Polynomial Rings And Affine Algebraic Geometry

Algebraic geometry is more advanced with the completeness condition for projective or complete varieties. Many geometric properties are well described by the finiteness or the vanishing of sheaf cohomologies on such varieties. For non-complete varieties like affine algebraic varieties, sheaf cohomology does not work well and research progress used to be slow, although affine spaces and polynomial rings are fundamental building blocks of algebraic geometry. Progress was rapid since the Abhyankar-Moh-Suzuki Theorem of embedded affine line was proved, and logarithmic geometry was introduced by Iitaka and Kawamata.Readers will find the book covers vast basic material on an extremely rigorous level:

This proceedings volume gathers selected, peer-reviewed works presented at the Polynomial Rings and Affine Algebraic Geometry Conference, which was held at Tokyo Metropolitan University on February 12-16, 2018. Readers will find some of the latest research conducted by an international group of experts on affine and projective algebraic geometry. The topics covered include group actions and linearization, automorphism groups and their structure as infinite-dimensional varieties, invariant theory, the Cancellation Problem, the Embedding Problem, Mathieu spaces and the Jacobian Conjecture, the Dolgachev-Weisfeiler Conjecture, classification of curves and surfaces, real forms of complex varieties, and questions of rationality, unirationality, and birationality. These papers will be of interest to all researchers and graduate students working in the fields of affine and projective algebraic geometry, as well as on certain aspects of commutative algebra, Lie theory, symplectic geometry and Stein manifolds.

This volume contains expository lectures from the Conference Board of the Mathematical Sciences Regional Conference held at Northern Illinois University on July 25-29, 1977.

This proceedings volume gathers together selected, peer-reviewed works presented at the Polynomial Rings and Affine Algebraic Geometry conference which was held at the Tokyo Metropolitan University on February 12-26, 2018, in Tokyo, Japan. In this book, the reader will find some of the latest research conducted by an international group of experts in affine and projective algebraic geometry. Topics covered include group actions and linearization, automorphism groups and their structure as infinite-dimensional varieties, invariant theory, the Cancellation Problem, the Embedding Problem, Mathieu spaces and the Jacobian Conjecture, the Dolgachev-Weisfeiler Conjecture, classification of curves and surfaces, real forms of complex varieties, and questions of rationality, unirationality, and birationality. The articles contained in this volume will be of interest to all researchers and graduate students working in the fields of affine and projective algebraic geometry, as well as in certain aspects of commutative algebra, Lie theory, symplectic geometry and Stein manifolds.

A Special Session on affine and algebraic geometry took place at the first joint meeting between the American Mathematical Society (AMS) and the Real Sociedad Matematica Espanola (RSME) held in Seville (Spain). This volume contains articles by participating speakers at the Session. The book contains research and survey papers discussing recent progress on the Jacobian Conjecture and affine algebraic geometry and includes a large collection of open problems. It is suitable for graduate students and research mathematicians interested in algebraic geometry.

By studying algebraic varieties over a field, this book demonstrates how the notion of schemes is necessary in algebraic geometry. It gives a definition of schemes and describes some of their elementary properties.

Originally published in 1985, this classic textbook is an English translation of Einführung in die kommutative Algebra und algebraische Geometrie. As part of the Modern Birkhäuser Classics series, the publisher is proud to make Introduction to Commutative Algebra and Algebraic Geometry available to a wider audience. Aimed at students who have taken a basic course in algebra, the goal of the text is to present important results concerning the representation of algebraic varieties as intersections of the least possible number of hypersurfaces and—a closely related problem—with the most economical generation of ideals in Noetherian rings. Along the way, one encounters many basic concepts of commutative algebra and algebraic geometry and proves many facts which can then serve as a basic stock for a deeper study of these subjects.

The present volume grew out of an international conference on affine algebraic geometry held in Osaka, Japan during 3–6 March 2011 and is dedicated to Professor Masayoshi Miyanishi on the occasion of his 70th birthday. It contains 16 refereed articles in the areas of affine algebraic geometry, commutative algebra and related fields, which have been the working fields of Professor Miyanishi for almost 50 years. Readers will be able to find recent trends in these areas too. The topics contain both algebraic and analytic, as well as both affine and projective, problems. All the results treated in this volume are new and original which subsequently will provide fresh research problems to explore. This volume is suitable for graduate students and researchers in these areas. Contents:Acyclic Curves and Group Actions on Affine Toric Surfaces (Ivan Arzhantsev and Mikhail Zaidenberg)Hirzeburch Surfaces and Compactifications of ℂ2 (M Furushima and A Ishida)Cyclic Multiple Planes, Branched Covers of Sn and a Result of D L Goldsmith (R V Gurjar)𝔸1*-Fibrations on Affine Threefolds (R V Gurjar, M Koras, K Masuda, M Miyanishi and P Russell)Miyanishi's Characterization of Singularities Appearing on 𝔸1-Fibrations Does Not Hold in Higher Dimensions (Takashi Kishimoto)A Galois Counterexample to Hilbert's Fourteenth Problem in Dimension Three with Rational Coefficients (Ei Kobayashi and Shigeru Kuroda)Open Algebraic Surfaces of Logarithmic Kodaira Dimension One (Hideo Kojima)Some Properties of ℂ* in ℂ2 (M Koras and P Russell)Abhyankar-Sathaye Embedding Conjecture for a Geometric Case (Tomoaki Ohta)Some Subgroups of the Cremona Groups (Vladimir L Popov)The Gonality of Singular Plane Curves II (Fumio Sakai)Examples of Non-Uniruled Surfaces with Pre-Tango Structures Involving Non-Closed Global Differential 1-Forms (Yoshifumi Takeda)Representations of 𝔾a of Codimension Two (Ryuji Tanimoto)The Projective Characterization of Genus Two Plane Curves Which Have One Place at Infinity (Keita Tono)Sextic Variety as Galois Closure Variety of Smooth Cubic (Hisao Yoshihara)Invariant Hypersurfaces of Endomorphisms of the Projective 3-Space (De-Qi Zhang) Readership: Graduate students and researchers in affine algebraic geometry. Keywords:Affine algebraic geometry;Commutative algebra;Polynomial algebra;Group actions;FibrationsKey Features:Many active researchers such as M Zaidenberg, R V Gurjar, M Koras, P Russell, F Sakai, V Popov, H Yoshihara, and D-Q Zhang contributed articles to this proceedingsMany viewpoints are taken into consideration to study surfaces and threefolds. These will give hints for further research in neighboring fields