Birational Geometry of Hypersurfaces

Birational Geometry of Hypersurfaces
Author: Andreas Hochenegger
Publisher: Springer Nature
Total Pages: 301
Release: 2019-10-08
Genre: Mathematics
ISBN: 3030186385

Originating from the School on Birational Geometry of Hypersurfaces, this volume focuses on the notion of (stable) rationality of projective varieties and, more specifically, hypersurfaces in projective spaces, and provides a large number of open questions, techniques and spectacular results. The aim of the school was to shed light on this vast area of research by concentrating on two main aspects: (1) Approaches focusing on (stable) rationality using deformation theory and Chow-theoretic tools like decomposition of the diagonal; (2) The connection between K3 surfaces, hyperkähler geometry and cubic fourfolds, which has both a Hodge-theoretic and a homological side. Featuring the beautiful lectures given at the school by Jean-Louis Colliot-Thélène, Daniel Huybrechts, Emanuele Macrì, and Claire Voisin, the volume also includes additional notes by János Kollár and an appendix by Andreas Hochenegger.

Birational Geometry of Foliations

Birational Geometry of Foliations
Author: Marco Brunella
Publisher: Springer
Total Pages: 140
Release: 2015-03-25
Genre: Mathematics
ISBN: 3319143107

The text presents the birational classification of holomorphic foliations of surfaces. It discusses at length the theory developed by L.G. Mendes, M. McQuillan and the author to study foliations of surfaces in the spirit of the classification of complex algebraic surfaces.

Automorphisms in Birational and Affine Geometry

Automorphisms in Birational and Affine Geometry
Author: Ivan Cheltsov
Publisher: Springer
Total Pages: 509
Release: 2014-06-11
Genre: Mathematics
ISBN: 3319056816

The main focus of this volume is on the problem of describing the automorphism groups of affine and projective varieties, a classical subject in algebraic geometry where, in both cases, the automorphism group is often infinite dimensional. The collection covers a wide range of topics and is intended for researchers in the fields of classical algebraic geometry and birational geometry (Cremona groups) as well as affine geometry with an emphasis on algebraic group actions and automorphism groups. It presents original research and surveys and provides a valuable overview of the current state of the art in these topics. Bringing together specialists from projective, birational algebraic geometry and affine and complex algebraic geometry, including Mori theory and algebraic group actions, this book is the result of ensuing talks and discussions from the conference “Groups of Automorphisms in Birational and Affine Geometry” held in October 2012, at the CIRM, Levico Terme, Italy. The talks at the conference highlighted the close connections between the above-mentioned areas and promoted the exchange of knowledge and methods from adjacent fields.

Birational Geometry of Algebraic Varieties

Birational Geometry of Algebraic Varieties
Author: Janos Kollár
Publisher: Cambridge University Press
Total Pages: 254
Release: 2010-03-24
Genre: Mathematics
ISBN: 9780511662560

One of the major discoveries of the past two decades in algebraic geometry is the realization that the theory of minimal models of surfaces can be generalized to higher dimensional varieties. This generalization, called the minimal model program, or Mori's program, has developed into a powerful tool with applications to diverse questions in algebraic geometry and beyond. This book provides the first comprehensive introduction to the circle of ideas developed around the program, the prerequisites being only a basic knowledge of algebraic geometry. It will be of great interest to graduate students and researchers working in algebraic geometry and related fields.

Explicit Birational Geometry of 3-folds

Explicit Birational Geometry of 3-folds
Author: Alessio Corti
Publisher: Cambridge University Press
Total Pages: 364
Release: 2000-07-27
Genre: Mathematics
ISBN: 9780521636414

This volume, first published in 2000, is an integrated suite of papers centred around applications of Mori theory to birational geometry.

Birational Geometry and Moduli Spaces

Birational Geometry and Moduli Spaces
Author: Elisabetta Colombo
Publisher: Springer Nature
Total Pages: 204
Release: 2020-02-25
Genre: Mathematics
ISBN: 303037114X

This volume collects contributions from speakers at the INdAM Workshop “Birational Geometry and Moduli Spaces”, which was held in Rome on 11–15 June 2018. The workshop was devoted to the interplay between birational geometry and moduli spaces and the contributions of the volume reflect the same idea, focusing on both these areas and their interaction. In particular, the book includes both surveys and original papers on irreducible holomorphic symplectic manifolds, Severi varieties, degenerations of Calabi-Yau varieties, uniruled threefolds, toric Fano threefolds, mirror symmetry, canonical bundle formula, the Lefschetz principle, birational transformations, and deformations of diagrams of algebras. The intention is to disseminate the knowledge of advanced results and key techniques used to solve open problems. The book is intended for all advanced graduate students and researchers interested in the new research frontiers of birational geometry and moduli spaces.

Rationality of Varieties

Rationality of Varieties
Author: Gavril Farkas
Publisher: Springer Nature
Total Pages: 433
Release: 2021-10-19
Genre: Mathematics
ISBN: 3030754219

This book provides an overview of the latest progress on rationality questions in algebraic geometry. It discusses new developments such as universal triviality of the Chow group of zero cycles, various aspects of stable birationality, cubic and Fano fourfolds, rationality of moduli spaces and birational invariants of group actions on varieties, contributed by the foremost experts in their fields. The question of whether an algebraic variety can be parametrized by rational functions of as many variables as its dimension has a long history and played an important role in the history of algebraic geometry. Recent developments in algebraic geometry have made this question again a focal point of research and formed the impetus to organize a conference in the series of conferences on the island of Schiermonnikoog. The book follows in the tradition of earlier volumes, which originated from conferences on the islands Texel and Schiermonnikoog.

Algebraic Geometry for Scientists and Engineers

Algebraic Geometry for Scientists and Engineers
Author: Shreeram Shankar Abhyankar
Publisher: American Mathematical Soc.
Total Pages: 311
Release: 1990
Genre: Mathematics
ISBN: 0821815350

Based on lectures presented in courses on algebraic geometry taught by the author at Purdue University, this book covers various topics in the theory of algebraic curves and surfaces, such as rational and polynomial parametrization, functions and differentials on a curve, branches and valuations, and resolution of singularities.