Osaka Journal of Mathematics

The boundary of the Milnor fiber for some non-isolated singularities of complex surfaces

Françoise Michel, Anne Pichon, and Claude Weber

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Abstract

We study the boundary $L_{t}$ of the Milnor fiber for the non-isolated singularities in $\mathbf{C}^{3}$ with equation $z^{m} - g(x,y) = 0$ where $m \geq 2$ and $g(x,y)=0$ is a non-reduced plane curve germ. We give a complete proof that $L_{t}$ is a Waldhausen graph manifold and we provide the tools to construct its plumbing graph. As an example, we give the plumbing graph associated to the germs $z^{2} - (x^{2} - y^{3})y^{l} = 0$ with $l$ odd and $l \geq 3$. We prove that the boundary of the Milnor fiber is a Waldhausen manifold new in complex geometry, as it cannot be the boundary of a normal surface singularity.

Article information

Source
Osaka J. Math., Volume 46, Number 1 (2009), 291-316.

Dates
First available in Project Euclid: 25 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1235574049

Mathematical Reviews number (MathSciNet)
MR2531151

Zentralblatt MATH identifier
1165.32012

Subjects
Primary: 14J17: Singularities [See also 14B05, 14E15] 32S25: Surface and hypersurface singularities [See also 14J17] 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

Citation

Michel, Françoise; Pichon, Anne; Weber, Claude. The boundary of the Milnor fiber for some non-isolated singularities of complex surfaces. Osaka J. Math. 46 (2009), no. 1, 291--316. https://projecteuclid.org/euclid.ojm/1235574049


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