Related papers: A^1-connected components of ruled surfaces

In this paper, we proved two results regarding the arithmetics of separably $\mathbb{A}^1$-connected varieties of rank one. First we proved over a large field, there is an $\mathbb{A}^1$-curve through any rational point of the boundary, if…

A conjecture of Colliot-Th\'{e}l\`{e}ne predicts that for a smooth projective variety $X$ over a finite extension $k$ of $\mathbb{Q}_p$ the kernel of the Albanese map $\text{CH}_0(X)^{\text{deg}=0}\to Alb_X(k)$ is the direct sum of a…

In this paper, we study the Nisnevich sheafification $\mathcal{H}^1_{\acute{e}t}(G)$ of the presheaf associating to a smooth scheme the set of isomorphism classes of $G$-torsors, for a reductive group $G$. We show that if $G$-torsors on…

We prove that on separated algebraic surfaces every coherent sheaf is a quotient of a locally free sheaf. This class contains many schemes that are neither normal, reduced, quasiprojective or embeddable into toric varieties. Our methods…

We prove that a smooth scheme of dimension $n$ over a perfect field is A^1-weakly equivalent to a point if it is A^1-n-connected. We also prove an excision result for A^1-homotopy sheaves over a perfect field.

A standard question in real algebraic geometry is to compute the number of connected components of a real algebraic variety in affine space. By adapting an approach for determining connectivity in complements of real hypersurfaces by Hong,…

We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to…

We start to study the problem of classifying smooth proper varieties over a field k from the standpoint of A^1-homotopy theory. Motivated by the topological theory of surgery, we discuss the problem of classifying up to isomorphism all…

In the framework of Abstract Differential Geometry, we show that to a given principal sheaf and a representation of its stuctural sheaf in $A^n$, where A is a sheaf of associative, commutative, unital algebras (over R or C), we associate a…

Given a graded monoid A with 1, one can construct a projective monoid scheme MProj(A) analogous to Proj(R) of a graded ring R. This paper is concerned with the study of quasicoherent sheaves (of pointed sets) on MProj(A), and we prove…

The paper contains a general construction which produces new examples of non simply-connected smooth projective surfaces. We analyze the resulting surfaces and their fundamental groups. Many of these fundamental groups are expected to be…

Let $S$ be a smooth projective surface with $p_g=q=0$. We show how to use derived categorical methods to study the geometry of certain special iterated Hilbert schemes associated to $S$ by showing that they contain a smooth connected…

We study smooth morphisms $f \colon X \to S$ that are $\mathbb{A}^1$-contractible in the unstable $\mathbb{A}^1$-homotopy category $\mathcal{H}(S)$. For base schemes $S$ of finite Krull dimension, we show that $\mathbb{A}^1$-contractibility…

We study the relationship between several notions of connectedness arising in ${\mathbb A}^1$-homotopy theory of smooth schemes over a field $k$: ${\mathbb A}^1$-connectedness, stable ${\mathbb A}^1$-connectedness and motivic connectedness,…

We prove that any geometrically connected curve $X$ over a field $k$ is an algebraic $K(\pi,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible…

This paper describes connected components of the strata of holomorphic abelian differentials on marked Riemann surfaces with prescribed degrees of zeros. Unlike the case for unmarked Riemann surfaces, we find there can be many connected…

We give a method for constructing maps from a non-commutative scheme to a commutative projective curve. With the aid of Artin-Zhang's abstract Hilbert schemes, this is used to construct analogues of the extremal contraction of a…

We show that the generic fiber of a family of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base. In the particular case where the general fibers of the…

We consider a combinatorial reconfiguration problem on a subclass of quadrangulations of surfaces called square-tiled surfaces. Our elementary move is a shear in a cylinder that corresponds to a well-chosen sequence of diagonal flips that…

Using the theory of cohomology support locus, we give a necessary condition for the Albanese map of a smooth projective surface being a submersion. More precisely, assuming the cohomology support locus of any finite abelian cover of a…