Toric embedded resolutions of Quasi-ordinary Hypersurface singularities

Abstract

A germ of a complex analytic variety is quasi-ordinary if there exists a finite projection to the complex affine space with discriminant locus contained in a normal crossing divisor. Some properties of complex analytic curve singularities generalize to quasi-ordinary singularities in higher dimensions, for example the existence of fractional power series parametrization, as well as the existence of some distinguished, characteristic monomials in the parametrization. The paper gives two different affirmative solutions to a problem of Lipman: do the characteristic monomials of a reduced hypersurface quasi-ordinary singularity determine a procedure of embedded resolution of the singularity? The first procedure builds a sequence of toric morphisms depending only on the characteristic monomials. Along the way characteristic monomials are defined for toric quasi-ordinary hypersurface singularities, and their properties are studied. The second procedure generalizes a method of Goldin and Tessier for plane branches. A key step is the re-embedding of the germ in a larger affine space using certain approximate roots of a Weierstrass polynomial. In the last two sections the two procedures are compared and a detailed example is worked out.