Related papers: The {\L}ojasiewicz exponent of non-degenerate surf…

We prove Koll\'{a}r conjecture for weighted homogeneous surface singularities with big central node. More precisely, we show that every irreducible component of the deformation space of the singularity is parametrized by a certain partial…

Let $M$ be a smooth connected orientable closed surface and $f_0\in C^\infty(M)$ a function having only critical points of the $A_\mu$-types, $\mu\in\mathbb N$. Let ${\mathcal F}={\mathcal F}(f_0)$ be the set of functions $f\in C^\infty(M)$…

We study the geometry of surfaces in $\mathbb{R}^{4}$ with corank $1$ singularities. For such surfaces the singularities are isolated and at each point we define the curvature parabola in the normal space. This curve codifies all the second…

Let f = 0 be an implicit singular plane curve. When deforming f = 0, inflections and vertex emerge from the singularities. In this papper, we classify the deformations of f = 0 with respect to the inflections and the vertices in the cases…

In this article we analyze the global diffeomorphism property of polynomial maps $F:\mathbb{R}^n\rightarrow\mathbb{R}^n$ by studying the properties of the Newton polytopes at infinity corresponding to the sum of squares polynomials…

Let $f$ be a generically finite polynomial map $f: \mathbb{C}^n\to \mathbb{C}^m$ of algebraic degree $d$. Motivated by the study of the Jacobian Conjecture, we prove that the set $S_f$ of non-properness of $f$ is covered by parametric…

We show that the cylindrical tangent cone $C\times \mathbf{R}$ for an area-minimizing hypersurface is unique, where $C$ is the Simons cone $C_S= C(S^3\times S^3)$. Previously Simon proved a uniqueness result for cylindrical tangent cones…

We use orbifold structures to deduce degeneracy statements for holomorphic maps into logarithmic surfaces. We improve former results in the smooth case and generalize them to singular pairs. In particular, we give applications on nodal…

The notion of singular one-parameter deformation of a Lie algebra is introduced. It is shown that the complex infinite-dimensional Lie algebra of polynomial vector fields in C with trivial 1-jet at the origin has such singular deformation.

We obtain a generic regularity result for stationary integral $n$-varifolds with only strongly isolated singularities inside $N$-dimensional Riemannian manifolds, in absence of any restriction on the dimension ($n\geq 2$) and codimension.…

Consider a polynomial $f$ defined over a field $k$, the multiplicity is perhaps the most naive measurement of the singularities of $f$. This paper describes the first steps toward understanding a much more subtle measure of singularities…

Let $f:M\ra \erre^{m+1}$ be an isometrically immersed hypersurface. In this paper, we exploit recent results due to the authors in \cite{bimari} to analyze the stability of the differential operator $L_r$ associated with the $r$-th Newton…

We show that the Elliott invariant is a classifying invariant for the class of $C^*$-algebras that are simple unital infinite dimensional inductive limits of sequences of finite direct sums of building blocks of the form $$ \{f\in…

In this paper we study exponential maps ($\mathbb{G}_a$-actions) on the family of affine two dimensional surfaces of the form $f(x)y=\phi(x,z)$ over arbitrary fields, describe the Makar-Limanov invariant and Derksen invariant of these…

Normally one assumes isolated surface singularities to be normal. The purpose of this paper is to show that it can be useful to look at nonnormal singularities. By deforming them interesting normal singularities can be constructed, such as…

Let $K$ be a number field, and let $C$ be a hyperelliptic curve over $K$ with Jacobian $J$. Suppose that $C$ is defined by an equation of the form $y^{2} = f(x)(x - \lambda)$ for some irreducible monic polynomial $f \in \mathcal{O}_{K}[x]$…

We study isolated quotient singularities by finite and linearly reductive group schemes (lrq singularities for short) and show that they satisfy many, but not all, of the known properties of finite quotient singularities in characteristic…

We investigate several possibilities of obtaining a {\L}ojasiewicz inequality for definable multifunctions and give some examples of applications thereof. In particular, we prove that the Hausdorff distance and its extension to closed sets…

In their study of Jack polynomials, Nazarov-Sklyanin introduced a remarkable new graded linear operator ${\mathcal L} \colon F[w] \rightarrow F[w]$ where $F$ is the ring of symmetric functions and $w$ is a variable. In this paper, we (1)…

We provide some results of the smoothing of surface singularities by Looijenga-Wahl and study smoothing of isolated surface singularities induced by equivariant smoothing of locally complete intersection ($\lci$) singularities. We classify…