Related papers: Birational geometry of algebraic varieties
In this note we introduce generalised pairs from the perspective of the evolution of the notion of space in birational algebraic geometry. We describe some applications of generalised pairs in recent years and then mention a few open…
We complete the analysis on the birational rigidity of quasismooth Fano 3-fold deformation families appearing in the Graded Ring Database as a complete intersection. When such a deformation family $X$ has Fano index at least 2 and is…
Many classical results in algebraic geometry arise from investigating some extremal behaviors that appear among projective varieties not lying on any hypersurface of fixed degree. We study two numerical invariants attached to such…
This is a survey of some recent developments in the study of singularities related to the classification theory of algebraic varieties. In particular, the definition and basic properties of Du Bois singularities and their connections to the…
For a geometrically rational surface X over an arbitrary field of characteristic different from 2 and 3 that contains all roots of 1, we show that either X is birational to a product of a projective line and a conic, or the group of…
In this note, we reduce various conjectures in birational geometry, including Shokurov conjecture on singularities of the base of log Calabi-Yau fibrations of Fano type and boundedness conjecture for rationally connected Calabi-Yau…
We study endomorphism rings of principally polarized abelian surfaces over finite fields from a computational viewpoint with a focus on exhaustiveness. In particular, we address the cases of non-ordinary and non-simple varieties. For each…
It is shown that hypersurfaces of degree $M$ in ${\mathbb P}^M$, $M\geqslant 5$, with at most quadratic singularities of rank at least 3, satisfying certain conditions of general position, are birationally superrigid Fano varieties and the…
In this paper, we prove various results on boundedness and singularities of Fano fibrations and of Fano type fibrations. A Fano fibration is a projective morphism $X\to Z$ of algebraic varieties with connected fibres such that $X$ is Fano…
In this paper, we use (bi)semicosimplicial language to study the classical problem of infinitesimal deformations of a closed subscheme in a fixed smooth variety, defined over an algebraically closed field of characteristic 0. In particular,…
In this paper we provide new examples of geometrically trivial strongly minimal differential algebraic varieties living on nonisotrivial curves over differentially closed fields of characteristic zero. Our technique involves developing a…
We study finite $p$-subgroups of birational automorphism groups. By virtue of boundedness theorem of Fano varieties, we prove that there exists a constant $R(n)$ such that a rationally connected variety of dimension $n$ over an…
These lectures give a short introduction to the study of curves on algebraic varieties. After an elementary proof of the dimension formula for the space of curves, we summarize the basic properties of uniruled and of rationally connected…
Small representations of a group bring us to large symmetries in a representation space. Analysis on minimal representations utilises large symmetries in their geometric models, and serves as a driving force in creating new interesting…
In this paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying…
A generically generated vector bundle on a smooth projective variety yields a rational map to a Grassmannian, called Kodaira map. We answer a previous question, raised by the asymptotic behaviour of such maps, giving rise to a birational…
In a previous article, a universal linear algebraic model was proposed for describing homogeneous conformal geometries, such as the spherical, Euclidean, hyperbolic, Minkowski, anti-de Sitter and Galilei planes. This formalism was…
Over the past few years there has been considerable progress in the structural understanding of special Colombeau algebras. We present some of the main trends in this development: non-smooth differential geometry, locally convex theory of…
We introduce horizontal and vertical motivic invariants of birational maps between rational dominant maps and study their basic properties. As a first application, we show that the (usual) motivic invariants vanish for birational…
The relevance of the algebraic entropy in the study of birational discrete time dynamical systems highlights the need to relate it to other characteristics of these systems. In this letter, two complementary proofs are given that the…