Overview
- Leads a reader to far advanced topics widely used in modern research, using basic tools from the first two years of university studies
- From the very beginning, the study of algebraic curves is aimed at the construction of their moduli spaces in the final chapters
- Supplied with numerous exercises and problems both making the book a convenient base for a university lecture course and allowing the reader to control his/her progress
Part of the book series: Moscow Lectures (ML, volume 2)
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Table of contents (19 chapters)
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Front Matter
Keywords
About this book
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.Thebook 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
Reviews
“The book under review is an accessible introduction to the study of complex algebraic curves and their moduli spaces. … The book comes with sets of exercises in each of its chapters and can be used as a textbook for a graduate course or for self-study by a motivated reader.” (Felipe Zaldivar, MAA Reviews, April 22, 2019)
Authors and Affiliations
About the authors
Maxim Kazaryan is a researcher at the Steklov Mathematical Institute RAS. He also works as a professor of mathematics at the NRU Higher School of Economics since 2008 and at the Skolkovo Institute of Science and Technology since 2016.
Sergei Lando is a professor of mathematics at the NRU Higher School of Economics since 2008 and at the Skolkovo Institute of Science and Technology since 2016. He was the first Dean of the Department of Mathematics at the NRU HSE. He also is a Vice-President of the Moscow Mathematical Society.
Victor Prasolov is a permanent teacher of mathematics at the Independent University of Moscow.
Bibliographic Information
Book Title: Algebraic Curves
Book Subtitle: Towards Moduli Spaces
Authors: Maxim E. Kazaryan, Sergei K. Lando, Victor V. Prasolov
Translated by: Natalia Tsilevich
Series Title: Moscow Lectures
DOI: https://doi.org/10.1007/978-3-030-02943-2
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2018
Hardcover ISBN: 978-3-030-02942-5Published: 06 February 2019
eBook ISBN: 978-3-030-02943-2Published: 21 January 2019
Series ISSN: 2522-0314
Series E-ISSN: 2522-0322
Edition Number: 1
Number of Pages: XIV, 231
Number of Illustrations: 37 b/w illustrations
Topics: Algebraic Geometry, Functions of a Complex Variable, Mathematical Physics