Many singularities are not stably equivalent to Newton-non-degenerate singularities

2014/01/27 by Anna Gourevitch, Gourevitch, Anna, Dmitry Kerner +1
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG

paper · pdf · doi:10.48550/arxiv.1401.6781

This paper has been withdrawn by the authors. Many thanks to J.Stevens for pointing the error in the proof

arxiv created 2014/01/28 · arxiv updated 2014/01/29

Abstract

This paper has been withdrawn. Consider an isolated complex hypersurface singularity, f(x₁,..,x_n)=0. For Newton-non-degenerate singularities the local topology is completely determined by an associated polyhedral object, the Newton diagram. "Most" singularities are not Newton-non-degenerate, for any choice of local coordinates. An old question of Arnol'd asks whether for any hypersurface singularity there exists a stabilization, f(x₁,...,x_n)+z²₁+...+z²_r, that becomes Newton-non-degenerate after some change of coordinates. The answer is: "totally no". We give some simple obstructions and present particular examples of plane curve singularities that have no Newton-non-degenerate stabilization (in any local coordinates).

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