602 research outputs found
The Orevkov invariant of an affine plane curve
We show that although the fundamental group of the complement of an algebraic
affine plane curve is not easy to compute, it possesses a more accessible
quotient, which we call the Orevkov invariant.Comment: 20 page
Lipschitz geometry of complex surfaces: analytic invariants and equisingularity
We prove that the outer Lipschitz geometry of a germ of a normal
complex surface singularity determines a large amount of its analytic
structure. In particular, it follows that any analytic family of normal surface
singularities with constant Lipschitz geometry is Zariski equisingular. We also
prove a strong converse for families of normal complex hypersurface
singularities in : Zariski equisingularity implies Lipschitz
triviality. So for such a family Lipschitz triviality, constant Lipschitz
geometry and Zariski equisingularity are equivalent to each other.Comment: Added a new section 10 to correct a minor gap and simplify some
argument
Lipschitz geometry does not determine embedded topological type
We investigate the relationships between the Lipschitz outer geometry and the
embedded topological type of a hypersurface germ in . It is
well known that the Lipschitz outer geometry of a complex plane curve germ
determines and is determined by its embedded topological type. We prove that
this does not remain true in higher dimensions. Namely, we give two normal
hypersurface germs and in having the same
outer Lipschitz geometry and different embedded topological types. Our pair
consist of two superisolated singularities whose tangent cones form an
Alexander-Zariski pair having only cusp-singularities. Our result is based on a
description of the Lipschitz outer geometry of a superisolated singularity. We
also prove that the Lipschitz inner geometry of a superisolated singularity is
completely determined by its (non embedded) topological type, or equivalently
by the combinatorial type of its tangent cone.Comment: A missing argument was added in the proof of Proposition 2.3 (final 4
paragraphs are new
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