) and \(t_1\cdots t_n=[\lambda_1,\mu_1]\cdots [\lambda_g,\mu_g]\).
In the last section, presented is a useful tool for determining if two Hurwitz systems are braid-equivalent: Let \((t_i,\lambda_j,\mu_j)_{1\leq i\leq n,1\leq j\leq g}\) be a Hurwitz system for a (not necessarily finite) group \(G\). Suppose \(t_s t_{s+1}=1\) for a particular \(s\) (\(1\leq s\leq n-1\)). Then, for any element \(h\in G\) written as a product of the \(t_k^{\pm 1},\lambda_l^{\pm 1},\mu_l^{\pm 1}\) \((1\leq k\leq n,1\leq l\leq g; k\not\in\{s, s+1\})\), the system \((t_i,\lambda_j,\mu_j)_{1\leq i\leq n,1\leq j\leq g}\) is braid-equivalent to the system obtained by replacing \(t_s,t_{s+1}\) by their \(h\)-conjugates respectively.

14H10 Families, moduli of curves (algebraic)
14H30 Coverings of curves, fundamental group
20F36 Braid groups; Artin groups
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