Related papers: Introduction to birational anabelian geometry
We explore connections between birational anabelian geometry and abstract projective geometry. One of the applications is a proof of a version of the birational section conjecture.
We consider function fields of transcendence degree at least 2 over algebraic closures of finite fields, and describe a functorial way to recover such function fields form their pro-l Galois theory.
We study the structure of abelian subgroups of Galois groups of function fields of surfaces.
The aim of Bogomolov's programme is to prove birational anabelian conjectures for function fields $K|k$ of varieties of dimension $\geq 2$ over algebraically closed fields. The present article is concerned with the 1-dimensional case. While…
This is a report on some of the main developments in birational geometry in the last few years focusing on the minimal model program, Fano varieties, singularities and related topics, in characteristic zero.
A new tool for the model theory of differentially closed fields and of compact complex manifolds is here developed. In such settings, it is shown that a type internal to the field of constants (resp. to the projective line) admits a maximal…
In this paper, we develop the main step in the global theory for the mod-$\ell$ analogue of Bogomolov's program in birational anabelian geometry for higher-dimensional function fields over algebraically closed fields. More precisely, we…
By exploiting the arithmetic homotopy of the moduli spaces of curves, Galois-Teichm\"uller theory stands at the interface of braid-mapping class groups and of anabelian geometry. Starting from the classical braid-theoretic construction of…
One of the fundamental questions in current field theory, related to Grothendieck's conjecture of birational anabelian geometry, is the investigation of the precise relationship between the Galois theory of fields and the structure of the…
We construct and study universal spaces for birational invariants of algebraic varieties over algebraic closures of finite fields.
This a collection of about 100 exercises. It could be used as a supplement to the book Koll\'ar--Mori: Birational geometry of algebraic varieties.
We extend the work of M.Borovoi on the nonabelian Galois cohomology of linear reductive algebraic groups over number fields to a general base scheme. As an application, we obtain new results on the arithmetic of such groups over global…
The objective of this paper is to study the anabelian object referred to as \emph{pointed virtual curves}. Namely, given a family of curves $Y \rightarrow X$ over a field $k$ under suitable conditions, we consider the…
We study a dynamical system induced by the Artin reciprocity map for a global field. We translate the conjugacy of such dynamical systems into various arithmetical properties that are equivalent to field isomorphism, relating it to…
We present the abelianisation of the birational transformations of the real projective plane.
In this paper, we classify the possible group structures on the set of $R$-valued points of an abelian variety, where $R$ is any real closed field. We make use of a family of abelian varieties that, in effect, allows one to quantify over…
This is (mostly) a survey article. We use an information about Galois properties of points of small order on an abelian variety in order to describe its endomorphism algebra over an algebraic closure of the ground field. We discuss in…
The aim of this paper is to extend our old results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.
In this paper we study birational Kleinian groups, i.e.\ groups of birational transformations of complex projective varieties acting in a free, properly discontinuous and cocompact way on an open set of the variety with respect to the usual…
We study the behaviour of the topological fundamental group under totally ramified abelian covers (a special case of abelian Galois covers) of complex projective varieties of dimension at least 2.