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## Über dieses Buch

This book features state-of-the-art research on singularities in geometry, topology, foliations and dynamics and provides an overview of the current state of singularity theory in these settings.

Singularity theory is at the crossroad of various branches of mathematics and science in general. In recent years there have been remarkable developments, both in the theory itself and in its relations with other areas.

The contributions in this volume originate from the “Workshop on Singularities in Geometry, Topology, Foliations and Dynamics”, held in Merida, Mexico, in December 2014, in celebration of José Seade’s 60th Birthday.

It is intended for researchers and graduate students interested in singularity theory and its impact on other fields.

## Inhaltsverzeichnis

### Extending the Action of Schottky Groups on the Complex Anti-de Sitter Space to the Projective Space

In this article we show that if a complex Schottky group, acting on the complex anti-de Sitter space, acts on the corresponding projective space as a Schottky group, then the space has signature (k, k): As a consequence, we are able to show the existence of complex Schottky groups, acting on $${\mathbb{P}}_\mathbb{C}^n$$, such that the complement of whose Kulkarni's limit set is not the largest open set on which the group acts properly and discontinuously. This is the starting point towards the understanding of the notion of the role of limit sets in the higher-dimensional setting.
Vanessa Alderete, Carlos Cabrera, Angel Cano, Mayra Méndez

### Puiseux Parametric Equations via the Amoeba of the Discriminant

Given an algebraic variety we get Puiseux-type parametrizations on suitable Reinhardt domains. These domains are defined using the amoeba of hypersurfaces containing the discriminant locus of a finite projection of the variety.
Fuensanta Aroca, Víctor Manuel Saavedra

### Some Open Questions on Arithmetic Zariski Pairs

In this paper, complement-equivalent arithmetic Zariski pairs will be exhibited answering in the negative a question by Eyral-Oka [14] on these curves and their groups. A complement-equivalent arithmetic Zariski pair is a pair of complex projective plane curves having Galois-conjugate equations in some number field whose complements are homeomorphic, but whose embeddings in $${\mathbb{P}}^2$$ are not.
Most of the known invariants used to detect Zariski pairs depend on the étale fundamental group. In the case of Galois-conjugate curves, their étale fundamental groups coincide. Braid monodromy factorization appears to be sensitive to the difference between étale fundamental groups and homeomorphism class of embeddings.
Enrique Artal Bartolo, José Ignacio Cogolludo-Agustín

### Logarithmic Vector Fields and the Severi Strata in the Discriminant

The discriminant, D, in the base of a miniversal deformation of an irreducible plane curve singularity, is partitioned according to the genus of the (singular) fibre, or, equivalently, by the sum of the delta invariants of the singular points of the fibre. The members of the partition are known as the Severi strata. The smallest is the δ-constant stratum, D(δ), where the genus of the fibre is 0. It is well known, by work of Givental’ and Varchenko, to be Lagrangian with respect to the symplectic form Ω obtained by pulling back the intersection form on the cohomology of the fibre via the period mapping. We show that the remaining Severi strata are also co-isotropic with respect to Ω, and moreover that the coefficients of the expression of Ω with respect to a basis of Ω 2(log D) are equations for D(δ). Similarly, the coeδcients of Ω ^k with respect to a basis for Ω 2k (log D) are equations for D(δk + 1). These equations allow us to show that for E 6 and E 8, D(δ) is Cohen-Macaulay (this was already shown by Givental’ for A 2k ), and that, as far as we can calculate, for A 2k all of the Severi strata are Cohen-Macaulay.
Paul Cadman, David Mond, Duco van Straten

### Classification of Isolated Polar Weighted Homogeneous Singularities

Polar weighted homogeneous polynomials are real analytic maps which generalize complex weighted homogeneous polynomials. In this article we give classes of mixed polynomials in three variables which generalize Orlik and Wagreich classes of complex weighted homogeneous polynomials. We give explicit conditions for this classes to be polar weighted homogeneous polynomials with isolated critical point. We prove that under small perturbation of their coe_cients they remain with isolated critical point and the diffeomorphism type of their link does not change.
José Luis Cisneros-Molina, Agustín Romano-Velázquez

### Rational and Iterated Maps, Degeneracy Loci, and the Generalized Riemann-Hurwitz Formula

We consider a generalized Riemann-Hurwitz formula as it may be applied to rational maps between projective varieties having an indeterminacy set and fold-like singularities. The case of a holomorphic branched covering map is recalled. Then we see how the formula can be applied to iterated maps having branch-like singularities, degree lowering curves, and holomorphic maps having a fixed point set. Separately, we consider a further application involving the Chern classes of determinantal varieties when the latter are realized as the degeneracy loci of certain vector bundle morphisms.
James F. Glazebrook, Alberto Verjovsky

