Resumen: The author proves a generalization of the Teissier theorem [B. Teissier, Invent. Math. 40, 267-292 (1977; Zbl 0446.32002)] about expressing the topological determinacy order by the polar invariants. In this generalization the main assumption is that the projective tangent cone of the hypersurface has at most isolated singularities. Some consequences
of the theorem are studied in the last section of the paper.
Resumen: We describe the jacobian ideal of the fibers St of an equiresolvable deformation of a quasi-ordinary hypersurface singularity (S, 0). This kind of deformation, inspired by the work of Teissier, has generic _ber isomorphic to (S, 0) and special fiber a toric singularity. We show a formula, in terms of the logarithmic jacobian ideal, for the pull-back of the jacobian ideal of St in its normalization. The logarithmic jacobian ideal is studied in the normal
toric case by Lejeune and Reguera in relation with the study of motivic invariants and arc spaces. We deduce some equisingularity properties of the normalized Nash modification of St.
Resumen: This paper proposes some material towards a theory of general toric varieties without the assumption of normality. Their combinatorial description involves a fan to which is attached a set of semigroups subjected to gluing-up conditions. In particular it contains a combinatorial construction of the blowing up of a sheaf of monomial ideals on a toric variety. In the second part it is shown that over an algebraically closed base field of
zero characteristic the Semple-Nash modification of a general toric variety is isomorphic to the blowing up of the sheaf of logarithmic jacobian ideals and that in any characteristic this blowing-up is an isomorphism if and only if the toric variety is non singular.
Resumen: A polar hypersurface P of a complex analytic hypersurface germ f = 0 can be investigated by analyzing the invariance of certain Newton polyhedra associated with the image of P, with respect to suitable coordinates, by certain morphisms appropriately associated with f. We develop this general principle of Teissier when f = 0 is a quasi-ordinary hypersurface germ and P is
the polar hypersurface associated with any quasi-ordinary projection of f = 0. We show a decomposition of P into bunches of branches which characterizes the embedded topological types of the irreducible components of f = 0. This decomposition is also characterized by some properties of the strict transform of P by the toric embedded resolution of 0 given by the second author. In the plane curve case this result provides a simple algebraic proof of a theorem of Le et al.
Resumen: We give a method to construct a partial embedded resolution of a nonnecessarily normal affine toric variety Z(Gamma) equivariantly embedded in a normal affine toric variety Z(rho). This partial resolution is an embedded normalization inside a normal toric ambient space and a resolution of singularities of the ambient space, which always exists, provides
an embedded resolution. The advantage is that this partial resolution is completely determined by the embedding Z(Gamma) subset of Z(rho). As a by-product, the construction of the normalization is made without an explicit computation of the saturation of the semigroup Gamma of the toric variety (see [3]). This result is valid for a base field k algebraically closed of arbitrary characteristic.
