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Let us denote by ${\\mathcal S}_{d}$ the singular locus of ${\\rm M}_{d}$ and by ${\\mathcal B}_{d}$ the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify ${\\rm M}_2$ with ${\\mathbb C}^2$ and, within that identification, that ${\\mathcal B}_{2}$ is a cubic curve; so ${\\mathcal B}_{2}$ is connected and ${\\mathcal S}_{2}=\\emptyset$. 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Hidalgo, Saul Quispe","submitted_at":"2015-02-10T17:34:23Z","abstract_excerpt":"Milnor proved that the moduli space ${\\rm M}_{d}$ of rational maps of degree $d \\geq 2$ has a complex orbifold structure of dimension $2(d-1)$. Let us denote by ${\\mathcal S}_{d}$ the singular locus of ${\\rm M}_{d}$ and by ${\\mathcal B}_{d}$ the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify ${\\rm M}_2$ with ${\\mathbb C}^2$ and, within that identification, that ${\\mathcal B}_{2}$ is a cubic curve; so ${\\mathcal B}_{2}$ is connected and ${\\mathcal S}_{2}=\\emptyset$. 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