Related papers: Computing the equisingularity type of a pseudo-irr…

Let C be a supersingular genus-2 curve over an algebraically closed field of characteristic 3. We show that if C is not isomorphic to the curve y^2 = x^5 + 1 then up to isomorphism there are exactly 20 degree-3 maps phi from C to the…

Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet-Schuermann-Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern…

We report on the problem of the existence of complex and real algebraic curves in the plane with prescribed singularities up to analytic and topological equivalence. The question is whether, for a given positive integer $d$ and a finite…

We provide a positive answer to Zariski's conjecture for families of singular surfaces in $\mathbb C^3,$ under the condition that the family has a smooth normalisation. As a corollary of the result, we obtain a surprising characterization…

We give a description of the equisingularity of a family of normal surface singularities by numerical invariants belonging to them. By equisingularity we mean Whitney regularity or a more restrictive condition using the Nash modification.

The motivation for this paper are computer calculations of complete lists of weight systems of quasihomogeneous polynomials with isolated singularity at 0 up to rather large Milnor numbers. We review combinatorial characterizations of such…

We study the representations of a class of non-commutative polynomial algebras truncated at degree 3, with one additional relation. We determine the irreducible components of their varieties of representations. We do this by showing that…

Let $\{C_i : i=1,\ldots,r\}$ be a set of irreducible plane curve singularities. For an action of a finite group $G$, let $\Delta^{L}(\{t_{a i}\})$ be the Alexander polynomial in $r\vert G\vert$ variables of the algebraic link…

We investigate the regularising properties of singular kernels at the level of germs, i.e. families of distributions indexed by points in $\mathbb{R}^d$. First we construct a suitable integration map which acts on general coherent germs.…

We study an analytically irreducible algebroid germ (X, 0) of complex singularity by considering the filtrations of its analytic algebra, and their associated graded rings, induced by the divisorial valuations associated to the irreducible…

We characterize plane curve germes non-degenerate in Kouchnirenko's sense in terms of characteristics and intersection multiplicities of branches.

Let $\mathbb{W}$ be an irreducible subvariety a Hilbert scheme $Hilb_{p_W} (t) (\mathbb{P}^n )$. We show that under mild hypothesis there are polynomial formulas for the degrees of the loci of hypersurfaces in $\mathbb{P}^n$ with singular…

The paper is based on a talk. Complete exposition is given in "Equivariant Hirzebruch class for singular varieties". Starting from the classical theory we describe Hirzebruch class and the related Todd genus of a complex singular algebraic…

We consider maps on genus-$g$ surfaces with $n$ (labeled) faces of prescribed even degrees. It is known since work of Norbury that, if one disallows vertices of degree one, the enumeration of such maps is related to the counting of lattice…

Let $V$, $\tilde V$ be hypersurface germs in $\CC^m$, each having a quasi-homogeneous isolated singularity at the origin. We show that the biholomorphic equivalence problem for $V$, $\tilde V$ reduces to the linear equivalence problem for…

Let $(\bf {V,0})\subset (\mathbb{C}^n,0)$ be a germ of a complex hypersurface and let $f: (\mathbb{C}^n,0)\to(\mathbb{C}^n,0)$ be a germ of a finite holomorphic mapping. If germs $(\bf {V,0})$ and ${\bf W}:=(F^{-1}(\bf{ V})),0)$ are…

We compute the graded rank of the cohomology of the hyperplane complement associated with a quaternionic reflection group, and observe that it factors into irreducible factors with positive integer coefficients. For an irreducible group,…

A new general eigenvalue formula for the eigenvalues of Casimir invariants, for the type-I quantum superalgebras, is applied to the construction of link polynomials associated with {\em any} finite dimensional unitary irrep for these…

The kernel method is an essential tool for the study of generating series of walks in the quarter plane. This method involves equating to zero a certain polynomial, the kernel polynomial, and using properties of the curve, the kernel curve,…

We classify stably simple reducible curve singularities in complex spaces of any dimension. This extends the same classification of of irreducible curve singularities obtained by V.I.Arnold. The proof is essentially based on the method of…