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Erschienen in:
2014 | OriginalPaper | Buchkapitel
On the Fundamental Groups of Non-generic \(\mathbb{R}\)-Join-Type Curves
verfasst von : Christophe Eyral, Mutsuo Oka
Erschienen in: Bridging Algebra, Geometry, and Topology
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Abstract
An \(\mathbb{R}\)
-join-type curve is a curve in \(\mathbb{C}^{2}\) defined by an equation of the form \(a \cdot \prod _{j=1}^{\ell}(y -\beta _{j})^{\nu _{j}} = b \cdot \prod _{i=1}^{m}(x -\alpha _{i})^{\lambda _{i}},\) where the coefficients a, b, α
i
and β
j
are real numbers. For generic values of a and b, the singular locus of the curve consists of the points (α
i
, β
j
) with λ
i
, ν
j
≥ 2 (so-called inner singularities). In the non-generic case, the inner singularities are not the only ones: the curve may also have “outer” singularities. The fundamental groups of (the complements of) curves having only inner singularities are considered in Oka (J Math Soc Jpn 30:579–597, 1978). In the present paper, we investigate the fundamental groups of a special class of curves possessing outer singularities.