Filters

20082

More filters

20192
Reset filters

Settings

Author: Kerner, Dmitry × End date: 2009 × Start date: 2000 × Has files: Yes × Type: Text × WGL Institute: MFO ×

Search Results

Now showing 1 - 2 of 2
  • Thumbnail Image
    Item
    On the directionally Newton-non-degenerate singularities of complex hypersurfaces
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Kerner, Dmitry
    We introduce a minimal generalization of Newton-non-degenerate singularities of hypersurfaces. Roughly speaking, an isolated hypersurface singularity is called directionally Newton-non-degenerate if the local embedded topological singularity type can be restored from a collection of Newton diagrams. A singularity that is not directionally Newton-non-degenerate is called essentially Newton-degenerate . For plane curves we give an explicit and simple characterization of directionally Newton-non-degenerate singularities, for hypersurfaces we give some examples. Then we treat the question: is Newton-non-degenerate or directionally Newton-non-degenerate a property of singular types or of particular representatives. Namely, is the non-degeneracy preserved in an equi-singular family? This is proved for curves. For hypersurfaces we give an example of a Newton-non-degenerate hypersurface whose equi-singular deformation consists of essentially Newton-degenerate hypersurfaces. Finally, the classical formulas for the Milnor number (Kouchnirenko) and the zeta function (Varchenko) of the Newton-non-degenerate singularity are generalized to some classes of directionally Newton-non-degenerate singularities.
  • Thumbnail Image
    Item
    On the [delta] δ=const collisions of singularities of complex plane curves
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2008) Kerner, Dmitry
    We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total ± invariant is preserved. These are also known as equi-normalizable or equi-generic deformations. We restrict primarily to the deformations of singularities with smooth branches. A new invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in collisions. We consider in details the ± = const deformations of ordinary multiple point, the deformation of a singularity into the collection of ordinary multiple points and the deformation of the type xp + ypk into a collection of Ak's.