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 divisorial valuations associated to the irreducible components of the exceptional divisor of the normalized blow-up of the normalization of (X, 0), centered at the point. 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 ZΓ = Spec C[Γ], and we show that the semigroup Γ is 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 (solved by Gubeladze in the algebraic case). © 2004, Department of Mathematics, Tokyo Institute of Technology. All rights reserved.