Blanc, Jérémy Pan, Ivan Vust, Thierry
Published in
Geometriae Dedicata

This article deals with the study of the birational transformations of the projective complex plane which leave invariant an irreducible algebraic curve. We try to describe the state of the art and provide some new results on this subject.

Avritzer, Dan Gonzalez-Sprinberg, Gerard Pan, Ivan
Published in
Rendiconti del Circolo Matematico di Palermo

We study the relation between Cremona transformations in space and quadratic line complexes. We show that it is possible to associate a space Cremona transformation to each smooth quadratic line complex once we choose two distinct lines contained in the complex. Such Cremona transformations are cubo-cubic and we classify them in terms of the relati...

Mendes, Luís Gustavo
Published in
Geometriae Dedicata

The idea of the proof of the classical Noether–Fano inequalities can be adapted to the domain of codimension one singular holomorphic foliations of the projective space. We obtained criteria for proving that the degree of a foliation on the plane is minimal in the birational class of the foliation and for the non-existence of birational symmetries ...

Blanc, Jérémy Pan*, Ivan Vust, Thierry
Published in
Bulletin of the Brazilian Mathematical Society, New Series

Nous poursuivons l'étude faite par G. Castelnuovo en 1892 au sujet du groupe des transformations birationnelles du plan complexe qui fixent points par points une courbe de genre >1 ; nous nous servons comme lui des systèmes linéaires adjoints de la courbe fixe. Nous démontrons que ces groupes sont abéliens, et qu'ils sont soit finis, d'ordre 2 ou 3...

Zhang, De-Qi
Published in
Mathematische Annalen

We give an effective upper bound of |Bir(X)| for the birational automorphism group of an irregular n-fold (with n = 3) of general type in terms of the volume V = V(X) under an “albanese smoothness and simplicity” condition. To be precise, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssy...

Drensky, Vesselin Yu, Jie-Tai
Published in
Frontiers of Mathematics in China

We study z-automorphisms of the polynomial algebra K[x, y, z] and the free associative algebra K 〈x, y, z〉 over a field K, i.e., automorphisms that fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K 〈x, y, z〉 we include also results about the structure of the z-tame automorphisms and ...

Déserti, Julie

Gonzalez-Sprinberg, Gerard Pan, Ivan
Published in
Mathematische Annalen

We compute the multidegrees and the Segre numbers of general determinantal Cremona transformations, with generically reduced base scheme, by specializing to the standard Cremona transformation and computing its Segre class via mixed volumes of rational polytopes.

Déserti, Julie

We prove that an automorphism of the group of birational transformations of the complex projective plane is the composition of an interior automorphism and an automorphism of the field C. The proof is based on a study of maximal abelian subgroups of the Cremona group.

Kajiwara, Kenji 梶原, 健司 Masuda, Tetsu 増田, 哲 Noumi, Masatoshi 野海, 正俊 Ohta, Yasuhiro 太田, 泰広 Yamada, Yasuhiko 山田, 泰彦
...

A theoretical foundation for a generalization of the elliptic difference Painleve equation to higher dimensions is provided in the framework of birational Weyl group action on the space of point configurations in general position in a projective space. By introducing an elliptic parametrization of point configurations, a realization of the Weyl gro...