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On the topology of the image by a morphism of plane curve singularities

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Abstract

For a finite morphism \(\varphi =(f,g)\) from the plane to the plane we describe the topology of the image of a branch in the source by the use of iterated pencils of analytic functions, constructed inductively in a natural way starting from the components of the map. In the case of the study of the topology of the discriminant curve, image by \(\varphi \) of the critical locus of the map, we show that the special fibres of the pencil \( \langle f,g\rangle \) suffice to determine the topological type of each branch of the discriminant curve. This is due to the known relations that exist between the branches of the critical locus of \(\varphi \) and the special fibres.

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Correspondence to H. Maugendre.

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F. Delgado is partially supported by the Grants MTM2007-64704 and MTM2012-36917-C03-01 (both Grants with the help of FEDER Program). The author is thankful to the Institut Fourier, Université de Grenoble I for hospitality.

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Delgado, F., Maugendre, H. On the topology of the image by a morphism of plane curve singularities. Rev Mat Complut 27, 369–384 (2014). https://doi.org/10.1007/s13163-013-0141-3

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  • DOI: https://doi.org/10.1007/s13163-013-0141-3

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