# zbMATH — the first resource for mathematics

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.

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