Related papers: A propos de la conjecture de Nash

This paper considers an extremal version of the Erd\H{o}s distinct distances problem. For a point set $P \subset \mathbb R^d$, let $\Delta(P)$ denote the set of all Euclidean distances determined by $P$. Our main result is the following: if…

We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with…

Esnault-Viehweg developed the theory of cyclic branched coverings $\tilde X\to X$ of smooth surfaces providing a very explicit formula for the decomposition of $H^1(\tilde X,\mathbb{C})$ in terms of a resolution of the ramification locus.…

We consider the semilinear elliptic equation $-\Delta u =\lambda f(u)$ in a smooth bounded domain $\Omega$ of $R^{n}$ with Dirichielt boundary condition, where $f$ is a $C^{1}$ positive and nondeccreasing function in $[0,\infty)$ such that…

We study Nash valuations on 3-fold terminal singularities, especially in type cAx/2. We find that, in type cAx/2, exceptional prime divisors computing the minimal discrepancy (which is 1/2 in this case) induce Nash valuations. We conjecture…

We study orbits in a family of Markoff-like surfaces with extra off-diagonal terms over prime fields $\mathbb{F}_p$. It is shown that, for a typical surface of this form, every non-trivial orbit has size divisible by $p$. This extends a…

Let E be a plane rational curve defined over complex numbers which has only locally irreducible singularities. The Coolidge-Nagata conjecture states that E is rectifiable, i.e. it can be transformed into a line by a birational automorphism…

Let Y be a pure dimensional analytic variety in C^n with an isolated singularity at the origin such that the exceptional set X of a desingularization of Y is regular. The main objective of this paper is to present a technique which allows…

We investigate the following conjecture: all compact non-K\"ahler complex surfaces admit birational structures. After Inoue-Kobayashi-Ochiai, the remaining cases to study are essentially surfaces in class VII_0^+. In case of Kato surfaces…

The purpose of this paper is two-fold. We first prove a series of results, concerned with the notion of Zariski multiplicity, mainly for non-singular algebraic curves. These results are required in the paper "A Theory of Branches for…

In this paper we study the existence of multiple nontrivial positive weak solutions to the following system of problems. \begin{align*} \begin{split} -\Delta_{p}u-\Delta_q u &= \lambda f(x)|u|^{r-2}u+\nu\frac{1-\alpha}{2-\alpha-\beta}h(x)…

Let $\mathcal S\to\mathbb A^1$ be a smooth family of surfaces whose general fibre is a smooth surface of $\mathbb P^3$ and whose special fibre has two smooth components, intersecting transversally along a smooth curve $R$. We consider the…

We consider the problem of existence and uniqueness of strong solutions $u: \Omega \subset \mathbb{R}^n \longrightarrow \mathbb{R}^N$ in $(H^{2}\cap H^{1}_0)(\Omega)^N$ to the problem \[\label{1} \tag{1} \left\{ \begin{array}{l}…

A cusp singularity is a surface singularity whose minimal resolution is a cycle of smooth rational curves meeting transversely. Cusp singularities come in naturally dual pairs. Looijenga proved in 1981 that if a cusp singularity is…

For a double solid $V\to P_3(C)$ branched over a surface $B\subset P_3(C)$ with only ordinary nodes as singularities, we give a set of generators of the divisor class group $Pic(\tilde{V}})$ in terms of contact surfaces of $B$ with only…

Let $G$ be a usc decomposition of $S^n$, $H_G$ denote the set of nondegenerate elements and $\pi$ be the natural projection of $S^n$ onto $S^n/G$. Suppose that each point in the decomposition space has arbitrarily small neighborhoods with…

Let $B$ be a smooth projective curve of genus $g$, and $S \subset B$ be a finite subset of cardinality $s$. We give an effective upper bound on the number of deformation types of admissible families of canonically polarized manifolds of…

It is proved that the scattering amplitude $A(\beta, \alpha_0, k_0)$, known for all $\beta\in S^2$, where $S^2$ is the unit sphere in $\mathbb{R}^3$, and fixed $\alpha_0\in S^2$ and $k_0>0$, determines uniquely the surface $S$ of the…

This is the first of a series of papers, where we investigate hierarchies of generalized {L}\"{u}roth problems on the hierarchy of rationality, starting with the obvious hierarchy between the rationality and the ruledness. Our primary goal…

We investigate the weak solvability and properties of weak solutions to the Dirichlet problem for a scalar elliptic equation $-\Delta u + b^{(\alpha)}\cdot \nabla u= f$ in a bounded domain $\Omega\subset {\mathbb R^2}$ containing the…