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The boundary of the Milnor fiber of the singularity \(f(x,y) + zg(x,y) = 0\)

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Abstract

Let \(f,g\in \mathbb {C}\{x,y\}\) be germs of functions defining plane curve singularities without common components in and let \(\Phi (x,y,z) = f(x,y) + zg(x,y)\). We give an explicit algorithm producing a plumbing graph for the boundary of the Milnor fiber of \(\Phi \) in terms of a common resolution for f and g. We give an example of a choice for f and g yielding a boundary of a Milnor fiber having more than one irreducible component.

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Acknowledgements

I would like to thank Némethi András for suggesting this problem to me and for the many helpful discussions we have had.

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Correspondence to Baldur Sigurðsson.

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The author was partially supported by the ERCEA 615655 NMST Consolidator Grant, by the Basque Government through the BERC 2014–2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323 during the writing of this article.

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Sigurðsson, B. The boundary of the Milnor fiber of the singularity \(f(x,y) + zg(x,y) = 0\). European Journal of Mathematics (2020). https://doi.org/10.1007/s40879-019-00393-w

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Keywords

  • Milnor fiber boundary
  • Plumbed manifold
  • Constructive algorithm

Mathematics Subject Classification

  • 32S25
  • 14J17
  • 57M50