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We are interested in the branch locus ${\\mathcal{B}_{g}}$ for $g>2$, i.e., the subset of $\\mathcal{M}_{g}$ consisting of surfaces with automorphisms. It is well-known that the set of hyperelliptic surfaces (the hyperelliptic locus) is connected in $\\mathcal{M}_{g}$ but the set of (cyclic) trigonal surfaces is not. By contrast, we show that for $g\\geq 5$ the set of (cyclic) trigonal surfaces is connected in $\\bar{\\mathcal{M}_{g}}$. 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Costa, Hugo Parlier, Milagros Izquierdo","submitted_at":"2013-05-01T20:26:50Z","abstract_excerpt":"Consider the moduli space $\\mathcal{M}_{g}$ of Riemann surfaces of genus $g\\geq 2$ and its Deligne-Munford compactification $\\bar{\\mathcal{M}_{g}}$. We are interested in the branch locus ${\\mathcal{B}_{g}}$ for $g>2$, i.e., the subset of $\\mathcal{M}_{g}$ consisting of surfaces with automorphisms. It is well-known that the set of hyperelliptic surfaces (the hyperelliptic locus) is connected in $\\mathcal{M}_{g}$ but the set of (cyclic) trigonal surfaces is not. By contrast, we show that for $g\\geq 5$ the set of (cyclic) trigonal surfaces is connected in $\\bar{\\mathcal{M}_{g}}$. 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