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

Pedro Daniel Gonzalez Perez, Gerard Gonzalez-Sprinberg

Published 2003-11-27Version 1

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 divisorial valuations associated to the irreducible components of the exceptional divisor of the normalized blow-up of the normalization (Y, 0) of (X, 0), centered at the point 0 of Y. If (X, 0) is a quasi-ordinary hypersurface singularity, we obtain that the associated graded ring is an algebra of finite type over the field of complex numbers, namely the coordinate ring of a non necessarily normal affine toric variety defined by a semigroup, which is shown to be an analytical invariant of (X, 0). This provides another proof of the analytical invariance of the 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.