Pedro Daniel Gonzalez Perez

Bijectiveness of the Nash Map for Quasi-Ordinary Hypersurface Singularities

In this paper we give a positive answer to a question of Nash, concerning the arc space of a singularity, for the class of quasi-ordinary hypersurface singularities, extending to this case previous results and techniques of Shihoko Ishii.

THE SEMIGROUP OF A QUASI-ORDINARY HYPERSURFACE

An analytically irreducible hypersurface germ (S, 0) ⊂ (Cd+1, 0) is quasi-ordinary if it canbe defined by the vanishing of the minimal polynomial f ∈ C{X}[Y ] of a fractional power series in the variables X = (X1, . . . , Xd) which has characteristic monomials, generalizing the classical Newton–Puiseux characteristic exponents of the plane-branch case (d = 1). We prove that the set of vertices of Newton polyhedra of resultants of f and h with respect to the indeterminate Y , for those polynomials h which are not divisible by f, is a semigroup of rank d, generalizing the classical semigroup appearing in the plane-branch case.We show that some of the approximate roots of the polynomial f are irreducible quasiordinary polynomials and that, together with the coordinates X1, . . . , Xd, provide a set of generators of the semigroup from which we can recover the characteristic monomials and vice versa. Finally, we prove that the semigroups corresponding to any two parametrizations of (S, 0) are isomorphic and that this semigroup is a complete invariant of the embedded topological type of the germ (S, 0) as characterized by the work of Gau and Lipman.

Motivic Poincaré series, toric singularities and logarithmic Jacobian ideals

The geometric motivic Poincare series of a variety, which was introduced by Denef and Loeser, takes into account the classes in the Grothendieck ring of the sequence of jets of arcs in the variety. Denef and Loeser proved that this series has a rational form. We describe it in the case of an affine toric variety of arbitrary dimension. The result, which provides an explicit set of candidate poles, is expressed in terms of the sequence of Newton polyhedra of certain monomial ideals,which we call logarithmic Jacobian ideals, associated to the modules of differential forms with logarithmic poles outside the torus of the toric variety.

Ultrametric Spaces of Branches on Arborescent Singularities

Let S be a normal complex analytic surface singularity. We say that S is arborescent if the dual graph of any good resolution of it is a tree. Whenever A, B are distinct branches on S, we denote by A ⋅ B their intersection number in the sense of Mumford. If L is a fixed branch, we define UL(A, B) = (L ⋅ A)(L ⋅ B)(A ⋅ B)−1 when A ≠ B and UL(A, A) = 0 otherwise. We generalize a theorem of Ploski concerning smooth germs of surfaces, by proving that whenever S is arborescent, then UL is an ultrametric on the set of branches of S different from L. We compute the maximum of UL, which gives an analog of a theorem of Teissier. We show that UL encodes topological information about the structure of the embedded resolutions of any finite set of branches. This generalizes a theorem of Favre and Jonsson concerning the case when both S and L are smooth. We generalize also from smooth germs to arbitrary arborescent ones their valuative interpretation of the dual trees of the resolutions of S. Our proofs are based in an essential way on a determinantal identity of Eisenbud and Neumann.

Geometric motivic Poincaré series of quasi-ordinary singularities

Geometric motivic Poincare series of a germ at a singular point of complex algebraic variety describes the truncated images of the space of arcs through the singular point. Denef and Loeser proved that it has a rational form. In this paper, the authors study an irreducible germ of quasi-ordinary hypersurface singularities and introduce the notion of logarithmic Jacobian ideals. The main result of this paper is to give the explicit rational form of geometric motivic Poincare series of such a singularity in terms of the lattice and the Newton polyhedra of the logarithmic Jacobian ideals.