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Analytic theory of curvettes and dicriticals

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Abstract

The geometric theory of pencils in a germ of a smooth complex surface has been studied by several authors and their properties strongly lie on the structure of the dicritical components of their resolution, whereas the strict transform of the general element of the pencil by the resolution can be seen as a union of the so-called curvettes. This theory can be interpreted as the study of ideals (with two generators) in the ring of complex convergent power series in two variables, which is a local regular ring of dimension 2. In this work, we study properties of curvettes and dicriticals in an arbitrary local regular ring of dimension 2 (without restrictions on its characteristic or the one of its residue field). All the results are stated in purely algebraic terms though the ideas come from geometry.

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Acknowledgments

Second author is partially supported by MTM2010-21740-C02-02.

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Authors and Affiliations

  1. Mathematics Department, Purdue University, West Lafayette, IN, 47907, USA

    Shreeram S. Abhyankar

  2. Departamento de Matemáticas-IUMA, Universidad de Zaragoza, C/ Pedro Cerbuna 12, 50009 , Zaragoza, Spain

    Enrique Artal Bartolo

Authors
  1. Shreeram S. Abhyankar
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  2. Enrique Artal Bartolo
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Correspondence to Enrique Artal Bartolo.

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S. S. Abhyankar: Deceased.

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Abhyankar, S.S., Artal Bartolo, E. Analytic theory of curvettes and dicriticals. Rev Mat Complut 27, 461–499 (2014). https://doi.org/10.1007/s13163-014-0148-4

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  • DOI: https://doi.org/10.1007/s13163-014-0148-4

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