A polyhedral characterization of quasi-ordinary singularities
Abstract
Given an irreducible hypersurface singularity of dimension $d$ (defined by a polynomial $f\in K[[ {\bf x} ]][z]$) and the projection to the affine space defined by $K[[ {\bf x} ]]$, we construct an invariant which detects whether the singularity is quasi-ordinary with respect to the projection. The construction uses a weighted version of Hironaka's characteristic polyhedron and successive embeddings of the singularity in affine spaces of higher dimensions. When $ f $ is quasi-ordinary, our invariant determines the semigroup of the singularity and hence it encodes the embedded topology of the singularity $ \{ f = 0 \} $ in a neighbourhood of the origin when $ K = \mathbb{C};$ moreover, the construction yields the approximate roots, giving a new point of view on this subject.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2015
- DOI:
- 10.48550/arXiv.1512.07507
- arXiv:
- arXiv:1512.07507
- Bibcode:
- 2015arXiv151207507M
- Keywords:
-
- Mathematics - Algebraic Geometry;
- Mathematics - Complex Variables;
- 14B05;
- 32S05;
- 13F25;
- 14E15
- E-Print:
- 30 pages, corrected typos, improved some explanations and clarified minimizing process for the polyhedron along the the construction