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

Abstract

Let (X, 0) be an irreducible germ of complex analytic space and b: (B,E) ! (X, 0) be its normalized blow-up centered at 0 2 X, that is, the map obtained by first blowing-up 0 on X and then normalizing the new space. To each irreducible component D of the exceptional divisor E is associated a divisorial valuation _D of K, the field of fractions of the local analytic algebra R of the germ (X, 0). Namely, if h 2 K, then _D(h) denotes the vanishing order of h _ b along D. The valuation _D defines a filtration of R by the ideals pk := {' 2 R| _D(') _ k} for k _ 0 and an associated graded algebra grD(X, 0) := L k_0 pk/pk+1 with distinguished graded maximal ideal mD(X, 0) := L k_1 pk/pk+1. In this way, one obtains a finite set of pairs (grD(X, 0),mD(X, 0)), canonically associated to the analytic germ (X, 0).