Analytical invariants of quasi-ordinary hypersurface singularities associated to divisorial valuations

Abstract

We study an analytically irreducible algebroid germ $(X, 0)$ of complex singularity by considering the filtrations of its analytic algebra, and their associated graded rings, induced by the {\it divisorial valuations} associated to the irreducible components of the exceptional divisor of the normalized blow-up of the normalization $(\bar{X}, 0)$ of $(X, 0)$, centered at the point $0 \in \bar{X}$. If $(X, 0)$ is a quasi-ordinary hypersurface singularity, we obtain that the associated graded ring is a $\C$-algebra of finite type, namely the coordinate ring of a non necessarily normal affine toric variety of the form $Z^\Gamma = \mbox{\rm Spec} \C [\Gamma]$, and we show that the semigroup $\Gamma$ is an analytical invariant of $(X, 0)$. This provides another proof of the analytical invariance of the {\it normalized characteristic monomials} of $(X, 0)$. If $(X, 0)$ is the algebroid germ of non necessarily normal toric variety, we apply the same method to prove a local version of the isomorphism problem for algebroid germs of non necessarily normal toric varieties (solved by Gubeladze in the algebraic case).