Related papers: Specialization of birational types

We prove that stable rationality specializes in regular families whose fibers are integral and have at most ordinary double points as singularities. Our proof is based on motivic specialization techniques and the criterion of Larsen and…

In this paper we use families of finite subgroups to study Grothendieck rings associated to certain discrete groups, such as the arithmetic ones.

We study the special fibers of a certain class of absolutely simple abelian varieties over number fields with endomorphism rings $\bz$ and possessing $l$-adic monodromy groups of the least possible rank. We also study the Dirichlet density…

We give a description of the algebraic families of birational transformations of an algebraic variety X. As an application, we show that the morphisms to Bir(X) given by algebraic families satisfy a Chevalley type result and a certain…

Inspired by the work of Ellenberg, Elsholtz, Hall, and Kowalski, we investigate how the property of the generic fiber of a one-parameter family of abelian varieties being geometrically simple extends to other fibers. In \cite{EEHK09}, the…

We compute the Grothendieck group of the category of abelian varieties over an algebraically closed field $k$. We also compute the Grothendieck group of the category of $A$-isotypic abelian varieties, for any simple abelian variety $A$,…

We construct an upgrade of the motivic volume by keeping track of dimensions in the Grothendieck ring of varieties. This produces a uniform refinement of the motivic volume and its birational version introduced by Kontsevich and Tschinkel…

We describe a general method, originated by Ofer Gabber, for showing that a very general fiber in a family has certain properties. We illustrate this method with concrete examples taken from algebraic dynamics, the rationality problem for…

We introduce a family of generalized Broughton polynomials and compute the characteristic varieties of complement of a curve arrangement defined by fibers of some generalized Broughton polynomials

We survey new results on finite groups of birational transformations of algebraic varieties.

A priori, the set of birational transformations of an algebraic variety is just a group. We survey the possible algebraic structures that we may add to it, using in particular parametrised family of birational transformations.

We introduce a new stable birational invariant, which takes the form of a functor sending a degenerating variety to the homotopy type of a chain complex. Our invariant is a categorification of the motivic volume of Nicaise and Shinder. From…

We show how to treat families of $\infty$-categories fibered in categorical patterns (e.g., $\infty$-operads and monoidal $\infty$-categories) in terms of fibrations by relativizing the Grothendieck construction. As applications, we…

The goal of this paper is to define families of toric varieties and to study their properties. These families are locally trivial fibrations over some base, whose fibres are isomorphic to a fixed complete toric variety. The study is…

Grothendieck fibrations provide a unifying algebraic framework that underlies the treatment of various form of logics, such as first order logic, higher order logics and dependent type theories. In the categorical approach to logic proposed…

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.

We give a complete description of the category of smooth complex representations of the multiplicative group of a central simple algebra over a locally compact nonarchimedean local field. More precisely, for each inertial class in the…

Building on previous work of Kollar, Ein, Lazarsfeld, and Hacon, we show that ample divisors of low degree on an abelian variety have mild singularities in case the abelian variety is simple or the degree of the polarization is two.

It is known by results of Koll\'ar, Ein, Lazarsfeld, Hacon and Debarre that divisors representing principal and other low degree polarizations on abelian varieties have mild singularities. In this note we extend such results to…

We prove that a family of varieties is birationally isotrivial if all the fibers are birational to each other.