CR deformations for rational homogeneous surface singularities

In this paper, we will present a CR-construction of the versal deformations of the singularities Vn(?)C2/Zn, n∈{2,3,4,…} defined by the immersions of C2 into Cn 1 Xn:(z,w)→(zn,zn-1w…,zwn-1,wn).     

CR deformations for rational homogeneous surface singularities

In this paper, we will present a CR-construction of the versal deformations of the singularitiesV _n ? ?²/? _n ,n ∈ {2,3,4,?} defined by the immersions of ?² into ? ⁿ⁺¹ X ⁿ : (z, w) → (z ⁿ ,z ^n?1 w, ?,zw ^n?1 ,w ⁿ )     

On the equi-normalizable deformations of singularities of complex plane curves

We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total δ invariant is preserved. These are also known as equi-normalizable or equi-generic deformations. We restrict primarily to the deformations of singularities with smooth branches. A natural invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in equi-normalizable families. We consider in details deformations of ordinary multiple point, the deformations of a singularity into the collections of ordinary multiple points and deformations of the type x ^p  + y ^pk into the collections of A _k ’s. The research was constantly supported by the Skirball postdoctoral fellowship of the Center of Advanced Studies in Mathematics (Mathematics Department of Ben Gurion University, Israel). Part of the work was done in Mathematische Forschungsinsitute Oberwolfach, during the author’s stay as an OWL-fellow. Some results were published in the preprint [17].     

Hamiltonian flow over deformations of ordinary double points

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On the number of Morse points for deformations of real hyperplane singularities

A relation for the algebraic number of Morse points for deformations of real hyperplane singularities is obtained. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 43, Topology and Its Applications, 2006.     

Free deformations of hypersurface singularities

The article is devoted to the study of the classification problem for Saito free divisors making use of the deformation theory of varieties. In particular, in the quasihomogeneous case, we describe an approach for computation of free deformations of quasicones over quasismooth varieties based on properties of deformations of varieties with $ {\mathbb{G}_m} $ -action. We also discuss some applications including the problem of compactification of modular spaces and computation of free deformations for certain simple, unimodal, and unimodular singularities.     

Equimultiple deformations of isolated singularities

We study equimultiple deformations of isolated hypersurface singularities, introduce a blow-up equivalence of singular points, which is intermediate between topological and analytic ones, and give numerical sufficient conditions for the blow-up versality of the equimultiple deformation of a singularity or multisingularity induced by the space of algebraic hypersurfaces of a given degree. For singular points, which become Newton nondegenerate after one blowing up, we prove that the space of algebraic hypersurfaces of a given degree induces all the equimultiple deformations (up to the blow-up equivalence) which are stable with respect to removing monomials lying above the Newton diagrams. This is a generalization of a theorem by B. Chevallier. This work was partially supported by Grant No.6836-1-9 of the Israeli Ministry of Sciences. The second author thanks the Max-Planck Institut (Bonn) for hospitality and financial support.     

Jet schemes and minimal toric embedded resolutions of rational double point singularities

Using the structure of the jet schemes of rational double point singularities, we construct “minimal embedded toric resolutions” of these singularities. We also establish, for these singularities, a correspondence between a natural class of irreducible components of the jet schemes centered at the singular locus and the set of divisors which appear on every “minimal embedded toric resolution”. We prove that this correspondence is bijective except for the E₈ singulartiy. This can be thought as an embedded Nash correspondence for rational double point singularities.     

Combinatorics of rational singularities

A normal surface singularity is rational if and only if the dual intersection graph of a desingularization satisfies some combinatorial properties. In fact, the graphs defined in this way are trees. In this paper we give geometric features of these trees. In particular, we prove that the number of vertices of valency 3 in the dual intersection tree of the minimal desingularization of a rational singularity of multiplicity m 3 is at most m - 2.     

On the number of rational points of a plane algebraic curve

The number of F_q -rational points of a plane non-singular algebraic curve defined over a finite field F_q is computed, provided that the generic point of is not an inflexion and that is Frobenius non-classical with respect to conics. Received: 18 March 2003