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Special loci in moduli spaces of curves. (English) Zbl 1071.14028
Schneps, Leila (ed.), Galois groups and fundamental groups. Cambridge: Cambridge University Press (ISBN 0-521-80831-6/hbk). Math. Sci. Res. Inst. Publ. 41, 217-275 (2003).
Let $${\mathcal M}_{g, n}$$ be the moduli space of Riemann surfaces with $$n$$ ordered marked points; the permutation group $$S_n$$ acts naturally on this space, and the quotient $${\mathcal M}_{g, n}/S_n =: {\mathcal M}_{g, [n]}$$ classifies the Riemann surfaces with $$n$$ unordered marked points. One may look at $${\mathcal M}_{g, n}$$ (respectively, $${\mathcal M}_{g, [n]}$$) as the quotient of $${\mathcal T}_{g, n}$$ by the action of the mapping class group $$\Gamma_{g, n}$$ (respectively, $$\Gamma_{g, [n]}$$), where $${\mathcal T}_{g, n}$$ is the Teichmüller space. Let $$\varphi$$ be an element of finite order of the mapping class group, then the image, in the moduli space, of the points of $${\mathcal T}_{g, n}$$ fixed by $$\varphi$$ is called the special locus of $$\varphi$$.
The paper under review presents several results on special loci. The author starts by reviewing results on the moduli spaces of Riemann surfaces and on the geometry of $${\mathcal M}_{g, n}$$ and $${\mathcal M}_{g, [n]}$$ for $$(g, n)$$ in $$\{(0,4), (0,5), (1,1), (1,2)\}$$. After that, she proves some results describing the special loci in $${\mathcal M}_{0, n}$$ and $${\mathcal M}_{0, [n]}$$, for arbitrary $$n$$. The author then goes on to present many results on the relation of the special locus of $$\varphi$$ and the moduli space of the topological quotient $$S/\varphi$$.
For the entire collection see [Zbl 1021.00013].

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14H55 Riemann surfaces; Weierstrass points; gap sequences 14H37 Automorphisms of curves 14H30 Coverings of curves, fundamental group 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 14E20 Coverings in algebraic geometry
##### Keywords:
automorphisms of curves
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