© 2013

Birational Geometry, Rational Curves, and Arithmetic

  • Fedor Bogomolov
  • Brendan Hassett
  • Yuri Tschinkel

Table of contents

  1. Front Matter
    Pages i-ix
  2. Ivan Arzhantsev, Hubert Flenner, Shulim Kaliman, Frank Kutzschebauch, Mikhail Zaidenberg
    Pages 1-13
  3. Fedor Bogomolov, Christian Böhning
    Pages 57-76
  4. Fedor Bogomolov, Ilya Karzhemanov, Karine Kuyumzhiyan
    Pages 77-92
  5. Ivan Cheltsov, Ludmil Katzarkov, Victor Przyjalkowski
    Pages 93-132
  6. Olivier Debarre
    Pages 133-145
  7. Stefan Kebekus
    Pages 147-162
  8. Sándor J. Kovács
    Pages 163-169
  9. Vladimir Lazić
    Pages 171-203
  10. Max Lieblich
    Pages 205-227
  11. Christian Liedtke
    Pages 229-292
  12. Anthony Várilly-Alvarado
    Pages 293-319

About this book


​​​​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.


arithmetic of higher-dimensional algebraic varieties birational geometry cone of curves rational curves rationality unirationality

Editors and affiliations

  • Fedor Bogomolov
    • 1
  • Brendan Hassett
    • 2
  • Yuri Tschinkel
    • 3
  1. 1.Department of Mathematics, Courant Institute of Math. SciencesNew York UniversityNew YorkUSA
  2. 2., Department of MathematicsRice UniversityHoustonUSA
  3. 3.Department of Mathematics, Courant Institute of Math. SciencesNew York UniversityNew YorkUSA

About the editors

F. Bogomolov is Professor at the Courant Institute, NYU. He is best known for his pioneering work on hyperkähler manifolds. B. Hassett is Professor and Chair of the department of Mathematics at Rice University. He published two books and around 50 papers on Algebraic and Arithmetic Geometry. Yuri Tschinkel is Professor at the Courant Institute, NYU and Director of the Mathematics and the Physical Sciences Division at the Simons Foundation.

Bibliographic information

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