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On the Milnor Formula in Arbitrary Characteristic

In: Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

Author

Listed:
  • Evelia R. García Barroso

    (Estadística e I.O. Sección de Matemáticas, Universidad de La Laguna, Departamento de Matemáticas)

  • Arkadiusz Płoski

    (Kielce University of Technology, Department of Mathematics and Physics)

Abstract

The Milnor formula μ = 2δ − r + 1 relates the Milnor number μ, the double point number δ and the number r of branches of a plane curve singularity. It holds over the fields of characteristic zero. Melle and Wall based on a result by Deligne proved the inequality μ ≥ 2δ − r + 1 in arbitrary characteristic and showed that the equality μ = 2δ − r + 1 characterizes the singularities with no wild vanishing cycles. In this note we give an account of results on the Milnor formula in characteristic p. It holds if the plane singularity is Newton non-degenerate (Boubakri et al. Rev. Mat. Complut. 25:61–85, 2010) or if p is greater than the intersection number of the singularity with its generic polar (Nguyen Annales de l’Institut Fourier, Tome 66(5):2047–2066, 2016). Then we improve our result on the Milnor number of irreducible singularities (Bull. Lond. Math. Soc. 48:94–98, 2016). Our considerations are based on the properties of polars of plane singularities in characteristic p.

Suggested Citation

  • Evelia R. García Barroso & Arkadiusz Płoski, 2018. "On the Milnor Formula in Arbitrary Characteristic," Springer Books, in: Gert-Martin Greuel & Luis Narváez Macarro & Sebastià Xambó-Descamps (ed.), Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, pages 119-133, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-96827-8_5
    DOI: 10.1007/978-3-319-96827-8_5
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