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We introduce a formal integral on the system of varieties mapping properly and birationally to a given one, with value in an associated Chow group. Applications include comparisons of Chern numbers of birational varieties, new birational invariants, ‘stringy’ Chern classes, and a ‘celestial’ zeta function specializing to the topological zeta function.

In its simplest manifestation, the integral gives a new expression for Chern-Schwartz-MacPherson classes of possibly singular varieties, placing them into a context in which a ‘change of variable’ formula holds.

The formalism has points of contact with motivic integration.

Given an -dimensional germ of analytic hypersurface , a finite projection : and a valuation on the ring of convergent series in variables, we study the valuations on the ring that extend . All these valuations are described when is a monomial valuation whose weight vector is not orthogonal to any of the faces of the Newton Polyhedron of the discriminant of the projection . This description is done in terms of the Puiseux parameterizations of with exponents in a cone.

A classical theorem of Briançon, Speder and Teissier states that a family of isolated hypersurface singularities is Whitney equisingular if, and only if, the *-sequence for a hypersurface is constant in the family. This paper shows that the constancy of relative polar multiplicities and the Euler characteristic of the Milnor fibres of certain families of non-isolated singularities is equivalent to the Whitney equisingularity of a family of corank 1 maps from -space to + 1-space. The number of invariants needed is 4 − 2, which greatly improves previous general estimates.

It is a very old and interesting open problem to characterize those collections of embedded topological types of local plane curve singularities which may appear as singularities of a projective plane curve of degree . The goal of the present article is to give a complete (topological) classification of those cases when is rational and it has a unique singularity which is locally irreducible (i.e., is unicuspidal) with one Puiseux pair.

Given an -dimensional germ of analytic hypersurface , a finite projection : and a valuation on the ring of convergent series in variables, we study the valuations on the ring that extend . All these valuations are described when is a monomial valuation whose weight vector is not orthogonal to any of the faces of the Newton Polyhedron of the discriminant of the projection . This description is done in terms of the Puiseux parameterizations of with exponents in a cone.

A rigid body, three of whose points are constrained to move on the coordinate planes, has three degrees of freedom. Bottema and Roth [] showed that there is a point whose trajectory is a solid tetrahedron, the vertices representing corank 3 singularities. A theorem of Gibson and Hobbs [] implies that, for general 3-parameter motions, such singularities cannot occur generically. However motions subject to this kind of constraint arise as interesting examples of parallel motions in robotics and we show that, within this class, such singularities can occur stably.

We discuss the behavior of vertices and inflexions of one-parameter families of plane curves which include a singular member. These arise as sections of smooth surfaces by families of planes parallel to the tangent plane at a given point. We cover all the generic cases, namely elliptic, umbilic, hyperbolic, parabolic and cusp of Gauss points. This work is preliminary to an investigation of symmetry sets and medial axes for these families of curves, reported elsewhere.

We show that the universal abelian cover of the complement to a germ of a reducible divisor on a complex space with isolated singularity is (dim − 2)-connected provided that the divisor has normal crossings outside of the singularity of . We apply this result to obtain a vanishing property for the cohomology of local systems of rank one and also study vanishing in the case of local systems of higher rank.

We generalize the definition of a flecnode on a surface in ℝ to a definition for a general immersed manifold in Euclidean space. Instead of considering a flecnode as a point on the manifold, we consider it as a pair of a normal vector and a tangent vector, called the flecnodal pair. The structure of this set is considered, as well as its connection to binormals and singularities in the set of height functions. The specific case of a surface immersed in ℝ is studied in more detail, with the generic singularities of the flecnodal normals and the flecnodal tangents classified. Finally, the connection between the flecnodals and bitangencies are studied, especially in the case where the dimension of the manifold equals the codimension.

M. Manoel and I. Stewart ([]) classify ℤ ⊕ ℤ-equivariant bifurcation problems up to codimension 3 and 1 modal parameter, using the classical techniques of singularity theory of Golubistky and Schaeffer []. In this paper we classify these same problems using an alternative form: the path formulation (Theorem 6.1). One of the advantages of this method is that the calculates to obtain the normal forms are easier. Furthermore, in our classification we observe the presence of only one modal parameter in the generic core. It differs from the classical classification where the core has 2 modal parameters. We finish this work comparing our classification to the one obtained in [].