One of the great successes of twentieth century mathematics has been the remarkable qualitative understanding of rational and integral points on curves, gleaned in part through the theorems of Mordell, Weil, Siegel, and Faltings. It has become clear that the study of rational and integral points has deep connections to other branches of mathematics: complex algebraic geometry, Galois and etale cohomology, transcendence theory and diophantine approximation, harmonic analysis, automorphic forms, and analytic number theory.
This text, which focuses on higher dimensional varieties,...
One of the great successes of twentieth century mathematics has been the remarkable qualitative understanding of rational and integral points on cu...
Rationality problems link algebra to geometry, and the difficulties involved depend on the transcendence degree of $K$ over $k$, or geometrically, on the dimension of the variety. A major success in 19th century algebraic geometry was a complete solution of the rationality problem in dimensions one and two over algebraically closed ground fields of characteristic zero. Such advances has led to many interdisciplinary applications to algebraic geometry.
This comprehensive book consists of surveys of research papers by leading specialists in the field and gives indications for future...
Rationality problems link algebra to geometry, and the difficulties involved depend on the transcendence degree of $K$ over $k$, or geometrically, ...
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...
This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometr...
This text offers a collection of survey and research papers by leading specialists in the field documenting the current understanding of higher dimensional varieties. Recently, it has become clear that ideas from many branches of mathematics can be successfully employed in the study of rational and integral points. This book will be very valuable for researchers from these various fields who have an interest in arithmetic applications, specialists in arithmetic geometry itself, and graduate students wishing to pursue research in this area.
This text offers a collection of survey and research papers by leading specialists in the field documenting the current understanding of higher dim...
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.
Topics that will be addressed include: birational properties such as rationality, unirationality, and rational connectedness, existence of rational cu...
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.
Based on the Simons Symposia held in 2015, the proceedings in this volume focus on rational curves on higher-dimensional algebraic varieties and appli...
