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Base points of polar curves on a surface of type \(z^n = f(x,y)\). (English) Zbl 1100.14030

The paper deals with base points of polar curves on surfaces with resolved singularities of special types.   For a (normal) surface singularity \((S,0)\subset(\mathbb{C}^N,0)\), consider a projection \((\mathbb{C}^N,0)\rightarrow(\mathbb{C}^2,0)\) generic with respect to the tangent cone of \(S\). When restricted to \(S\), the projection has a critical locus: the polar curve. Thus for generic projections one gets a linear system of polar curves on the surface. One of the problems related to the resolution of surfaces is to determine the base points of polar curves on the resolved surface. This question depends essentially on the type of resolution, e.g. for Nash modification (Nash blow-up) there are no base points.
The author deals with the resolution of singularities by Jung’s method. The idea is first to project the polar curves to the plane \(\mathbb{C}^2_{xy}\) and then to consider the (minimal) embedded resolution of the image curves. For the singularities of the type \(z^n=f(x,y)\) the author proposes an effective criterion on the existence of base points (theorem 4.4 and corollary 4.5). The criterion is in terms of the embedded resolution tree of the plane curve \(f(x,y)=0\) and that of its polar \(\alpha\partial_x f+\beta\partial_yf=0\).   At the end several examples are considered.

MSC:

14J17 Singularities of surfaces or higher-dimensional varieties
32S45 Modifications; resolution of singularities (complex-analytic aspects)
32S25 Complex surface and hypersurface singularities
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