### On Singular Varieties with Smooth Subvarieties

Let k be an algebraically closed field of characteristic 0. Let C be an irreducible nonsingular curve such that $${rC} = {S} \cap {F}, \ {r} \in \mathbb{N}$$, where S and F are two surfaces and all the singularities of F are of the form z 3 = x s y s , s prime, with gcd(3, s) = 1. We prove that C can never pass through such kind of singularities of a surface, unless r = 3a, $${a} \ {\in} \ \mathbb{N}$$. The case when the singularities of F are of the form z 3 = x 3s y 3s , $${s} \ {\in} \ \mathbb{N}$$; were studied in [3]. Next, we study multiplicity-r structures on varieties for any positive integer r. Let Z be a reduced irreducible nonsingular (n − 1)-dimensional variety such that $${rZ} = {X} \cap F$$, where X is a normal n-fold with certain type of singularities, F is a (N − 1)-fold in $$\mathbb{P}^{N}$$; such that $${Z} \ \cap$$ Sing(X) ≠ ∅ We study the singularities of X through which Z passes.
María del Rosario González-Dorrego

### On Polars of Plane Branches

It is well known that the equisingularity class, which is the same as the topological type, of the general polar curve of a plane branch is not the same for all branches in a given equisingularity class, but depends upon the analytic type of the branch. It was shown in [1] that, for sufficiently general branches in a given equisingularity class, the topology of the general polar is constant. The aim of this paper is to go beyond generality and show how one could describe the topology of the general polars of all branches in a given equisingularity class, making use of the analytic classification of branches as described in [5]. We will show how this works in some particular equisingularity classes for which one has the complete explicit analytic classification, and in particular for all branches of multiplicity less or equal than four, based on the classification given in [4].
A. Hefez, M. E. Hernandes, M. F. Hernández Iglesias

### Singular Intersections of Quadrics I

Let $${Z} \ \subset \ \mathbb{R}^{n}$$ be given by k + 1 equations of the form $$\sum\limits_{i=1}^{n} {A}_{i}{{x}_{i}^{2}} = {0}, \qquad\qquad \sum\limits_{i=1}^{n} {{x}_{i}^{2}} = 1,$$ where $${A}_{i} {\in} \mathbb{R}^{k}$$. It is well known that the condition for Z to be a smooth variety (known as weak hyperbolicity) is that the origin in $$\mathbb{R}^{k}$$ is not a convex combination of any collection of k of the vectors A i . We interpret this condition as a transversality property in order to approach the case when it is singular and we extend some results known for the smooth case, in particular the computation of the homology groups of Z in terms of the combinatorics of the natural quotient polytope. We show that Z cannot be an exotic homotopy sphere nor a non-simply connected homology sphere and use this to show that, except for some clearly characterized degenerate cases, when Z is not smooth it cannot be a topological or even a homological manifold.
Santiago López de Medrano

### A New Conjecture, a New Invariant, and a New Non-splitting Result

We prove a new non-splitting result for the cohomology of the Milnor fiber, reminiscent of the classical result proved independently by Lazzeri, Gabrielov, and Lê in 1973-74.
We do this while exploring a conjecture of Fernández de Bobadilla about a stronger version of our non-splitting result. To explore this conjecture, we define a new numerical invariant for hypersurfaces with 1-dimensional critical loci: the beta invariant. The beta invariant is an invariant of the ambient topological-type of the hypersurface, is non-negative, and is algebraically calculable. Results about the beta invariant remove the topology from Bobadilla’s conjecture and turn it into a purely algebraic question.
David B. Massey

### Lipschitz Geometry Does not Determine Embedded Topological Type

We investigate the relationships between the Lipschitz outer geometry and the embedded topological type of a hypersurface germ in ($${\mathbb{C}}^n$$, 0). It is well known that the Lipschitz outer geometry of a complex plane curve germ determines and is determined by its embedded topological type.We prove that this does not remain true in higher dimensions. Namely, we give two normal hypersurface germs ($${{X}}_1$$, 0) and ($${{X}}_2$$, 0) in ($${\mathbb{C}}^3$$, 0) having the same outer Lipschitz geometry and different embedded topological types. Our pair consist of two superisolated singularities whose tangent cones form an Alexander-Zariski pair having only cusp-singularities. Our result is based on a description of the Lipschitz outer geometry of a superisolated singularity. We also prove that the Lipschitz inner geometry of a superisolated singularity is completely determined by its (non-embedded) topological type, or equivalently by the combinatorial type of its tangent cone.
Walter D. Neumann, Anne Pichon

### Projective Transverse Structures for Some Foliations

We construct examples of regular foliations of holomorphic surfaces which are generically transverse to a compact curve and have a projective transverse structure.

### Chern Classes and Transversality for Singular Spaces

In this paper we compare different notions of transversality for possible singular complex algebraic or analytic subsets of an ambient complex manifold and prove a refined intersection formula for their Chern-Schwartz- MacPherson classes. In case of a transversal intersection of complex Whitney stratified sets, this result is well known. For splayed subsets it was conjectured (and proven in some cases) by Aluffi and Faber. Both notions are stronger than a micro-local “non-characteristic intersection“ condition for the characteristic cycles of (associated) constructible functions, which nevertheless is enough to imply the asked refined intersection formula for the Chern-Schwartz-MacPherson classes. The proof is based the multiplicativity of Chern-Schwartz-MacPherson classes with respect to cross products, as well as a new Verdier-Riemann-Roch theorem for “non-characteristic pullbacks“.
Jörg Schürmann
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