Related papers: Nash problem for surfaces

We provide a characterisation of Schur multiplicative maps on both finite and infinite dimensional matrix spaces, and show that every surjective Schur multiplicative contraction is automatically an isometry. We also generalise this result…

We prove an analog of Belyi's theorem for the algebraic surfaces. Namely, any non-singular algebraic surface can be defined over a number field if and only it covers the complex projective plane with ramification at three knotted…

We consider bijective maps $\phi$ on the full operator algebra $\mathcal{B}(\mathcal{H})$ of an infinite dimensional Hilbert space with the property that, for every $A,B,X\in \mathcal{B}(\mathcal{H})$, $X$ is the Douglas solution of the…

We construct an explicit bijection between bipartite pointed maps of an arbitrary surface $\mathbb{S}$, and specific unicellular blossoming maps of the same surface. Our bijection gives access to the degrees of all the faces, and distances…

We show that on integral normal separated schemes whose function field is separably closed, for each pair of points the intersection of the resulting local schemes is local. This extends a result of Artin from rings to schemes. The argument…

The aim of this paper is to give a description of simply connected abelian locally Nash groups of dimension $2$. Along the way we prove that, for any $n\geq 2$, a locally Nash structure over $(\mathbb{R}^n,+)$ can be characterized via a…

We prove a microlocal characterisation of character sheaves on a reductive Lie algebra over an algebraically closed field of sufficiently large positive characteristic: a perverse irreducible G-equivariant sheaf is a character sheaf if and…

We extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, that is orientable and non-orientable as well. This…

We construct a normal projective $\mathbb{Q}$-Gorenstein surface over an algebraically closed field whose canonical ring is not finitely generated. Moreover, we provide a counterexample to the minimal model program for…

Extending Enriques' characterization to algebraically closed fields of characteristic $p \geq 7$, we show that every smooth projective surface $X$ with $h^1(X, \mathcal{O}_X) = 2$ and $p_1(X) = p_2(X) = 1$ is birational to an Abelian…

The paper surveys several results on the topology of the space of arcs of an algebraic variety and the Nash problem on the arc structure of singularities.

Let F be a holomorphic map whose components satisfy some polynomial relations. We present an algorithm for constructing Nash maps locally approximating F, whose components satisfy the same relations.

We prove that a regular projective surface $S$ over a field $k$ of characteristic $p \ge 7$, with $H^0(S,\mathcal{O}_S) = k$ and $-K_S$ being nef, is geometrically integral over $k$.

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…

A bielliptic surface (or hyperelliptic surface) is a smooth surface with a numerically trivial canonical divisor such that the Albanese morphism is an elliptic fibration. In the first part of this paper, we study the structure of bielliptic…

It has been recently shown that the iteration of Nash modification on not necessarily normal toric varieties corresponds to a purely combinatorial algorithm on the generators of the semigroup associated to the toric variety. We will show…

We prove the Nagata compactification theorem for any separated map of finite type between quasi-compact and quasi-separated algebraic spaces, generalizing earlier results of Raoult. Along the way we also prove (and use) absolute noetherian…

We discuss the birational geometry of singular surfaces in positive characteristic. More precisely, we establish the minimal model program and the abundance theorem for Q-factorial surfaces and for log canonical surfaces. Moreover, in the…

In this paper, we prove that Bloch's conjecture holds for all smooth, complex, projective surfaces with $p_g=q=0$ and $K^2=9$.

We prove that any biLipschitz mapping of the boundary of the unit disk onto the boundary of the domain of the same area can be extended to a biLipschitz mapping of the whole plane which preserves the area of any measurable subset.