On the existence of connected components of dimension one in the branch locus of moduli spaces of riemann surfaces

Antonio F. Costa, Milagros Izquierdo

Abstract


Let $g$ be an integer $\geq3$ and let $B_{g}=\{X\in\mathcal{M}_{g}: \mathrm{Aut}(X)\neq Id\}$ be the branch locus of $M_{g}$, where $M_{g}$ denotes the moduli space of compact Riemann surfaces of genus $g$. The structure of $B_{g}$ is of substantial interest because $B_{g}$ corresponds to the singularities of the action of the modular group on the Teichm├╝ller space of surfaces of genus $g$ (see [14]). Kulkarni ([15], see also [13]) proved the existence of isolated points in the branch loci of the moduli spaces of Riemann surfaces. In this work we study the isolated connected components of dimension 1 in such loci. These isolated components of dimension one appear if the genus is $g=p-1$ with $p$ prime $\geq11$. We use uniformization by Fuchsian groups and the equisymmetric stratification of the branch loci.

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DOI: http://dx.doi.org/10.7146/math.scand.a-15213

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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