Using the Broughton’s equisymmetric stratification of the moduli space of compact Riemann surfaces of genus g≥2\documentclass[12pt]{minimal}
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\begin{document}$$g \ge 2$$\end{document}, Bartolini, Costa and Izquierdo have shown that its singular locus Sg\documentclass[12pt]{minimal}
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\begin{document}$$S_{g}$$\end{document} is disconnected except for g=3,4,7,13,17,19,59\documentclass[12pt]{minimal}
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\begin{document}$$g = 3, 4, 7, 13, 17, 19, 59$$\end{document}. One of a lot of problems motivated by this result concerns the study of its connected components, both from the quantitative and qualitative points of view. In 2012, Bartolini and Izquierdo have shown that all strata corresponding to the actions of Z2\documentclass[12pt]{minimal}
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\begin{document}$$Z_{2}$$\end{document} and Z3\documentclass[12pt]{minimal}
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\begin{document}$$Z_{3}$$\end{document} are contained in a single connected component Cg2,3\documentclass[12pt]{minimal}
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\begin{document}$$C^{2,3}_g$$\end{document}. The other components which were known to these authors were those composed by single strata and they asked whether there are components different from Cg2,3\documentclass[12pt]{minimal}
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\begin{document}$$C^{2,3}_ g $$\end{document} being the union at least two equisymmetric strata. The aim of this paper is to give an affirmative answer to this question, by showing that there are connected components composed of precisely two strata in Sg\documentclass[12pt]{minimal}
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\begin{document}$$S_{g}$$\end{document} for g=rp(p-1)/2\documentclass[12pt]{minimal}
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\begin{document}$$g = rp(p -1)/2$$\end{document} where r≥6\documentclass[12pt]{minimal}
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\begin{document}$$r \ge 6$$\end{document} and p>5r+3\documentclass[12pt]{minimal}
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\begin{document}$$p > 5r + 3$$\end{document} is a prime for which pr+2\documentclass[12pt]{minimal}
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\begin{document}$$pr + 2$$\end{document} is also a prime.
