Algebraic Curves: Towards Moduli Spaces

This book provides a concise yet thorough introduction to the notion of moduli spaces of complex algebraic curves, which have become central in algebraic geometry and mathematical physics. It studies individual smooth algebraic curves and families of curves, highlighting the discovery that curves with only mild singularities form smooth compact moduli spaces. The book is suitable for undergraduate and graduate coursework and includes a wealth of problems.

Format: Hardback
Length: 231 pages
Publication date: 06 February 2019
Publisher: Springer Nature Switzerland AG

This comprehensive book provides a concise yet thorough introduction to the fascinating realm of moduli spaces of complex algebraic curves. Over the past few decades, this notion has gained immense importance not only in algebraic geometry but also in mathematical physics, particularly in the field of string theory.

The book begins by delving into the study of individual smooth algebraic curves, showcasing some of the most exquisite examples. It then moves on to explore families of curves, recognizing that studying these families can be more efficient than examining individual curves. Even if some members of the family contain singular curves, the overall spaces can still be smooth. A significant breakthrough of the 20th century, attributed to P. Deligne and D. Mumford, was the discovery that curves with only mild singularities can form smooth compact moduli spaces. This remarkable finding had an unexpected side effect: it revealed that analyzing more complex curve singularities is not a prerequisite for understanding the geometry of moduli spaces.

The book does not employ 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 deep understanding of algebraic geometry. However, this basic knowledge is sufficient to extend these concepts to vector bundles and other geometric objects associated with moduli spaces. Despite its simplicity, the book provides clear explanations on the construction of moduli spaces and offers readers practical tools for working with this notion.

This book is based on several lecture courses delivered by the authors at the Independent University of Moscow and Higher School of Economics. In addition to its theoretical content, it includes a wealth of problems, making it an invaluable resource for both individual research and classroom instruction. Whether you are a student, researcher, or enthusiast of algebraic geometry and mathematical physics, this book will deepen your understanding of moduli spaces and their applications.

Weight: 524g
Dimension: 242 x 163 x 16 (mm)
ISBN-13: 9783030029425
Edition number: 1st ed. 2018