Related papers: Vari'et'es presque rationnelles, leurs points rati…

We introduce and develop the model-theoretic notions of absolute connectedness and type-absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups (that is, Chevalley groups) over arbitrary…

We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $\pi : X \to A$, where $A$ is an abelian…

We investigate properties of the Albanese map and the fundamental group of a complex projective variety with many rational points over some function field, and prove that every linear quotient of the fundamental group of such a variety is…

We study the 0-th stable A^1-homotopy sheaf of a smooth proper variety over a field k assumed to be infinite, perfect and to have characteristic unequal to 2. We provide an explicit description of this sheaf in terms of the theory of…

Fix a number field k. We prove that if there is an algorithm for deciding whether a smooth projective geometrically integral k-variety has a k-point, then there is an algorithm for deciding whether an arbitrary k-variety has a k-point and…

We prove a Witt vector version of the usual Grauert-Riemenschneider vanishing theorem over fields of positive characteristic, solving a question raised by Blickle, Esnault, Chatzistamatiaou and R\"ulling. We then deduce some rationality…

We study unirational algebraic varieties and the fields of rational functions on them. We show that after adding a finite number of variables some of these fields admit an infinitely transitive model. The latter is an algebraic variety with…

A theorem of Graber, Harris, and Starr states that a rationally connected fibration over a curve has a section. We study an analogous question in symplectic geometry. Namely, given a rationally connected fibration over a curve, can one find…

(On the fundamental group of rationnally connected varieties.) I show that the fundamental group of a normal variety which is rationally chain connected is finite. The proof holds in non-zero characteristic. Je d\'emontre que le groupe…

We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field. This bound depends only on the dimensions and degrees of the irreducible components and holds for very…

Let k be a p-adic field. Some time ago, D. Harbater [9] proved that any finite group G may be realized as a regular Galois group over the rational function field in one variable k(t), namely there exists a finite field extension $F/k(t)$,…

The classical Chevalley-Weil theorem asserts that for an \'etale covering of projective varieties over a number field K, the discriminant of the field of definition of the fiber over a K-rational point is uniformly bounded. We obtain a…

We develop a sequential-topological study of rational points of schemes of finite type over local rings typical in higher dimensional number theory and algebraic geometry. These rings are certain types of multidimensional complete fields…

Let k be an algebraically closed field of characteristic 0, and let f be a morphism of smooth projective varieties from X to Y over the ring k((t)) of formal Laurent series. We prove that if a general geometric fiber of f is rationally…

We revisit the abstract framework underlying the fibration method for producing rational points on the total space of fibrations over the projective line. By fine-tuning its dependence on external arithmetic conjectures, we render the…

The projective orthogonal and symplectic groups $PO_n(F)$ and $PSp_n(F)$ have a natural action on the $F$ vector space $V' = M_n(F) \oplus ... \oplus M_n(F)$. Here we assume $F$ is an infinite field of characteristic not 2. If we assume…

This text is devoted to the theory of varieties, which provides an important tool, based in universal algebra, for the classification of regular languages. In the introductory section, we present a number of examples that illustrate and…

Let $R$ be the field of real Puiseux series. It is a real closed field. We construct the first examples of smooth intersections of two quadrics in $\mathbb{P}_R^5$ and smooth cubic hypersurfaces in $\mathbb{P}_R^4$ which are not stably…

This is a small note meant to be published in a Conference Proceedings. We discuss elementary rationality questions in the Grothendieck ring of varieties for the quotient of a finite dimensional vector space over a characteristic 0 field by…

For a smooth projective variety X over an arbitrary field k, we discuss the surjectivity of the Albanese map from the Chow group of zero-cycles of degree zero on X to the group of rational points of the Albanese variety of X. Over…