# Abstract Schober

A polyhedral characterization of quasi-ordinary singularities

Let $X$ be an irreducible hypersurface given by a polynomial $f \in K[[x_1, \ldots, x_d]][z]$, where $K$ denotes an algebraically closed field of characteristic zero. The variety $X$ is called quasi-ordinary with respect to the projection to the affine space defined by $K[[x_1, \ldots, x_d]]$ if the discriminant of $f$ is a monomial times a unit. In my talk I am going to present the construction of an invariant that allows to detect whether a given polynomial $f$ (with fixed projection) defines a quasi-ordinary singularity. This involves a weighted version of Hironaka's characteristic polyhedron and successive embeddings of the singularity in affine spaces of higher dimensions. Further, I will explain how the construction permits to view $X$ as an "overweight deformation" of a toric variety which leads then to the proof of our characterization.