Related papers: Specialization to the tangent cone and Whitney Equ…

We consider weighted Hardy inequalities involving the distance function to the boundary of a domain in the $N$-dimensional Euclidean space with nonempty boundary. We give a lower bound for the corresponding best Hardy constant for a domain…

The set of volumes of stable surfaces does have accumulation points. In this paper, we study this phenomenon for surfaces with one cyclic quotient singularity, towards answering the question under which conditions we can still have…

We show that any smooth closed immersed curve in $\mathbb R^n$ with a one-to-one convex projection onto some $2$-plane develops a Type~I singularity and becomes asymptotically circular under Curve Shortening flow in $\mathbb R^n$. As an…

Let $X$ be a very general degree $d\geq 5$ hypersurface in $\mathbb{P}^3$. We compute the ample cone of the Hilbert scheme $X^{[n]}$ of $n$ points on $X$ for various small values of $n$ (the answer is already known for large $n$). We obtain…

We show that if f: X --> Y is a finite, separable morphism of smooth curves defined over a finite field F_q, where q is larger than an explicit constant depending only on the degree of f and the genus of X, then f maps X(F_q) surjectively…

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…

Despite remarkable success in describing supergravity reductions and backgrounds, generalized geometry and the closely related exceptional field theory are still lacking a fundamental object of differential geometry, the Riemann tensor. We…

On an affine variety $X$ defined by homogeneous polynomials, every line in the tangent cone of $X$ is a subvariety of $X$. However there are many other germs of analytic varieties which are not of cone type but contain ``lines'' passing…

We give sharp lower bounds for the degree of the syzygies involving the partial derivatives of a homogeneous polynomial defining an even dimensional nodal hypersurface. This implies the validity of formulas due to M. Saito, L. Wotzlaw and…

If $X$ is an algebraic variety with at worst canonical singularities and $S$ is a $\Q$-Cartier hypersurface in $X$, the canonical threshold of the pair $(X,S)$ is the supremum of $c\in\R$ such that the pair $(X,cS)$ is canonical. We show…

\noindent We study the Zariski tangent cone $T_X\stackrel{\pi}{\lar} X$ to an affine variety $X$ and the closure $\bar{T}_X$ of $\pi^{-1}({\rm Reg}(X))$ in $T_X$. We focus on the comparison between $T_X$ and $\bar{T}_X$, giving sufficient…

We consider surfaces $X$ defined by plane divisorial valuations $\nu$ of the quotient field of the local ring $R$ at a closed point $p$ of the projective plane $\mathbb{P}^2$ over an arbitrary algebraically closed field $k$ and centered at…

We extend the results of Riemannian geometry over finite groups and provide a full classification of all linear connections for the minimal noncommutative differential calculus over a finite cyclic group. We solve the torsion-free and…

We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The surfaces exist for any sufficiently large prescribed genus $g$, and are non-compact…

We give a new criterion for when a resolution of a surface of general type with canonical singularities has big cotangent bundle and a new lower bound for the values of $d$ for which there is a surface with big cotangent bundle that is…

In [5], S\'aez and Schn\"urer studied the graphical mean curvature flow of complete hypersurfaces defined on subsets of Euclidean space. They obtained long time existence. Moreover, they provided a new interpretation of weak mean curvature…

We study conformal metrics with prescribed Gaussian curvature on surfaces with conical singularities and geodesic boundary in supercritical regimes. Exploiting a variational argument, we derive a general existence result for surfaces with…

We give necessary and sufficient conditions of infinite determinacy for smooth function germs whose critical locus contains a given set. This set is assumed to be the zero variety X of some analytic map germ having maximal rank on a dense…

For finite difference discretizations with linear complexity and provably convergent to weak solutions of the second boundary value problem for the Monge-Amp\`ere equation, we give the first proof of uniqueness. The boundary condition is…

This paper explores local second-order weak sharp minima for a broad class of nonconvex optimization problems. We propose novel second-order optimality conditions formulated through the use of classical and lower generalized support…