Birational Geometry, Rational Curves, and Arithmetic - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Birational Geometry, Rational Curves, and Arithmetic write by Fedor Bogomolov. This book was released on 2013-05-17. Birational Geometry, Rational Curves, and Arithmetic available in PDF, EPUB and Kindle. This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometry. It focuses on the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions. Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry is the study of lines and conics. From the modern standpoint, arithmetic is the study of rational and integral points on algebraic varieties over nonclosed fields. A major insight of the 20th century was that arithmetic properties of an algebraic variety are tightly linked to the geometry of rational curves on the variety and how they vary in families. This collection of solicited survey and research papers is intended to serve as an introduction for graduate students and researchers interested in entering the field, and as a source of reference for experts working on related problems. Topics that will be addressed include: birational properties such as rationality, unirationality, and rational connectedness, existence of rational curves in prescribed homology classes, cones of rational curves on rationally connected and Calabi-Yau varieties, as well as related questions within the framework of the Minimal Model Program.
Geometry Over Nonclosed Fields
Geometry Over Nonclosed Fields - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Geometry Over Nonclosed Fields write by Fedor Bogomolov. This book was released on 2017-02-09. Geometry Over Nonclosed Fields available in PDF, EPUB and Kindle. Based on the Simons Symposia held in 2015, the proceedings in this volume focus on rational curves on higher-dimensional algebraic varieties and applications of the theory of curves to arithmetic problems. There has been significant progress in this field with major new results, which have given new impetus to the study of rational curves and spaces of rational curves on K3 surfaces and their higher-dimensional generalizations. One main recent insight the book covers is the idea that the geometry of rational curves is tightly coupled to properties of derived categories of sheaves on K3 surfaces. The implementation of this idea led to proofs of long-standing conjectures concerning birational properties of holomorphic symplectic varieties, which in turn should yield new theorems in arithmetic. This proceedings volume covers these new insights in detail.
Higher Dimensional Varieties and Rational Points
Higher Dimensional Varieties and Rational Points - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Higher Dimensional Varieties and Rational Points write by Károly Jr. Böröczky. This book was released on 2013-12-11. Higher Dimensional Varieties and Rational Points available in PDF, EPUB and Kindle. Exploring the connections between arithmetic and geometric properties of algebraic varieties has been the object of much fruitful study for a long time, especially in the case of curves. The aim of the Summer School and Conference on "Higher Dimensional Varieties and Rational Points" held in Budapest, Hungary during September 2001 was to bring together students and experts from the arithmetic and geometric sides of algebraic geometry in order to get a better understanding of the current problems, interactions and advances in higher dimension. The lecture series and conference lectures assembled in this volume give a comprehensive introduction to students and researchers in algebraic geometry and in related fields to the main ideas of this rapidly developing area.
Rational Curves on Algebraic Varieties
Rational Curves on Algebraic Varieties - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Rational Curves on Algebraic Varieties write by Janos Kollar. This book was released on 2013-04-09. Rational Curves on Algebraic Varieties available in PDF, EPUB and Kindle. The aim of this book is to provide an introduction to the structure theory of higher dimensional algebraic varieties by studying the geometry of curves, especially rational curves, on varieties. The main applications are in the study of Fano varieties and of related varieties with lots of rational curves on them. This Ergebnisse volume provides the first systematic introduction to this field of study. The book contains a large number of examples and exercises which serve to illustrate the range of the methods and also lead to many open questions of current research.
Arithmetic Geometry
Arithmetic Geometry - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Arithmetic Geometry write by Clay Mathematics Institute. Summer School. This book was released on 2009. Arithmetic Geometry available in PDF, EPUB and Kindle. Based on survey lectures given at the 2006 Clay Summer School on Arithmetic Geometry at the Mathematics Institute of the University of Gottingen, this tile is intended for graduate students and recent PhD's. It introduces readers to modern techniques and conjectures at the interface of number theory and algebraic geometry.
