Eckl, Thomas Pukhlikov, Aleksandr
Published in
Arnold Mathematical Journal

We prove birational superrigidity of Fano double hypersurfaces of index one with quadratic and multi-quadratic singularities, satisfying certain regularity conditions, and give an effective explicit lower bound for the codimension of the set of non-rigid varieties in the natural parameter space of the family. The lower bound is quadratic in the dim...

Wolter, Jonas
Published in
European Journal of Mathematics

We prove that there are exactly two G-minimal surfaces which are G-birational to the quintic del Pezzo surface, where . These surfaces are the quintic del Pezzo surface itself and the surface .

Reichstein, Zinovy
Published in
Archiv der Mathematik

We use a recent advance in birational geometry to prove new lower bounds on the essential dimension of some finite groups.

Avilov, Artem
Published in
European Journal of Mathematics

We classify three-dimensional singular cubic hypersurfaces with an action of a finite group G, which are not G-rational and have no birational structure of G-Mori fiber space with the base of positive dimension. Also we prove the A5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \us...

Grivaux, Julien
Published in
Mathematische Zeitschrift

If X is a rational surface without nonzero holomorphic vector field and f is an automorphism of X, we study in several examples the Zariski tangent space of the local deformation space of the pair (X, f).

Staglianò, Giovanni
Published in
Advances in Geometry

A famous result of Crauder and Katz (1989) concerns the classification of special Cremona transformations whose base locus has dimension at most two. They also proved that a special Cremona transformation with base locus of dimension three has to be one of the following: 1) a quinto-quintic transformation of ℙ5; 2) a cubo-quintic transformation of ...

Decaup, Julie Dubouloz, Adrien
Published in
Bollettino dell'Unione Matematica Italiana

We classify closed curves isomorphic to the affine line in the complement of a smooth rational projective plane conic Q. Over a field of characteristic zero, we show that up to the action of the subgroup of the Cremona group of the plane consisting of birational endomorphisms restricting to biregular automorphisms outside Q, there are exactly two s...

Cantat, Serge

We survey a few results concerning groups of birational transformations. The emphasis is on the Cremona group in two variables and methods coming from geometric group theory.

Krylov, Igor
Published in
European Journal of Mathematics

Varieties of Fano type are very well behaved with respect to the MMP, and they are known to be rationally connected. We study a relation between the classes of rationally connected varieties and varieties of Fano type. It is known that these classes are birationally equivalent in dimension 2. We give examples of rationally connected varieties of di...

Déserti, Julie
Published in
European Journal of Mathematics

We give examples of new degree growths for polynomial automorphisms of Ck\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^k$$\end{document} and birational ...