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Algebraic Curves

Towards Moduli Spaces

  • Maxim E. Kazaryan
  • Sergei K. Lando
  • Victor V.  Prasolov
Textbook

Part of the Moscow Lectures book series (ML, volume 2)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 1-11
  3. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 13-31
  4. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 33-49
  5. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 51-57
  6. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 59-69
  7. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 71-90
  8. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 91-102
  9. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 103-111
  10. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 113-123
  11. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 125-130
  12. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 131-138
  13. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 139-156
  14. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 157-162
  15. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 163-175
  16. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 177-192
  17. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 193-200
  18. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 201-212
  19. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 213-220
  20. Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
    Pages 221-226
  21. Back Matter
    Pages 227-231

About this book

Introduction

This book offers a concise yet thorough introduction to the notion of moduli spaces of complex algebraic curves. Over the last few decades, this notion has become central not only in algebraic geometry, but in mathematical physics, including string theory, as well.

The book begins by studying individual smooth algebraic curves, including the most beautiful ones, before addressing families of curves. Studying families of algebraic curves often proves to be more efficient than studying individual curves: these families and their total spaces can still be smooth, even if there are singular curves among their members. A major discovery of the 20th century, attributed to P. Deligne and D. Mumford, was that curves with only mild singularities form smooth compact moduli spaces. An unexpected byproduct of this discovery was the realization that the analysis of more complex curve singularities is not a necessary step in understanding the geometry of the moduli spaces.

The book does not use the sophisticated machinery of modern algebraic geometry, and most classical objects related to curves – such as Jacobian, space of holomorphic differentials, the Riemann-Roch theorem, and Weierstrass points – are treated at a basic level that does not require a profound command of algebraic geometry, but which is sufficient for extending them to vector bundles and other geometric objects associated to moduli spaces. Nevertheless, it offers clear information on the construction of the moduli spaces, and provides readers with tools for practical operations with this notion.

Based on several lecture courses given by the authors at the Independent University of Moscow and Higher School of Economics, the book also includes a wealth of problems, making it suitable not only for individual research, but also as a textbook for undergraduate and graduate coursework

Keywords

algebraic curves Riemann-Roch theorem Weierstrass points Abel theorem moduli spaces compactification of moduli spaces stable curves

Authors and affiliations

  • Maxim E. Kazaryan
    • 1
  • Sergei K. Lando
    • 2
  • Victor V.  Prasolov
    • 3
  1. 1.Steklov Mathematical Institute of RASNational Research University Higher School of Economics, Skolkovo Institute of Science and TechnologyMoscowRussia
  2. 2.National Research University Higher School of Economics, Skolkovo Institute of Science and TechnologyMoscowRussia
  3. 3.Independent University of MoscowMoscowRussia

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-030-02943-2
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-030-02942-5
  • Online ISBN 978-3-030-02943-2
  • Series Print ISSN 2522-0314
  • Series Online ISSN 2522-0322
  • Buy this book on publisher's site