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Register a new userClassification of Complex Singularities with Non-Degenerate Newton Boundary
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In his groundbreaking work on classification of singularities with regard to right and stable equivalence of germs, Arnold has listed normal forms for all isolated hypersurface singularities over the complex numbers with either modality less than or equal to two or Milnor number less than or equal to 16. Moreover, he has described an algorithmic classifier, which determines the type of a given such singularity. In the present paper, we extend Arnolds work to a large class of singularities which is unbounded with regard to modality and Milnor number. We develop an algorithmic classifier, which determines a normal form for any singularity with corank less than or equal to two which is equivalent to a germ with non-degenerate Newton boundary in the sense of Kouchnirenko. In order to realize the classifier, we prove a normal form theorem: Suppose K is a mu-constant stratum of the jet space which contains a germ with a non-degenerate Newton boundary. We first observe that all germs in K are equivalent to some germ with the same fixed non-degenerate Newton boundary. We then prove that all right-equivalence classes of germs in K can be covered by a single normal form obtained from a regular basis of an appropriately chosen special fiber. All algorithms are implemented in the library arnold.lib for the computer algebra system Singular.
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We characterize plane curve germes non-degenerate in Kouchnirenkos sense in terms of characteristics and intersection multiplicities of branches.
Let $f$ be an isolated singularity at the origin of $mathbb{C}^n$. One of many invariants that can be associated with $f$ is its {L}ojasiewicz exponent $mathcal{L}_0 (f)$, which measures, to some extent, the topology of $f$. We give, for generic surface singularities $f$, an effective formula for $mathcal{L}_0 (f)$ in terms of the Newton polyhedron of $f$. This is a realization of one of Arnolds postulates.
The decomposition of a two dimensional complex germ with non-isolated singularity into semi-algebraic sets is given. This decomposition consists of four classes: Riemannian cones defined over a Seifert fibered manifold, a topological cone over thickened tori endowed with Cheeger-Nagase metric, a topological cone over mapping torus endowed with Hsiang-Pati metric and a topological cone over the tubular neighbourhoods of the links singularities. In this decomposition there exist semi-algebraic sets that are metrically conical over the manifolds constituting the link. The germ is reconstituted up to bi-Lipschitz equivalence to a model describing its geometric behavior.
The prolongation g^{(k)} of a linear Lie algebra g subset gl(V) plays an important role in the study of symmetries of G-structures. Cartan and Kobayashi-Nagano have given a complete classification of irreducible linear Lie algebras g subset gl(V) with non-zero prolongations. If g is the Lie algebra aut(hat{S}) of infinitesimal linear automorphisms of a projective variety S subset BP V, its prolongation g^{(k)} is related to the symmetries of cone structures, an important example of which is the variety of minimal rational tangents in the study of uniruled projective manifolds. From this perspective, understanding the prolongation aut(hat{S})^{(k)} is useful in questions related to the automorphism groups of uniruled projective manifolds. Our main result is a complete classification of irreducible non-degenerate nonsingular variety with non zero prolongations, which can be viewed as a generalization of the result of Cartan and Kobayashi-Nagano. As an application, we show that when $S$ is linearly normal and Sec(S) eq P(V), the blow-up of P(V) along S has the target rigidity property, i.e., any deformation of a surjective morphism Y to Bl_S(PV) comes from the automorphisms of Bl_S(PV).
We study singularities of Gauss maps of fronts and give characterizations of types of singularities of Gauss maps by geometric properties of fronts which are related to behavior of bounded principal curvatures. Moreover, we investigate relation between a kind of boundedness of Gaussian curvatures near cuspidal edges and types of singularities of Gauss maps of cuspidal edges. Further, we consider extended height functions on fronts with non-degenerate singular points.
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