Related papers: The Nash problem from a geometric and topological …

We survey some aspects of the theory of elliptic surfaces and give some results aimed at determining the Picard number of such a surface. For the surfaces considered, this will be equivalent to determining the Mordell-Weil rank of an…

We study finite morphisms of varieties and the link between their top multiplicity loci under certain assumptions. More precisely, we focus on how to determine that link in terms of the spaces of arcs of the varieties.

Here we simplify the proof of the de Rham theorem for Schwartz functions on affine Nash manifolds and generalize the result to the case of non affine Nash manifolds.

Two of the main open problems in the theory of contact loci are the arc-Floer conjecture, which states that the compactly supported cohomology of the restricted $m$-contact locus and the fixed-point Floer cohomology of the $m$-th iterate of…

We show that the hessian map of quartic plane curves is a birational morphism onto its image, thus bringing new evidence for a very interesting conjecture of Ciro Ciliberto and Giorgio Ottaviani. Our new approach also yields a simpler proof…

The Nash problem on arc families is affirmatively answered for a toric variety by Ishii and Kollar's paper which also shows the negative answer for general case. The Nash problem is one of questions about the relation between arc families…

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most…

We compute the Hochschild cohomology of Hilbert schemes of points on surfaces and observe that it is, in general, not determined solely by the Hochschild cohomology of the surface, but by its "Hochschild-Serre cohomology": the bigraded…

This paper aims to develop a theory for linear-quadratic Nash systems and Master equations in possibly infinite-dimensional Hilbert spaces. As a first step and motivated by the recent results in [31], we study a more general model in the…

We give an example of a real algebraic manifold embedded in a complex space that does not satisfy the Nash-Artin approximation Property. This Nash-Artin approximation Property is closely related to the problem of determining when the…

We study the geometry and topology of immersed surfaces in Euclidean 3-space whose Gauss map satisfies a certain two-piece-property, and solve the ``shadow problem" formulated by H. Wente.

In this short note, we present new observations and examples concerning the existence and rigidity of solutions to the Allen-Cahn equation with degenerate minimal hypersurfaces as their limit interfaces.

This article deals with the existence of hypersurfaces minimizing general shape functionals under certain geometric constraints. We consider as admissible shapes orientable hypersurfaces satisfying a so-called reach condition, also known as…

The goal of this paper is to prove Bloch's conjecture for the numerical Godeaux surface constructed by P. Craighero and R. Gattazzo.

In this paper, we perform a detailed investigation on the various geometrical properties of trapped surfaces and the boundaries of trapped region in general relativity. This treatment extends earlier work on LRS II spacetimes to a general 4…

In this paper, we explore the structure of the Hitchin map for higher dimensional varieties with emphasis on the case of algebraic surfaces.

We study necessary conditions on the geometry and the topology of domains in $\mathbb{R}^2$ that support a positive solution to a classical overdetermined elliptic problem. The ideas and tools we use come from constant mean curvature…

Beyond normal surfaces there are several open questions concerning 2- dimensional spaces. We present some results and conjectures along this line.

We develop the notion of a geometric covering of a rigid space X, which yields a much larger class of covering spaces than that studied previously by de Jong. Geometric coverings of X are closed under disjoint unions and are \'etale local…

A celebrated theorem in Real Algebraic and Analytic Geometry (originally due to Bruhat-Cartan and Wallace and stated later in its current form by Milnor) is the (Nash) curve selection lemma. It states that each point in the closure of a…