arXiv subjects
Publications and source records attributed to Jawad Snoussi.
We show that iterating Nash blowups resolve the singularities of normal toric surfaces satisfying the following property: the minimal generating set of the corresponding semigroup is contained in one or two segments. We also provide examples with an arbitrary number of segments for which the same result holds.
From a procedure to calculate the $C_5$-cone of a reduced complex analytic curve $X \subset \mathbb{C}^n$ at a singular point $0 \in X$, we extract a collection of integers that we call {\it auxiliary multiplicities} and we prove they characterize the Lipschitz type of complex curve singularities. We then use them to improve the known bounds for the number of irreducible components of the $C_5$-cone. We finish by giving an example showing that in a Lipschitz equisingular family of curves the number of planes in the $C_5$-cone may not be constant.
We prove fibration theorems à la Milnor for differentiable real maps with non isolated critical values. We study the situation for maps with linear discriminant, and prove that the concept of d-regularity is the key point for the existence of a Milnor fibration on the sphere. We also explain how one can modify the target space by homeomorphisms to linearize a general discriminant. Whenever the composed map is d-regular one has fibration on the sphere. Plenty of examples are discussed along the text.
We study the behavior of limits of tangents in topologically equivalent spaces. In the context of families of generically reduced curves, we introduce the $s$-invariant of a curve and we show that in a Whitney equisingular family with the property that the $s$-invariant is constant along the parameter space, the number of tangents of each curve of the family is constant. In the context of families of isolated surface singularities, we show through examples that Whitney equisingularity is not sufficient to ensure that the tangent cones of the family are homeomorphic. We explain how the existence of exceptional tangents is preserved by Whitney equisingularity but their number can change.
We explore some equisingularity criteria in one parameter families of generically reduced curves. We prove the equivalence between Whitney regularity and Zariski's discriminant criterion. We prove that topological triviality implies smoothness of the normalized surface. Examples are given to show that Witney regularity and equisaturation are not stable under the blow-up of the singular locus nor under the Nash modification.
Let $X$ be an analytic subset of an open neighbourhood $U$ of the origin $\underline{0}$ in $\mathbb{C}^n$. Let $f\colon (X,\underline{0}) \to (\mathbb{C},0)$ be holomorphic and set $V =f^{-1}(0)$. Let $\B_ε$ be a ball in $U$ of sufficiently small radius $ε>0$, centred at $\underline{0}\in\mathbb{C}^n$. We show that $f$ has an associated canonical pencil of real analytic hypersurfaces $X_θ$, with axis $V$, which leads to a fibration $Φ$ of the whole space $(X \cap \mathbb{B}_ε) \setminus V$ over $\mathbb{S}^1 $. Its restriction to $(X \cap \mathbb{S}_ε) \setminus V$ is the usual Milnor fibration $ϕ= \frac{f}{|f|}$, while its restriction to the Milnor tube $f^{-1}(\partial \D_η) \cap \mathbb{B}_ε$ is the Milnor-Lê fibration of $f$. Each element of the pencil $X_θ$ meets transversally the boundary sphere $\mathbb{S}_ε= \partial \B_ε$, and the intersection is the union of the link of $f$ and two homeomorphic fibers of $ϕ$ over antipodal points in the circle. Furthermore, the space ${\tilde X}$ obtained by the real blow up of the ideal $(Re(f), Im(f))$ is a fibre bundle over $\mathbb{R} \mathbb{P}^1$ with the $X_θ$ as fibres. These constructions work also, to some extent, for real analytic map-germs, and give us a clear picture of the differences, concerning Milnor fibrations, between real and complex analytic singularities.