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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
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