Simons Symposia

Birational Geometry, Rational Curves, and Arithmetic

Editors: Bogomolov, Fedor, Hassett, Brendan, Tschinkel, Yuri (Eds.)

  • ​Leading experts report on recent advances in higher-dimensional birational geometry, with special regard to arithmetic applications
  • Highlights the tight connections between arithmetic and geometry
  • Documents the central role of the theory of rational curves
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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.

About the authors

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.

Table of contents (12 chapters)

  • Infinite Transitivity on Affine Varieties

    Arzhantsev, Ivan (et al.)

    Pages 1-13

  • The Birational Geometry of the Hilbert Scheme of Points on Surfaces

    Bertram, Aaron (et al.)

    Pages 15-55

  • Isoclinism and Stable Cohomology of Wreath Products

    Bogomolov, Fedor (et al.)

    Pages 57-76

  • Unirationality and Existence of Infinitely Transitive Models

    Bogomolov, Fedor (et al.)

    Pages 77-92

  • Birational Geometry via Moduli Spaces

    Cheltsov, Ivan (et al.)

    Pages 93-132

Buy this book

eBook $109.00
price for USA (gross)
  • ISBN 978-1-4614-6482-2
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $149.00
price for USA
  • ISBN 978-1-4614-6481-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $149.00
price for USA
  • ISBN 978-1-4939-0158-6
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
Birational Geometry, Rational Curves, and Arithmetic
Editors
  • Fedor Bogomolov
  • Brendan Hassett
  • Yuri Tschinkel
Series Title
Simons Symposia
Copyright
2013
Publisher
Springer-Verlag New York
Copyright Holder
Springer Science+Business Media New York
eBook ISBN
978-1-4614-6482-2
DOI
10.1007/978-1-4614-6482-2
Hardcover ISBN
978-1-4614-6481-5
Softcover ISBN
978-1-4939-0158-6
Series ISSN
2365-9564
Edition Number
1
Number of Pages
IX, 319
Topics