Fedor Bogomolov

On the density of rational points on elliptic fibrations

1. Introduction Let X be an algebraic variety defined over a number field F. We will say that rational points are potentially dense if there exists a finite extension K/F such that the set of K-rational points X(K) is Zariski dense in X. The main problem is to relate this property to geometric invariants of X. Hypothetically, on varieties of general type rational points are not potentially dense. In this paper we are interested in smooth projective varieties such that neither they nor their unramified coverings admit a dominant map onto varieties of general type. For these varieties it seems plausible to expect that rational points are potentially dense (see [2]).

On Two Conjectures in Birational Algebraic Geometry

In this article I want to formulate and prove a synthetic version of two well known conjectures. One of them is the so-called Bloch-Kato conjecture. It provides a description of the torsion cohomology groups for any field in terms of Milnor K-functor. Another one was formulated by A. Grothendieck and concerns only the fields of rational functions on algebraic varieties over number fields. Namely, it claims that the Galois group of the algebraic closure of such field considered as an abstract profinite group defines the field in a functorial way. In fact, the Bloch-Kato conjecture can also be reformulated in terms of some quotient of the Galois group above.

Constructing rational curves on K3 surfaces

We develop a mixed-characteristic version of the MoriMukai technique for producing rational curves on K3 surfaces. We reduce modulo p, produce rational curves on the resulting K3 surface over a finite field, and lift to characteristic zero. As an application, we prove that all complex K3 surfaces with Picard group generated by a class of degree two have an infinite number of rational curves.

Remarks on endomorphisms and rational points

Let X be an algebraic variety and let f : X 99K X be a rational self-map with a fixed point q, where everything is defined over a number field K. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold, originally proved by Claire Voisin and the first author in 2007.

Rational curves on foliated varieties

The article refines and generalises the study of deformations of a morphism along a foliation begun by Y. Miyaoka, [Mi2]. The key ingredients are the algebrisation of the graphic neighbourhood, see Fact 3.3.1, which reduces the problem from the transcendental to the algebraic, and a p-adic variation of Mori’s bend and break in order to overcome the “naive failure”, see Remark 3.2.3, of the method in the required generality. Qualitatively the results are optimal for foliations of all ranks in all dimensions, and are quantitatively optimal for foliations by curves, for which the further precision of a cone theorem is provided.

Hyperbolicity of nodal hypersurfaces

Abstract We show that a nodal hypersurface X in ℙ3 of degree d with a sufficiently large number l of nodes, , is algebraically quasi-hyperbolic, i.e. X can only have finitely many rational and elliptic curves. Our results use the theory of symmetric differentials and algebraic foliations and give a very striking example of the jumping of the number of symmetric differentials in families.

Isoclinism and stable cohomology of wreath products

Using the notion of isoclinism introduced by P. Hall for finite p-groups, we show that many important classes of finite p-groups have stable cohomology detected by abelian subgroups (see Theorem 11). Moreover, we show that the stable cohomology of the n-fold wreath product \(G_{n} = \mathbb{Z}/p \wr \ldots \wr \mathbb{Z}/p\) of cyclic groups \(\mathbb{Z}/p\) is detected by elementary abelian p-subgroups and we describe the resulting cohomology algebra explicitly. Some applications to the computation of unramified and stable cohomology of finite groups of Lie type are given.

Ordinary reduction of K3 surfaces

Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.

UNIRATIONALITY AND EXISTENCE OF INFINITELY TRANSITIVE MODELS

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 the given field of rational functions and an infinitely transitive regular action of a group of algebraic automorphisms generated by unipotent algebraic subgroups. We expect that this property holds for all unirational varieties and in fact is a peculiar one for this class of algebraic varieties among those varieties which are rationally connected.

Birationally isotrivial fiber spaces

We compute the dynamical degrees of certain compositions of reflections in points on a smooth cubic fourfold. Our interest in these computations stems from the irrationality problem for cubic fourfolds. Namely, we hope that they will provide numerical evidence for potential restrictions on tuples of dynamical degrees realisable on general cubic fourfolds which can be violated on the projective four-space.We prove that a family of varieties is birationally isotrivial if all the fibers are birational to each other.